136324487-Konrad-Knopp-Theory-and-Application-of-Infinite-Series-Complete-1954 OCR

THEORY ANDAPPLICATION OFINFINITE SERIES BLACKTE &SONLTMITEO 16/18Wliham IVStreet,Ch,lnng: Cros~. !.ONDO"l.l \\1C2­ 17Stanhope Street, GLASb·OW BLACKIE &SON(INDIA) LIMITED 103/5FortStreet, BOMBAY BLACKI~: &SON(CANADA) LIMIrED TORO>lTO THEORY AND APPLICATION OF INFINI1-'E SERIES BY DR.I(ONRAD I(NoPp PROFESSOR OFMATHEMATIC~ AI"THE UNIVLRSITY OFTUBINGEN Trims/atI'd fromthe."ecolldGerman Edition andrevised inaccordance withtheFour/Itby AlissR.C.H.Young,Ph.D.,L.i:sSe. BLACKIE &SONLIMITED LONDON ANDGLASGOW Fm!mued1?S Reprl1ltpd 1?44,1946 StCond Fm.r./tsh ,.iht1On, tJeUls!aledjH..l, theFDlJ.rthGerman Edatoll, 19S1 R~fJ",,,,,.d lQ')4 p,,,,ttdinGreatBritainbyBlackltf!!fSon,Ltd.,01"'1/11& Fromthepreface tothefirst(German) edition. Thereisnogeneralagreement astowhereanaccount ofthetheory ofinfiniteseriesshouldbegin,whatitsmflin'outlines shouldbc,orwhat itshouldinclude. Ontheonchand,thewholeofhigheranalysis may beregarded asafieldfortheapplication ofthistheory,foralllimiting processes --including differentIation andintegration -arebasedon theinvestigation ofinfinitesequences or ofinfiniteseries.Ontheother hand,inthestrictest (andtherefore narrowest) sense,theonlymatters thatareinplaceinatextbook oninfinite series aretheirdefinitIOn, the manipulation ofthesymbolism connected withthem,andthetheory ofconvergence. Inhis"Vorlesungen uberZahlen- undFunktionenlehre", Vo!.1, Part2,A.Pringsheim hastreated thesubject withtheselimitations. Therewasnoquestion ofoffering anything similarinthepresent book. Myaimwasquitedifferent: namely, togiveacomprehensive account ofalltheinvestigations ofhigh~ranalysis inwhichinfiniteseries arethechiefobjectofinterest, thetreatment tobeasfreefromassump­ tionsaspossible andtostartattheverybegmning andleadontothe extensive frontiers ofpresent-day research. Tosetallthisforthinas interesting andintelligIble awayaspossible, butofcoursewithout in theleastabandoning exactness, withtheobjectofprovidmg thestudent withac(~venient introduction tothesubjectandofgivinghimanidea ofitsrichjandfascinating variety-suchwasmyvision. Thematerial grewinmyhands,however, andresisted myefforts toputitintoshape.Inordertomakeaconvenient andusefulbook, thefieldhadtoberestricted. ButIwasguidedthroughout bytheex­ perience Ihavegainedinteaching--Ihavecovered thewholeofthe ground severaltimesinthegeneral courseofmyworkandinlectures attheuniversities ofBerlinandKonigshcrg -andalsobytheaim ofthebook.Itwastogiveathorough andreliabletreatment 1IJhicll1IJould beofassistance tothestudentattending lecturesandwhichwouldatthe sametimebeadapted forprivatestudy. Thelatteraimwasparticularly deartome,andthisaccounts for theforminwhichIhavepresented thesubject-matter. Sinceitisgener­ allyeasier-especially forbeginners -toproveadeduction inpure mathematics thantorecognize therestrictions towhichthetrainof reasoning issubject, Ihavealwaysdweltontheoretical difficulties. and VI Preface. havetriedtoremove thembymeansofrepeated illustrations; and although Ihavethereby deprived myselfofagooddealofspacefor important matter, Ihopetowinthegratitude ofthestudent. Iconsidered thatanintroduction tothetheoryofrealnumbers wasindispensable asabeginning, inorderthatthefirstfacts relatin~ toconvergence mighthaveafirmfoundation. Tothisintroduction I haveaddedafairlyextensive account ofthetheoryofsequences, and, finally,theactualtheoryofinfinite series.Thelatteristhenconstructed intwostoreys, sotospeak: aground-floor, inwhichtheclassical part ofthetheory(uptoaboutthestageofCauchy's Analyse algcbrique) isexpounded, thoughwiththehelpofverylimitedresources, andasupcr­ structure, inwhichIhaveattemptcd togiveanaccount ofthelater developments oftheI~)lhcentury. Forthereasons mentioned above,Ihavehadtoomitmanyparts ofthesubjecttowhichIwouldgladlyhavegivenaplacefortheirown sake.Semi-convergent series,Euler'ssummation formula, adetailed treatment oftheGamma-function, problems arising fromthehyp.:r­ geometric series,thetheoryofdoubleseries,thenewerworkonpower seru::s,and,inparticular, amorethorough development ofthelastchapter, thatondivergent series--alltheseIwasreluctantly ohliged toset aside.Ontheotherhand,Iconsidered thatitwasessential todealwith sequences andseriesofcompkx terms.Asthetheoryrunsalmostparallel withthatforrealvariables, however, Ihave,fromthebeginning, for­ mulated allthedefinitions andproved allthetheorems concerned in suchawaythattheyremainvalidwithout alteration, whether the"arbi­ trary"numbers involved arerealorcomplex. Thesedefinitions and theorems arcfurtherdistinguished bythesigno. Inchoosing theexamples--inthisrespect, however, Ilayno claimtooriginahty; onthecontrary, incollecting themIhavemade extensive useoftheliterature - Ihavetakenpainstoputpractical applications inthefore-front andtoleavemereplaying withtheoretical niceties alone.Ifencetherearce.g.aparticularly largenumber ofexer­ cisesonChapter VIIIandonlyveryfewonChapter IX.Unfortunately therewasnoroomforsolutions orevenforhintsforthesolution of theexamples. Alistofthemostimportant papers, comprchensive accounts, and textbooks oninfinite seriesisgivenatthecndofthebook,immediately infrontoftheindex. Konigsberg, September 1921. Preface. Fromthepreface tothesecond (German) edition.VII Thefactthatasecondeditionwascalledforaftersucharemarkably shorttimecouldbetakentomeanthatthefirsthadonthewholebeen ontherightlines.Hencethegeneral planhasnotbeenaltered, but ithasbeenimproved inthedetailsofexpression anddemonstration on almosteverypage. Thelastchapter, thatdealingwithdivergent series,hasbeenwholly rewritten, withimportant extensions, sothatitnowinsomemeasure provides anintroduction tothetheoryandgivesanIdeaofmodern work onthesubject. Konigsberg, December 1!J23. Preface tothethird(German) edition. Themaindifference between thethirdandseconJedItIOns isthat ithasbecome possible toaddanewchapteronEuler'ssummation formula andasymptotic expansions, whichIhadreluctantly omitted fromthe firsttwoeditions. Thisimportant chapter hadmeanwhile appeared in asimtl.lr formintheEnglish translation pubhshed byBlaclu'e f:JjSon [,imitfd, J,ondonandGlasgow, inI!)2H. Inaddition, thewholeofthebookhasagainbeencarefully revised, andtheproofshavebeenimproved orSImplified inaccordance withthe progress ofmathematIcal knowledge orteaching experience. Thisapplies especially totheor~ms 269and287. Dr.W.SchobeandHerrP.Securius havegivenmevaluable assist­ anceincorrecting theproofs,forwhichIthankthemheartily. Tilbingen, March1931. Preface tothefourth(German) edition. Inviewofpresent difficulties nolargechanges havebeenmadefor thefcurthedition, butthebookhasagainbeenrevisedandnumerous detailshavebeenimproved, discrepancies removed, andseveralproofs ,implified. Thereferences totheliterature havebeenbrought upto date. TUhingen, July1947. VIII Preface. Preface tothefirstEnglish edition. Thistranslation ofthesecondGerman edition hasbeenveryskil­ fullyprepared byMissR.C.H.Young,L.esSe.(Lausanne), Research Student, GirtonCollege, Cambridge. Thepublishers, Messrs. Blackie andSon,I,td.,Glasgow, havecarefully superintended theprinting. Inaddition, thepublishers werekindenough toaskmetoadda chapteronElller'ssummation formula andasymptotic expansions. Iagreed todosoallthemoregladlybecause, asImentioned intheoriginal prt:­ face,itwasonlywithgreatreluctance thatIomitted thispartofthesub­ jectintheGerman edition. Thischapter hasbeentranslated byMiss W.M.Deans,B.Sc.(Aberdeen), M.A.(Cantab.), withequalskill. Iwishtotakethisopportunity ofthanking thetranslators andthe publishers forthetroubleandcaretheyhavetaken.If-asIhope-­ mybookmeetswithafavourable reception andisfoundusefulbyEnglish­ speaking students ofMathematics, thecreditwilllargelybetheirs. Tiibingen, February 1928. Konrad Knopp. Preface tothesecondEnglish edition. ThesecondEnglish editionhasbeenproduced tocorrespond tothe fourthGerman edition(1947). Although mostofthechanges areindividually small,theyhavenone­ thelessinvolved acon~iderable number ofalterations, abouthalfofthe workhavingbeenre-set. Thetranslation hasbeencarriedoutbyDr.R.C.H.Youngwho wasresponsible fortheoriginal work. Contents. [ntroduction . • . • . • • . . . ... . ... ........... PartI.Page Realnumbers andsequences. Chapter 1. Principles ofthetheory ofrealnumbers. §1.Thesy"tem ofrational numbers andItsgaps ~2Sequences ofrational numbers . . . . ~3Irrational numbers . . . . . . . . . ~4.Completene~s anduniqueness ofthesystem ofrealnumbers §5.Radixfractions andtheDcd,kmd sectlOn Exercises onChapter I(l-fl).••••••••••••• Chapter H.3 14 23 33 37 42 Sequences ofrealnumbers. §6.Arbitrary sequences andarbitrary nullsequences 4:l §7.Powers, roots,andlogarithms "pecIRl null sequencl'~ 4!l §fl.Convergent sequences. . 64 ~!lThetwomaincriteria. . . 7M ~10L1I11Iting pOIntsandupperandlowerhauts BB §11.fnfil1lte series,mfmite products, andinfinite contlllued fractions 98 ExerCises onChapter Jl(9-3'3) • • ••• • •••106 Part11. Foundations ofthetheoryofinfinite series. /.' Chapter Ill. Seriesofpositive terms. §12.Thefirstprincipal criterion andthetwocomparison tests ~1::1.TheroottestandtheTatintest. . . . . . • ~14Senesofpositive, monotone decreasing terms. Exercises onChapter III(34-44).... . . I-(G51)110 116 120 125 x Contents. Ch3pter IV. Page Seriesofarbitrary terms. §15.Thesecondprincipal cnterion andthealgebra ofconvergent series126 §16.Absolute convergence. Derangement ofsenes. 136 §17.MultiplIcation ofinfinite serie" 146 Exercises onChapter IV(4iJ~()3) . • . 149 Chapter V. /Powersenes. §18.Theradiusofconvergence. . . li19.Functions ofarealvariable. . §20.Principal pioIJt:rties offunctIOns IepI"Cscllted byPO\\ersenes §21.Thealgebra ofpowerseries. . Exercises onChapter V(64-73) ChapterVI Theexpansions oftheso-called elementary functions. §22.Therational functions. . . . §23.Theexponential functIon. . . §24.Thetrigonometrical functIOns §25.Thebinomtal serit·s. . . . §26.Thelogarithmic senes §27.Thecyclometrical functIOns ExerCises onChapter VI(74-84).151 158 171 179 188 189 191 198 208 211 213 215 Chapter VII. Infinite products. §28.Products withpositive lerJl]~. . . . . . . . . . . • • • • • . . 218 §29.Products witharbItrary terms.Absolllle COJ1VCIgCJ1LC 221 §30.ConnecltoJ1 between senesandprod"cts.Conditional andunconditional convergence 226 Exercises onChapter VlIl~5~\HJ) . • • • • • • • • • • • • • . 228 Chapter VIII. 'LClosedandnumerical expressions forthesumsofseries. §31.Statement oftheproblem . . . . . . . . . . . . . . ~30 §32.Evaluation ofthesumofaseriesbymeansofaclosedexpression ~32 §33.Transformation ofseries. . • • • • • . . . . • . • • • • • • . 240 §34.Numerical evaluations . . . . . . . . . . . . . . . . . . . . . 247 §35.Applications ofthetrall~lormation of~eIlLS 10numerical evaluations 260 Exercises onChapter VHr (100~132). •••••••••••. 267 Chapter X.Contents. PartIII Development ofthetheory. j Chapter [X. Seriesofpositive terms. §36.Detailed studyofthetwocomparison tests 274 §37.Theloganthmlc scales. . . . . . . . . . . . 278 §38.Special compari_on testsofthesecondkind 284 §39.Theorems ofAbe!,Dmi,and Prin-J~heim. andtheirapphc.ltlOn toa freshdeductIOn ofthelogarithmic scaleofcompanson test~ 290 §40.Seriesofmonotone1y diminishing positive terms . • . . • . .. 2D4 ~41.General remarks onthetheory oftheconverg-ence anddivergence ofseriesofpositive terms. . . . . . . . . . . . 298 §42.SystematLmtion ofthegeneral theory ofconvergence 305 Exercises onChapter IX(133-141). . . • • • • • • 311 ,/ Seriesofarbitrary terms. §43.Testsofconvergence forseriesatarbitrary terms §44.Rearrangement ofconditionally convergent series §45.Multiplication ofconditionally convergent series Exercises onChapter X(14~--l[):3) . Chapter Xl. /series ofvariable terms(Sequences offunctions). §46.Uniform convergence . . . . . §47.Passage tothelimittermbyterm §48Testsofuniform convergence ~49.Fourierseries. . . . . A.Euley's formulae . . . . B.Dlnchlefs Illtegral . . . C.ConditIOns ofconvergence §50.Applications ofthetheory ofFouyier series §51.Products withvariable terms Exercises onChapter XI(154-173) . Chapter XIf. Seriesofcomplex terms. §52.Complex number., andsequences . §53.Seriesofcomplex terms. . . . . §54.Powerserie'i. Analytic functions.312 318 320 324 326 338 344 350 350 356 364 372 380 385 388 :396 401 XII Contents. §55.Theelementary analytic functIOns I.Rational functions . . . •n.Theexponential function Ill.Thefunctions coszandsinz IV.ThefunctIOns cotzandt.mz V.ThclogarithmIc scnes VI.Theinverse sincseries VII.Theinverse tangcnt sencs VIII.Thebinomial senes. . . .Page. 410 410 411 414 417 419 421412 42:3 XIII.§56.Senesofvariable terms. Uniform convergence. Weierstra~s' theo- f('mondoubleseries . . . . . . . . .. .0-. 428 §57.Products withcomplex terms . . . . . . • 434 §58.SpeCial classes ofseriesofanalytic functIOns 441 A.Dtrtchlet's senes. 441 B.Faculty series. . 446 C.Lambert's series. 448 E:s:erclses onChapter XII(174-199) 452 JChapter Divergent series. §59. §60. §61. §62. §63.General remarks ondivergent sene~andtheprocesses oflinlltation TheC-andH-processes Application ofCl-summation tothetheory ofFouYler senes TheA-procc~s . . . ThcE-proccss , . . Exercises onChapter XIII(200-216) . Chapter XIV.457 478 4B2 4U~ 507 516 Euler's summation formula andasymptotic expansions. §64.Euler'ssummation formula • A.Thesummation formula B.ApplicatIOns C.TheevaluatIOn ofrem'linders §65.Asymptotic series..•. . . . . §66.Special casesofasymptotic expansIOns A.Examples oftheexpansion problem. B.Examples ofthesummation problem Exercises onChapter XIV(217-225) • Bibliography Nameandsubject inde~• . , • • • •518 518 525 531 5:35 543 543 548 553 556 557 Introduction. Thefoundation onwhichthestructure ofhigheranalysis restsis thetheoryofrealnumbers. Anystricttreatment ofthefoundations of thedifferential andintegral calculus andofrdatedsubjects mustin­ evitably startfromthere;andthesameistrueevenfore.g.thecal­ culation ofrootsandlogarithms. Thetheoryofrealnumbers firstcreates thematerial onwhichArithmetic andAnalysis cansubseyuently build, andwithwhichtheydealalmostexclusively. Thenecessity forthishasnotalwaysbeenrealized. Thegreat creators oftheinfinitesimal calculus -Leibniz andNewton 1--and thenolessfamousmenwhodeveloped it,ofwhomEuler 2isthechief, weretoointoxicated bythemightystreamoflearning springing fro:n thenewly-discovered sources tofeelobliged tocnticize fundamentals. Tothemtheresultsofthenewmethods weresufficient evidence for thesecurity oftheirfoundatIOns. ItwasonlywhL:nthestrL:affibegan toebbthatcritiL:alanalysis ventured toexamine thefUIlllamL:ntal con­ ceptions. AbouttheendofthL:18thcenturysucheffortsbecamestronger andstronger, chieflyowingtothepowerful influence ofGauss 3.Nearly acentury hadtopass,however, beforethemostessential matters could beconsidered thoroughly clearedup. Nowadays rigourinconnection withtheunderlying number concept isthemostimportant requirement inthetreatment ofanymathematical subject. Eversincethelaterdecades ofthepastcentury thelastword onthematterhasbeenuttered, sotospeak,-byfVeierstrass 4inthe sixties,andbyCantor 5andDedekind 6in1872.Nolectureortreatise 1Gott/Tled Wilhelm Leibniz, borninLeipZIg in1646,diedinHanover In 17Hl.lsaacNewton, bornatWoolsthorpe inlti42,dIedinLondon in1727.E3ch discovered thefoundations oftheinfiniteSImal calculus independently oftheother. 2Leonhard Euler,borninBasleIn1707,dIedinSt.Petersburg In1is:l. 3KarlFriedrich Gauss,bornatBrunswIck In1777,dIedatGottlngen In11'5.1. •KarlWeierstrass, bornatOstenfclde in1815,diedinBerlinin1897.The firstrigorous account ofthetheoryofrealnumbers whIchlVeierstrass hadexpounded inhislectures since1860wasgivenbyG.Mittag-LefJler, oncofhispupIls, Inhis c~say:DIeZahl,Elnleitung zurTheone deranalytischen FunktIonen, TheTtJhoku MathematIcal Journal, Vo!.17,pp.157-209. 1920. •GeorgCantor, borninSt.Petersburg In1845,dIedatHalleIn1918;cf. Mathem. Annalen, Vo!.5,p.12a.1872. "Richard Dedekind, bornatBrunswick inIS:H,dIedthereIn191G:cf.hIS book:Stetlgkelt undIrratlOnale Zahlen, Bruns\\lck 1872. 1 2 Introduction. dealingwiththefundamental partsofhigheranalysis canclaimvalidity unlessittakestherefinedconcept oftherealnumber asitsstarting­ point. Hencethetheoryofrealnumbers hasbeenstatedsooftenand insomanydifferent wayssincethattimethatitmightseemsuperfluous togiveanother verydetailed exposition 7:forinthisbook(atleastin thelaterchapters) wew~htoaddress ourselves onlytothosealready acquainted withtheelements ofthedifferential andintegral calculus. Yetitwouldscarcely sufficemerelytopointtoaccounts givenelsewhere. Foratheoryofinfinite series,aswillbesufficiently clearfromlater developments, wouldbeupinthecloudsthroughout, ifitwerenot firmlybasedonthesystemofrealnumbers, theonlypossible foundation. Onaccount ofthis,andinordertoleavenottheslightest uncertainty astothehypotheses onwhichweshIIIbuild,wcshalldiscuss inthe following pagesthose id~asanddatafromthetheoryofrealnumbers whichweshallneedfurtheron.vVehavenointention, however, ofcon­ structing astatement ofthetheory compr~s3ed intosmaller spacebut otherwise complete. Wemerelywishtomakethemainideas,themost important questions, andtheanswers tothem,asclearandprominent aspossible. Sofarasthelatterareconcerned, ourtreatment throughout willcertainly bedetailed andwithout omIssions; itisonlyinthecases ofdetailsofsubsidiary importance, andofquestions astothecomplete­ nessanduniqueness ofthesystemofrealnumbers whichlieoutsidethe planofthisbook,thatweshallcontent ourselves withshorterindications. 7Anaccount whichiseasytofollowandwhichIllcludcs alltheessentials isgivenbyH.v.Man!<o!dt, Einfuhrung IIIdil'hohereMath..matlk,Vo!.I,H"'edition (byK.Knopp), LeipZig 1944.-Thetreatment ofG.KowlI!ewski, Grundzuge derDifferential- undIntegralrechnung, 6thedition, LeipZig I!J29, ISaccurate and concise. -Arigorous con~tructlon ofthesystemofrealnumbers, whichgoesinto themlllutest details, istobefound IIIA.[,oewy,Lehrbuch dcrAlgebra, PartI, LeipZig 1915,IIIA.Pnngsheim, Vorlesungen uberZahlen- undFunktlOnenlehre, Vo!.I,PartI,2ndedition, LClpzlg 192:3(cf.alsothereviewofthelatterworkby H.Hahn,Gott.gel.Anzelgen I!l19,pp.321-47), andIIIabookbyE.Lmzdall exclu"vely devoted tothiSpurpose, Grundlagen derAnalysis (DasReehnen mtt ganzen, ratlOnalen, Irrationalen, komplexen Zahlen), Lcipzlg 1\J30.Acnticalaccount ofthewholeproblem istobefoundinthearticlebyF.Bachmann, Aufbau des Zahlensystems, IIItheEnzyklopadlc d.math.Wlssensch., Vo!.I,2""editIOn, PartI, article :3,LeipZig andBerlIn1\J3M. PartI. Realnumbers andsequences. Chapter 1. Principles ofthetheoryofrealnumbers. §1.Thesystem ofrational numbers anditsgaps. Whatdowemeanbysayingthataparticular number is"known" or"gIven" ormaybe"calculated"? \Vhatdoesonemeanbysaying thathelmowsthevalueofv2"or:Tt,orthathecancalculateV'5? AquesllOn lIkethistseaSIertoaskthantoanswer. WereItosay thatV2=1·414, Ishould obviously bewrong, since,onmulti­ plying-out,1·414X1·414doesnotgive2.IfIassert, withgreater caution, thaty~=1·4142135 andsoon,eventhatisnotenable answer, andindeed inthetirstinstance itisentirely meaningles'i. The quesIJon is,afterall,howwearetogoon,andthlS,without further indIcatIOn, wecannot tell.Noristhepositton Improved bycarrying thedeClmal further, eventohundreds ofplaces. Inthissenseit maywellbesaidthatnoonehaseverbeheldthewholeofy2",­ nothelditcompletely inhisownhands, sotospeak----whilst a statement thatY9=3orthat35--;-7=5hasafinished andthorough­ lysatisfactory appearance. Theposition isnobetterasregards thenumber :Tt,oralogarithm orsineorcosine fromthetables. Yetwefeelcertainthatv2"andJ1andlog5reallydohatJeqUItedefinite values, andeventhatweactually knowthesevalues. Butaclear notionofwhattheseimpreSSIOns exactlyamount toorimplywedo notasyetpossess. Letusendeavour toformsuchanidea. Having raiseddoubtsastothejustification forsuchstatements as"Iknowv2"",wemust,tobeconsistent, proceed toexamine nowfaroneisjustIfied eveninasserting thatheknowsthenumber -gl-orisgiven(forsomespeCIfic calcul.llion) thenumberf.Nay more,lhesignificance ofsuchstatements as"Iknowthenumber 97" or"forsuchandsuchacalculation IamgivC1Za=2andb=0"would 4 ChapterI.Principles ofthetheoryofrealnumbers. requirescrutiny. 'Veshouldhavetoenquire intothewholesignificance orconceptofthenaturalnumhers 1,2,3,... Thislastquestion, however, strikesusatonceasdistinctly trans­ gressing thebounds ofMathematics andasbelonging toanorderof ideasquiteapartfromthatwhichwepropose todevelop here. Noscience restsentirely withinitself:eachborrows thestrength ofitsultimate foundations fromstrataaboveorbelowit,suchasexperi­ ence,ortheoryofknowledge, orlogic,ormetaphysics, ...Everyscience mustacceptsomething assimplygiven,andonthatitmayproceed to build.Inthissenseneither mathematics noranyotherscience starts without assumptions. Theonlyquestion whichhastobesettledby acriticism ofthefoundation andlogicalstructure ofanyscienceiswhat shallbeassumed asinthissense"given"; orbetter,whatminimum of initialassumptions willsuffice,toserveasabasisforthesubsequent development ofalltherest. Fortheproblem wearedealingwith,thatofconstructing thesystem ofrealnumbers, thesepreliminary investigations aretediousandtrouble­ some,andhaveactually, itmustbeconfessed, notyetreached anyentirely satisfactory conclusion atall.Adiscussion adequate tothepresent position ofthesubjectwouldconsequently takeusfarbeyondthelimits oftheworkwearecontemplating. Instead, therefore, ofshouldering anobligation toassume asbasisonlyaminimum ofhypotheses, we propose toregardatonceasknown(or"given", or"secured") agroup ofdatawhosededucibility fromasmallerbodyofassumptions isfamiliar toeveryone -namely, thesystemofrational numbers, i.e.ofnumbers integral andfractional, positive andnegative, including zero.Speaking broadly, itisamatterofcommon knowledge howthissystemmaybe constructed, if--asasmaller bodyofassumptions -onlytheordered sequence ofnatural numbers 1,2,:3,...,andtheircombinations by addition andmultiplication, areregarded as"given". Foreveryone knows -andwemerelyindicate itinpassing-howfractional numbers arise fromtheneedofinverting theprocess ofmultiplication, --negative numbers andzerofromthatofinverting theprocessofaddition 1. Thetotality, oraggregate, ofnumbers thusobtained iscalledthe system(orset)ofrational numbers. Eachofthesecanbecompletely and literally "given" or"yvritten down"or"madeknown" withthehelpofat mosttwonatural numbers, adividing barandpossibly aminussign. Forbrevity, werepresent thembysmallitaliccharacters; a,b,..., x,y,...Thefollowing aretheessential properties ofthissystem: 1Secthework~ofLoewy, Pringsheim, andLandau mentioned intheIntro­ duction; al~oO.HOlder, DIeAnthmetlk instrenJ.(er Begrilndung, 2,,,1editIon, llerlin l!12!J;andO.StolzandJ.A.Gmeiner, Theoretlsche Arithmetlk, 3',1edition, LeipZig 1911. §1.Thesystemofrational numbers anditsgaps. 5 1.1.Rational numbers formanordered aggregate; meaning that between anytwo,sayaandb,oneandonlyoneofthethreerelations a<b. a=b, a>b necessarily holds2;andtheserelations of"order" between rational numbers arcsubjecttoasetofquitesimplelaws,whichweassumeknown, theonlyessential onesforourpurposes beingthe Fundamental LawsofOrder. 1.Invariably 3a=a. 2.a~balwaysimplies b-~a. 3.a=b,h=eimpliesa=e. 4.a;::-;b,b<e,-ora<b,b::;e,-implies 4a<c. 2.Anytworational numbers maybecombined infourdistinct ways,referred torespectively asthefourprocesses (orbasicoperations) ofAddition, Subtraction, Multiplication, andDivision. Theseoperations canalwaysbecarriedouttooncdefinite result,withthesingleexception ofdivision by0,whichisundefined andshouldberegarded asanentirely impossible ormeaningless process; thefourprocesses alsoobeyanumber ofsimplelaws,theso-called Fundamental LawsofArithmetic, andfurther rulesdeducible therefrom. Thesetooweshallregardasknown, andstate,concisely, those Fundamental LawsorAxiomsofArithmetic fromwhichalltheothersmay2. beinferred, bypurelyformalrules(i.e.bythelawsofpurelogic). I.Addition. ].Everypairofnumbers aandbhasinvariably associ- atedwithItathird,c,calledtheirsumanddenoted bya+b. 2.a=a',b~b'alwaysimplvaIb-~a'+b'. 3.Invariably, a+b=b+a(Commutative Law). '1.Invariably, (a+b)+c=II+(b+c)(Associative Law). 5.a<balwaysimpliesa+c<b+c(LawofMonotony). 11.Subtraction. Toeverypairofnumbers aandbtherecorresponds athirdnumber c,suchthata+c=b. • a>bandb<aaremerelytwodifferent expressIons ofthesamerelation. Stnctly speakmg, theonesymbol"<"wouldtherefore suffice. 3WithregardtothisseemIngly trivial"law"cf.footnote 11,p.Il,remarkI,p.28, andfootnote 24,p.21l. 4Toexpressthatoneoftherelation~ oforder:a<b,a==b,ora>b,does nothold,wewrite,respectively, a~b("grcater thanorequalto","atleastequal to","notlessthan"), {l-tob("unequal to","different from")ora'-b.Eachof thesestatements (negatlOns) definitely excludc~ oneofthethree relatton~ andleaves undecided whichoftheothertwoholdsgood. 6 ChapterI.Principles ofthetheoryofrealnumbers. Ill.Multiplication. 1.Toeverypairofnumbers aandbtherecorresponds athird number c,calledtheirproductanddenoted byab. 2.a=a',b=b'alwaysimpliesab=a'b'. 3.Inallcasesab=ba(Commutative Law). 4..Inallcases(ab)c=a(bc)(;\ssociative Law). 5.Inallcases(a+b)c-=a c+bc(Distributive L'lW). 6.a<bimplies, provided cISpositive, a c<b c(LawofMono- tony). IV.Division. Toeverypairofnumbers aandbofwhichthefirstisnot0there corresponds athirdnumber c,suchthata c=b. Asalready remarked, alltheknown rulesofarithmetic, -and henceultimately allmathematical results,--arededuced fromthese fewlaws,withthehelpofthelawsofpurelogicalone.Among these laws,oneisdistinguished byitsprimarily mathematical char,lcter, namely the V.LawofInduction, whichmaybereckoned amongthefundamental lawsofarithmetic andisnormally statedasfollows: Ifaset'Dlofnaturalnumbers includes thenumb~r 1,andif,every timeacertainnaturalnumber nandallthoselessth,m 11canbetakento belongtotheaggregate, thenumber(nf-1)mlybeinferred alsotobelong toit,then'.lJlincludes allthenaturalnumbers. Thislawofinduction itselffollows quiteeasilyfromthefollo\\'ing theorem, whichappears evenmoreobvious andistherefore normally calledthefundamental lawofthenaturalnumbers: LawoftheNatural Numbers. Ineverysetofnaturalnumbers that isnot"empty" thereisalwaysanumber lessthanalltherest. Forif,according tothehypotheses oftheInduction Law,wecon­ sidertheset'.llofnaturalnumbers notbelonging to'111,thisset ~llmust be"empty", thatis,IJJlmustcontain allthenaturalnumbers. Foroth~r­ wise,bythelawofthenaturalnumbers, ~llwouldinclude anumber less thanalltherest.Thisleastnumber wouldexceed1,foritwasassumed that1belongsto~JI;henceitcouldbedenoted byn+1.Thennwould belongto'Dj,but(n+1)wouldnot,whichcontradicts thehypotheses inthelawofinduction.5 Inapplications itisusuallyanadvantage tobeabletomakestate­ mentsnotmerelyaboutthenaturalnumbers butaboutanywholenumbers. bThefollowing rathermoregeneral formofthelawofinduction canbe deduced inexactlythesamewayfromthefundamental lawofthenaturalnumbers. Ifsetcl)1ofnatural numbers Include_ thenumber 1,andIfthenumber (n-I-1) canbeprovedtobelongtotheaggregate proVided thenumber 11does,then ~Icon' taIn'1111thf'naturalnumbers. §1.Thesystemofrationalnumbers anditsgaps. 7 Thelawsthentakethefollowing forms,obviously equivalent tothose above: LawofInduction. Ifastatement involves anaturalnumber n(e.g. "ifn~10,then2n>n3",orthelike)andif a)thisstatement iscorrectforn=p, and b)itscorrectness forn=p,p+1,...,k(wherekisanynatural number? p)always implies itscorrectness forn=k+1,thenthe statement iscorrectforeverynaturalnumber ~~~p. LawofIntegers. Ineverysetofintegers all-,pthatisnot"empty", thereisalwaysanumber lessthanalltherest.6 Wcwilllastlymention atheorem susceptible, inthedomain of rational numbers, ofimmediate proof,although itbecomes axiomatic incharacter verysoonafterthisdomain isleft;namelythe VI.Theorem ofEudoxus. Ifaandbareanytwopositive rational numbers, thenanatural number nalwaysexists 7suchthatnb>a. Thefourwaysofcombining tworational numbers giveinevery caseastheresultanother rational number. Inthissensethesystem ofrational numbers formsaclosedaggregate (naturlicher Rationalitats­ bereichornumbercorpus). Thisproperty offorming aclosedsystem"ith respecttothefourrulesisobviously notpossessed bytheaggregate of allnaturalnumbers, or ofallpositIve andnegative integers. Thesearc, sotospeak,toosparsely sowntomeetallthedemands whichthefour rulesmakeuponthem. Thisclosedaggregate ofallrational numbers andtirelawswhich'/Old init,arethenallthatweregardasgiven,knozlJn,secured. Asthattypeofargument whichm.lkesuseofII/equa"tle~ andabsolute t'lllues3. maybeahttleunfamilIar tosome,ItsmostImportant rule-,maybe,etdownhere, brieflyandWIthout proof: I.Inequalities. Hereallfolio....sfromtheIa\\sofordl'randmonotony. Inparticular 1.Thestatt'ments Inthelawsofmonotony arereversible; e.g.(l-+-C <b-+-caIway,Inlphes II<b;andsodoesIIc<bc,protHded c>O. 2.a<::b,c<da!way,nnphes a-+-c<b-+-cl. :1.a<b,c<dIlnphes, proVIded bandcareposltl\e, a c<bd. 4.a<ba:way~Impl:es -b< -a, andalso,prOVIded aispOSItive, ~<~. ----~ •Toreducetheseformsofthelawstotheprevious ones,weneedonlycon­ Siderthenaturalnumbers msuchthat,intheonecase,thestatement 111question iscorrect forn~(p-1)-+-m,or,intheother,that(p-1)-+-mbelongs tothe non-"elnpty" setunderconsideration. •ThiStheorem isusually,butincorrectly, ascnbed toArchl1l1edes; itisalready tobefoundinEuclId,Elements, BookV,Def.4. 8 Chapter 1.Principles ofthetheoryofrealnumbers. Alsothesetheorems, aswellasthelawsoforderandmonotony, hold(with appropnate 111odlficatlons) \vhenthes]gn~ u:~"J"~." I"-:"and"'i="arcsub­ ~tltutedfor"<",provldcd wenlaint,un thea~sumptions thatc,b,md{IarcPO~I­ tlve,III1,a,and4respectively. n.Absolute values. Definition: ByIaI,theabsolute value(ormodulus) ofa,ISmeantthatoncofthetwonumbers +aand-awhich ISPOSitive, sup­ posmga=\=0;andthenumber 0,Ifa~o.(HenceI0I.-0andIfa*0,IaI>0.) The follo~mg theorems hold, among~t others: 1.I{I\'.I-aI. 2.\abI~lal.IbI. Ill_I. jbj_lbl .3.Ia-raI'-a-IaI'provided a=\=O. J4.Ia+bI:..0IaI+IbI;Ia+bI~IaI- IbI,andmdcedIa+bI ~11al-Ibl[. 5.The t~orelations 1aI<rand-r<a<rarcexactly equivalent; similarly forIx-aI<randa-r<x<a+r. n.Ia-bIisthe<ils/mlce betwcen thePOl1ltsaandh,Withtherepresen­ tatIOnofnumbcrs onastraight hnedeSCribed nnmedlately bcluw. ProofofthefirstrclatlOn 104:±a~IaI,±b:::;IbI.soth'ltby3,I,2, ±(a-jb)£IaI+IbI,andhenceIa·1bI;S1a1+IbI. \Vealsoassume ittobeknownhowtherelations ofmagnitude betwccn rational numbers maybcillustrated graphically byrelations ofpositions between pointsonastraight line.Onastraight lineor 1IUmber-axis, anytwodistinct pointsarcmarked, one0,theorigin(0) andoneU,theunitpoint(1).ThepointPwhichistorepresent anumber a=P(q>0,P>:0,bothintegers) isobtained bymarking offontheq axis,IpItimesinsuccession, beginning at0,theqthpartofthedis­ tance0U(immediately constructed byelementary geometry) eitherin thedirection 0U,ifP>0,orifpisnegative, intheopposite direction. Thispoint 8wecallforbrevitythepointa,andthetotalityofpoints corresponding inthiswaytoallrational numbers weshallrefer toastherational pointsoftheaxis.-Thestraight lineisusually thought ofasdrawnfromlefttorightandUchosentotherightofO. Inthiscase,thewordspositive andnegative obviously become equiva­ lentsofthephrases: totherightof0andtotheleftof0,respectively; and,moregenerally, a<bsignifies thataliestotheleftofb,btothe rightofa.Thismodeofexpression mayoftenassistusinillustrating abstract relations between numbers. BTheposition ofthispointisindependent oftheparticular representatIOn ofthenumber a,i.e.ifa=p'!q'isanother representation Withq''>0andp'~0 bothintegers, andiftheconstruction isperformed withq',p'IIIplaceofq,P,the samepointPisobtamed. §1.Thesystemofrationalnumbers anditsgaps. Thiscompletes thesketchofwhatwepropose totakeasthe previously secured foundation ofoursubject. Weshallnowregard thedescription ofthesefoundations ascharacterizing theconcept of nun-zber; inotherwords,weshallcallanysystemofconceptually well­ distinguished objects(elements, symbols) anun-zber systen-z, andits elements nun-zbers, if-toputitquitebrieflyforthemoment -we canoperatewiththeminessentially thesamewaysaswedowithrational numbers. \Veproceed togivethissomewhat inaccurate statement aprecise formulation. Weconsider asystemSofwell-distinguished objects, whichwe'.I. denotebyex,f3,. ...Swillbecalledanumbersystemanditselements ex,f3,...willbecallednumbers if,besidesbeingcapableofdefimtion exclusively bymeansofrational numbcrs (i.c.ultimately bymcansof naturalnumbers alone) 9,thesesymbols ex,f3,...satisfythefollowing four conditions: 1.Between anytwoelements exandf3ofSoneandonlyoneofthe threerelations 10 et<f3,et=f3,rx.>f3 necessarily holds(thisisexpressed brieflybysayingthatSisanordered system)andtheserelationsoforderbetween theclements ofSaresubject tothesamefundamental laws1astheiranalogues inthesystemofrational numbers 11. 2.Fourdistinct methods ofcombining anytwoelements ofSare defined, calledAddition, Subtraction, Multiplication andDivision. With asingleexception, tobementioned immediately (3.),theseprocesses canalwaysbecarriedouttooncdefiniteresult,andoheythesameFun­ damental Laws2,I-IV,astheiranalogues inthesystemoftherational •Weshallcomeacrossactualexamples In§aand§5;forthemoment, we n.aythinkofdeCimal fractions, orsimIlarsymbols constructed fromrationalnumbers. Sec'11<0footnote W,p.12. 10Cf.alsofootnotes 2and4. 11Astowhatwemaycallthepractical meaning oftheserelatiOns, nothing isimplied;"<"mayasusualstandfor"Ie~sthan",but1tmayequally wellmean "before", "totheleftof","h1gher than","lowerthan","subsequent to",mfact mayexpress anyrelatIOnoforder(including "greater than"). Thismeaning merely hastobedefined w1thout ambigUity andkeptconsistent. SimIlarly, "equality" neednotimplyidentity. Thus,forexample, Wlthmthesystemofsymbols ofthe formp/q,wherep,qareIntegers andq=j=0,thesymbols a/4,li/8,-H/-12 are generally saidtobe"equal"; that1S,forcertampurposes (calculating, measuring, andsoon)wedefineequality withmoursystemofsymbols Insuchawaythat3/4 -~ li/8=-!J/-12, although a/4,(jIS,-H/-12 areInthefirstinstance d1fferent elements ofthatsystem(scealso14,noteI). 10 ChapterI.Principles ofthetheoryofrealnumbers. numhers 12.(The"zero"ofthesystem, whichmustbeknowninorder thattheelements canhedividedintopositiveandnegative, istobedefined asexplained infootnote 14below.) 3.Witheveryrational number wccanassociate anelement ofS (andallothers"equal" toit)insuchamanner that,ifaandbdenote rational numbers, (x,f3theirassociates fromS: a)therelation 1.holding between exandf3isofthesameformas thatholding between aandb. b)theclement resulting fromacombination ofexandf3(i.e.ex+f3, ex-f3,ex'f3,orex--:-f3)hasforitsassociated rational number theresult ofthesimilarcombination ofaandb(i.e.a+b,a-b,a.b,ora--:-b respectively). [Thisisalsoexpressed, moreshortly, bysayingthatthesystemS contains asub-system Sfsi1nilar andiso1norphous tothesystem ofrational numbers. Suchasub-system isinfactconstituted bythose elements ofSwhichwchaveassociated withrational numbers 13,] Insuchacorrespondence, anelement ofSassociated withtherational number zero,andallelements equaltoit,maybeshortlyreferred toas the"zero"ofthesystemofelements. Theexception mentioned in2. thenrelatestodivision byzero14. 12WIthreferencc tothesefour proee~~es Itshouldbenoted,asintheca,e ofthesynlbols <and--,thatnopractIcal InterpretatIOn I~Imphed. -'Veal", drawattentIon tothefactthatsubtractIOn ISalready completely defined Interms ofaddItIOn, andd1\'ISlon mtermsofmultIplIcatIOn, sothat,properly speakmg, onlytwomodesofcombmIng elements needbeas~umed known, 131'\\0ordered system, aresi1nilar IfItISpossIble toassociate eachelement oftheone\\Ithane1em..ntoftheotherInsuchawaythatthesameoncofthe relatIOns 4,Iasholdsbetween twoelements oftheoncsystem alsoholdsbetween thetwoasSOCiated clements oftheothcr.theyaretso1norphous relatIvely tothe pos~Ible modesofcombmmg theIrelements, Iftheelement resultmg fromacom­ binatIon oftwoelements oftheone~ystem I~associated WIththat n'~ultmg from theSimilarcombmatlOn ofthetwoassochltcd elements oftheother 'y~tem, ,.ThethIrdofthestIpulatIOns bymeansofwhIchwcherech.lractl'rise the concept ofnumber I~fulfIlled, moreover, asacon~equence ofthefirstandsecond. Forourpurposes, thl"factISnotessential; butasItIS,Iglllficant fromasvstematlc pomtofVIew,webneflyindicate itsproofasfollow,' By4,2,thereJ'anelement ,forwhich ex+,~oex.Fromthefundamental 1,1\",2,I,Itthen'1U1teea'll) follow~ thaoneandthesameelen1('nt ,of.'>'satl~fies ex-1-'"',foreveryex.Thiselement "\\Ithallelements equaltoIt,iscalledtheneutralelement relatIvely totheprocess of[.ddition, orforbreVItythe"zero" InS.Ifexisdlfferl'nt fromthiS"zero", there IS,further, anelement Eforwhich exE,="';andItagaInappearsthlt thl~element isthe~ameasthatsatisfYing 'lE"'"foranyother'" IIIS.This E,WIthallelements equaltoIt,IScalledtheneutralelement relatively totheprocess ofmultlphcatlOn, or,brJefly,the"UnIt"inS.TheelemenN ofSproduced byrepeated additIOn or subtractIOn ofthiS"umt", andanyothersequaltothem,arethencalled"integer," ofS.Allfurther elements ofS(andallequaltothem)whIch re~ultflomthese bytheprocess ofdl\lslOn thenformthe~ub-system S'ofSInque,tlOn; thatIt I~silllllarandiwmorpholls tothesystemofallrational number~ l'Infacteasily deduced from4,Iand4,2.-Tbu~,asasserted, ourconcept ofnumber ISalread) determined bythereqUirements of4,1,2and4. §1.Thesystemotrational numbers anditsgaps. 11 4.Foranytwoclements 'JI:andflofSbothstanding intherelation ">"tothe"zero"ofthesystem, thereexistsanaturalnumber 11for which 11fl>ex.Here 11fldenotes thesumfl-i--fl+...+flcontaining theclementfl11times.(Postulate ofEudoxus,' cf.2,VI.) Tothisabstract characterisation oftheconcept ofnumber we willappend thefollowing remark 1:;:IfthesystemScontains noother elements thanthosecorresponding torational numbers asspecified in3,thenoursystemdoesnotdifferinanyessential featurefromthe systemofrational numbers, hutonlyinthe(purely external) designation oftheelements bysymbols, orinthe(purely practical) interpretation whichwegivetothesesymbols; differences almost asirrelevant, atbottom, asthosewhichOl.cur",henwcwritefiguresatonctimein Arabic eharactel's, atanother, inRoman orChinese, ortakethemto denotenowtemperature, nowvelocity orelectric charge. Disregarding external characteristics ofnotation andpractical interpretation, wc shouldthusbeperfectly justified inconsidering thesystemSasidentical withthesystemofrational numbers andinthissensewemayputa=ex, b--,fl,... If,however, thesystemScont.lins otherelements besidestheabove mentioned, thenwcshallsaythatSlIlc/udes thesystemofrational numbers, andisanextension ofit.\\'hethcr asystemofthismorecom­ prehensive kindexistsatall,n:mains forthemoment anopenquestion; l'Wehavedefined theconcept ofnumber byasetofproperties characterising it.ACritical constructIOn ofthefoundatlOn~ ofarithmetiC, whIch ISqUiteout ofthequestIOn ....lthmthe11l11lt,ofth"volunle, wouldhavetocomprl~e astrict mvestlgatlOn astotheextenttowhichthc'"properties arcmdependent ofone another, i.e.whether anyone ofthemcanorcannotbededuced fromtherestas aprovable fad.Further, it\\emldhavetobe,h.....nthatnoneofthesefund.unental stipulallons ISIncontradictIOn Withanyother-andothernl.ltlers toowould reqUIre consideration. The~e inve~ugatlOns arcteulOus andhavenotyetreached a finalconclUSIOn. Inthetreatment byE.Lmu}au mentIOned onp.2,footnote 7.ItISproved WIth ab~olute rigourthatthefundamental la\\sofarithmetiC \\llIch \~eha\esetup canallbededuced frOlllthefollowmg 5aXlOnlS relating tothenatural numbers: AxiomI:IISanaturalnumber. Axiom2:Foreverynatural number nthere ISJustoneothernumber thatiscalledthesuccessor ofn.(LetIthedenoted by'1'.) Axiom3:Wehavealwaysn'{oI. AxIOm4:Fromm'~-r,',itfollowsthat111=n. Axiom5:Theinduction lawVisvalId(mitsfirstform). These (ja"iom" firstformulated asherebyG.Pell1lO,butInsubstance setup byR.Dedekmd, assume thatthenatural numbers asawholeareregarded asgl~'en, thatarelatIOn ofequalIty (andhenceabomequahty) ISdefined between them, andthatthiSequahty satisfies therelations 1,1,2,3(which belong topure logic). 12 Chapter I.Principles ofthetheoryofrealnumbers. butanexample willcomebeforeournoticepresently inthesystemof realnumbers 16. Having thusagreedastotheamount ofpreliminary assumption werequire, wemaynowdropallargument unthesubject, andagain raisethequestion: Whatdowemeanbysayingthatzveknowthenumber v'2orTT? Itmustinthefirstinstance betermed altu6etherparadoxical that anumber havingitssquareequalto2doesnotexistinthesystemso farconstructed 17,-or,ingeometrical language, thatthepointAof thenumber-axis, whosedistance from0equalsthediagonal ofthe squareofside0U,coincides withnoneofthe"rational points". For therational numbers aredense,i.e.between anytwoofthem(which arcdistinct) wecanpointoutasmanymoreasweplease(since,ifa.~b, b--athenrational numbers givenbya+v--,forv=1,2,...,n,eVI-n+1 dentlyallliebetween aandbandaredistinct fromtheseandfromone another); buttheyarenot,aswemightsay,denseenough tosymbohse allconceivable points. Rather, astheaggregate ofallintegers proved tooscantytomeettherequirements ofthefourprocesses ofarithmetic, 16Themodeofdefinmg thenumber-concept givenin4isofcoursenot theonly po~slbl" one.Frequently thedesignatIOn ofnumber isstdIa~enbed to objectswhIchfalltosatisfysomeoneorotherofthereqUIrements thereImddown. Thusformstance wcmayrelmqUlsh theconditIOn thattheobjects undercon­ Sideration should beconstructively developed fromratIOnal numbers, regardmg allYentities (forInstance pOints,ordistances, orsuchlike)asnutnbers, prOVided onlytheysatisfythecondItions 4,1-4,or,mshort,arcslmdarandI,omorphous tothe system ....ehavejustsetup.-ThisconceptIOn ofthenotIOnofnumber, inaccordance WithwhichallIsomOIphoussy,tems mustberegarded asmtheab­ stractsenseidentIcal, ISperfectly justified fromamathematlc.!1 pOIntofView,but objectIOns neeessanly ansemconnection withthethcoryofknowledge. -Wc shallencounter another modificatIOn ofthenumber -concept whenwecometo dealWithcomplex numbers. 17Proof: There IScertamly nonatural number ofsquare equalto2,as I"=Iandallothermtegers havetheirsquares ~4.Thusv'2couldonlyben (po~ltive) fractIOn P,whereqmaybetaken~2andpnmetop(1.e.thefractIOn q 2 . . . I ) B OfP. . I (P)P.Ph h IIS111ItSowe,tterms. utI -IS111It,owestterms,soIS- _._c,wICt1ere-q ,qq°q forecannotreducetothewholenumber 2.Inaslightly different form:Forany twonaturalnumbers pandgWithout common factor,wchavenecessanl:' p2+2g2. Fors1l1cetwo1I1tegers Without common factorscannotbothbeeven,eitherPIS odd,orelsepISevenandgodd.Inthefirstcasep2isagainodd,hencecannot equalanevenmtegcr 2q2.InthesecondcasepJ=(2p')2ISdiVISible by4,but2q2 ISnot,,inceitISdoubleanoddnumber. Sop2=F2'I'agam. Thl<;Pythtlf!oras IS saidtohavealready known(cf.M.Cantur, Gesch.d.Mathem., Vo\.I,2"<1ed.,pp. 142andHID.1894). §1.Thesystemofrationalnumbers andItsgaps. 13 soalsotheaggregate ofallrational numbers contains toomanygap.>18 tosatisfythemoreexacting demands ofrootextraction. Onefeels, nevertheless, thataperfectly definite numerical valuebelongs tothepoint Aandtherefore tothesymbolv'2.Whatarethetangible factswhich underlie thisfeeling? Obviously, inthefirstinstance, this:Wedo,itistrue,know perfectly wellthatthevalues1·4or1·41or1·414etc.forv'2arein­ accurate, infactthatthese(rational) numbers havesquares<:2,i.e. aretoosmall. Butwealsoknowthatthevaluesl·Gor1·42or 1·415etc.areinthesamesensetoolarge;thatthevaluewhichwe areattempting toreachwouldhavetherefore toliebetween thecorres­ ponding toolargeandtoosmallvalues. Wcthusreachthedefinite conviction thatthevalueof.,12iswithinourgrasp,although thegiven valuesareallincorrect. TherootofthisconvictIOn canonlyliein thefactthatwehaveatourcommand aprocess, bywhichtheabove valuesmaybecontinued asfarasweplease; wecan,thatis,form pairf..ofdecimal fractions, with1,2,3,...placesofdecimals, onefrac­ tionofeachpairbeingtoolarge,andtheothertoosmall,and thetwodiffering onlybyoneunitinthelastdecimal place,i.e.by<-loY', ifnisthenumber ofdecimal places. Asthisdifference maybemade 'ISsmallasweplease, bysufficiently increasing thenumber nofgiven jecimal places, wearetaughtthrough theaboveprocess toenclose thevaluewhichweareinsearchofbetween twonumbers asnear aswepleasetooneanother. Byametaphor, somewhat boldatthe present stage,wesaythatthrough thisprocessv'2itselfis"given", --­ invirtueofit,v'2is"known", --byit,v'2maybe"calculated", and soon. \Vehaveprecisely thesamesituation withregardtoanyothervalue whichcannotactually bedenoted byarational number, asforinstance TT,log2,sin100etc.Ifwesay,thesenumbers areknown,nothing more isimplied thanthatwcknowsomeproccss (inmostcasesanextremely laborious onc)bywhich,asdetailed inthecaseofv'2,thedesiredvalue maybeimprisoned, hemmed in,withinanarrower andnarrower space between rational numbers,--andthisspaceultimately narrowed down asmuchasweplease. Forthepurpose ofasomewhat moregeneral andmoreaccurate 18Thisistheparadox, scarcdy capable ofanydirectillustration, thataset ofpoints,dellSeInthesensejustexplained, mavalready bemarked onthenumber axis,andyetnotcomprise allthepointqofthestraight line.Thesituation may bedeSCribed thus:Integers formafirstroughpartition intocompartment,; r,ltlonal numhers fillthesecompartments aswithafinesand,whIchonminute inspectIOn Inevitably stilldiscloses gaps.TofilltheseWIllbeournextproblem. 14 ChapterI.Principles ofthetheoryofrealnumbers. statement ofthesematters, weinsertadiscussion ofsequences ofrational numbers, provisional incharacter, butnevertheless offundamental im­ portance forallthatcomesafter. §2.Sequences ofrational numbersl• Intheprocess indicated aboveforcalculating v'2,successive well­ defined rational numbers wereconstructed; theirexpression indecimal formwasmaterial inthedescription; fromthisformwenowpropose tofreeit,andstartwiththefollowing 6.Definition. If,bymeansofanysuitableprocessofconstruction, we canformsuccessively afirst,asecond,athird,...(rational) numberand iftoeverypositiveintegernoneandonlyollewell-drfined (rational) number Xnthuscorresponds, thenthenumbers (inthisorder,corresponding tothenaturalorderoftheintegers1,2,3,. , n,...)aresaidtoformasequence. Wedenoteitforbrevityby(xn) or(Xl'X2,•••). 6. Examples. 1.xn~I;i.e.thesequence (-~),orI,~'-31 ,...,!,...n n ~ n 2.xn---2n;I.e.thesequence 2,4,H,Ill,... 3.Xn=an;i.e.thesequence a,a',a3,•••,where IIisagivennumber. -4.xn--~{I-(_I)n}; 1.e.thesequence I,0,1,0,1,0,... 5.xn=thedeCimal fractIOn forv'2,terminated atthenthdigit. __(_I)n-l. . ._1__I _ 16.xn----n---,I.e.thesequence I,2'I3'4'... 7.LetXI=1,x,=I,X3=XI+X,=2and,generally, forn?:1, let X"--,Xn_t+xn_2•Wethusobtainthesequence I,1,2,3,5,8,13,21,..•,ubudlly calledFibonaccl's sequence. \ (1 I' Inwhichxn=1+2+...+7i)1 I 1 • 18.I,2,2'-2,-2':1,3'-3,-3'.•• 345 n-t-l9.2'"2'3'4'...J---n-,••• 1234 n-110.0,2'3'4'5'...,-ii--'··· 11.xn=thenthpnmenumber 2;i.e.thesequence 2,3,5,7,ll,13,.•• 12'rh 1:11125137. esequence'2'6'12'60"'" IInthissectionallliteralsymbols willcontinue tostandforrational numbers only. 2Euclidprovedthatthereisaninfinityofprimes.IfPI'P2'...,P.areany primenumbers, thentheintegerm-~(1',p,...P.)+1iseitherapnmedifferent fromPI'1'2'...,1'.,orelseaproduct ofsuchprimes. Hencenofinitesetofpnme numbers canmclude allpnmes. §2.Sequences ofrationalnumbers. 15 Remarks. 1.Thelawofformation maybeqUItearbitrary; itneednot,inparticular, beembodied inanyexplicit formula enablmg ustoobtain Xn,foragivenn,by directcalculation. Inexamples 6,6,7and11,clearlynosuchformula canbeim­ mediately wnttendown.Ifthetermsofthesequence aremdlvidually ~pven,neither thelawofformatIOn (cf.6,6and12)noranyotherkindofregulanty (cf.6,11) amongthesuccessive numbers isnecessanly apparent. 2.ItISsometimes advantageous tostartthesequence witha"Oth"termxo, orevenwitha(_I)'hor(_2)thterm, X_1>X_2'OccasIOnally, itpaysbettertostart mdexing With2or3.Theonlyessential isthatthereshouldbeanmtegerm~0 ,uchthatXnISdefined foreveryn~1/1.ThctermxmISthencalledtheinitIalterm ofthesequence. WeWillhowever, eventhen,continue todesignate asthenthterm Ihatwhichbearsthemdexn.In§6,2,3and4,formstance, wecanWithout further difficulties takeaOthtermoreven(-1)'"or(-2)'htoheadthesequence. The"first term"ofaseq\fence isthennotnece.sanly thetermWithwhichthesequence begms. Thenotation willbepreferably (xo,x..•..)or(x_I>Xo,•.•),etc.,asthecasemaybe, unlessItISeitherquiteclearorirrelevant whereourenumeration begms,andthe abbreViated notatIOn (xn)canbeadopted. 3.Asequence isfrequently charactensed asznfimte. Theepithet isthen merf'lyintended toemphaSize thefactthateverytermISsucceeded by'otherterms. ItisalsosaidthattbereISanmfimte numberoft..rms.Moregenerally, there" saidtobeafinitenumberoran711fimtenumberofthmg.underconsideration accord­ mgasthenumber ofthesethingscanbemdicated byadefinite mtegral number ornot.AndwemayremarkherethatthewordI1Ifimte, whenotherwise usedm thesequel,wl1lhaveasymbolic Significance only,mtended asaconcise expressIOn ofsomeperfectly defimte (andusuallyquiteSimple) circumstance. 4.Ifallthetermsofasequence haveoneandthesamevaluee,thesequence ISSaidtobeIdentically equaltoc,andinsymbols (xn)=c.Moregenerally, weshall wnte(x,.)==(xn')ifthetwosequences (xn)and('X'n')agreetermforterm,i.e.for everymdexmquestion xn=xn'. 5.Itisoftenhelpful andconvenient torepre.ent asequence graphically bymarkmg offItstermsonthenumber-aXIs, ortothmkofthemassomarked. Wethusobtamasequence ofpomtf. Butindomgthisitshouldbeborneinmind that,inasequence, oneandthesamenumber mayoccurrepeatedly, even"in­ finitelyoften"(cf.6,4);thecorrespomlmg pointhasthentobecounted (i.e.con­ Sidered asatermofthesequence ofpomts)repeatedly, ormfinltely often,a,the casemaybe. li.Agraphical representation ofadifferent kmdisobtamed bymarking, withrespect toapairofrectangular coordmate axes,thepointswhosecoordmates are(n,'X'n)forn=1,2,3,...andjoinmgconsecutive pomtsbystraight segments. Thebrokenlinesoconstructed givesapicture(diagram, orgraph)ofthesequence. Toconsider fromthemostdiversepointsofviewthesequences hereby introduced, andtherealsequences thatwillshortlybedefined, willbethe mainobjectofthefollowing chapters. Weshallbeinterested morepar­ ticularly inproperties whichhold,orarestipulated tohold,forallthe termsofthesequence, oratleastforalltermsbeyond(orfollowing) some definiteterm3.Withreference tothislastrestriction, itmaysometimes 3E.g.allthetermsofthesequence 6,9are>1.Or,allthetermsofthe sequence 6,2afterthe6thare>100(ormoreshortly: forn>6,xn>100).7. or16 ChapterI.Principles ofthetheoryofrealnumbers. besaidthatparticular considerations inhandarevalid"afinitenumber oftermsbeingdisregarded", oronlyconcern theultimate behaviour of thesequence. Ourfirstexamples ofconsiderations ofthekindreferred toareafforded bythefollowing definitions: ~.Definitions. 1.Asequence issaidtobebounded 4,Ifthereisa positive numberKsuchthateachtermXnofthesequence satisfies the inequality ThenumberKisthencalledaboundofthesequence. Remarks andExamples. 1.Indefinition 8,itisamatterofpractical IndlfTerence whether wewrite "~K"or"<K". ForIf1.'I'nI;":;Kholdsalways (I.e.forevery 11Inquestion), thenwecanalsofindaconstant K'suchthatIXnI<K'holdsalways; indeed, clearlyanyK'>KWIllservethepurpose. Conversely, ifIx"I<Ka!way<,then afortiOriIXnI:0::;K.Whentheexactmagnitude oftheboundcomesInofcourse thedIstinctIOn maybeessential. 2.IfKisaboundof(xn),thensoisanylargernumberK'. 3.Thesequences 6,1,4,5,G,9,10areeVIdently bounded; sois6,3,pro­ videdIaI;S;1.Thesequences 6,2,7,8,11arecertainly notso.Whether 6,:I foreveryIaI>1,or6,12,ISbounded ornot,bnotImmediately obVIOUS. 4.IfallweknowistheeXistence ofaconstant Kt>suchthatx"<Kt>for everyn,thenthesequence ISsaIdtohebounded ontheright(orabove)andKIIS calledaboundabove(oraTIghthandbound)ofthesequence. Ifthereisaconstant K2suchthatxn>K2always,then(xn)isSaIdtobe bounded ontheleft(orbelow)andK2iscalledaboundbelow(oralefthaudbound) ofthesequence. HereKIandK2neednotbeposlllve. 5.Supposing agivensequence isbounded ontheright,itmaystillhappen thatamongItSnumbers noneisthegreatest. ForInstance, 6,10ISbounded on thenght,yeteverytermofthISsequence isexceeded byallthatfollowIt,andnone canbethegreatest 5.Similarly, asequence boundcd ontheleftneedcontam no leastterm;cf.6,1andO.-(Wlth thisfact,whichwillappearatfirstsightpara­ dOXical, thebegmner ~houldmakehImself thoroughly famlhar.) Among afimtenumberofvaluesthereISofcoursealwaysbothagreatest and aleast,i.e.avaluenotexceeded byanyoftheothers.andonewhIchnoneofthe othersfallsbelow. (Theremay,however, beseveralequaltothisgreatest orleast value.) ti.Theproperty ofboundedness ofasequence xn(though nottheactualvalue ofoneofthebounds) ISaproperty ofthetail-endofthesequence; Itisunaffected byanyalteratIOn toanisolatedtermofthesequence. (Proof?) •ThiSnomenclature appears tohavebeenintroduced byC.Jordan. Cours d'analyse, Vol.I,p.22.Paris1803. 6Thebeginner should ~ardagainst modesofexpression suchasthese, whichmayoftenbeheard:"forninfinitely large, XfI=I"j"1isthegreatest number ofthesequence". Anything ofthissortISsheernonsense (cf.onthISpoint 7,3).Forthetermsofthesequence are0.i,J,I,...andnoneoftheseis_cc1,on thecontrary allofthemare<I.Andthereisnosuchthingasan"infil1ltely largen". §2.Sequences ofrational numbers. 17 n.Asequence issaidtobemonotone ascending orincreasing 9. if,foreveryvalueofn, itissaidtobemonotone descending ordecreasing 21,foreveryn, Bothhindswillalsobereferredtoasmonotone sequences. Remarks andExamples. 1.Asequence neednotofcoursebeeithermonotone increasing, ormono­ tonedecreaSing; cf.6,4,6,8.lVIonotone sequences are,however, extremely com­ mon,andusually easiertodealwiththanthosewhicharenotmonotone. That ISwhyItisconveruent togivethemadistlngUlshmg name. 2.Instead of"ascending" weshouldmorestnctly say"non-descending", andinsteadof"descendmg", "non-ascending". This,however, isnotcustomary. IfInanyspeCial Instance theSIgnofequabty isexcluded, sothatxn<xn+Ior "n>xn!"asthecascmaybe,foreveryn,thenthesequence ISsaidtobestrictly monotone (lncreasmg ordecreaSing). 3.Thesequences 6,2,5,7,](1,Il,12and6,I,9aremonotone; thefirst­ namedascending, theothersdescendmg. 6,3ISmonotone descendmg, If0~a-:;I, butmonotone ascendmg Ifa~>I;fora<0,ItISnotmonotone. 4.ThedesignatIOn of"monotonc" isduetoC.Neumann (Oberdicnach Kteis-, Kugel-undZylmdcrfunktlOnen fortschreltenden Entwlckelungen, pp.26, 27.LeipZig IS81). whichthereadershouldpay tomakehimself masterofIts IXn1<EWenowcometoadefinition to thegreatest attention, sparing noeffort meaning andallthatitimplies. Ill.Asequencewillbecalledanullsequence 2fitpossesses thefol-10 lowingproperty: givenanyarbitrary positive(rational) number E,thein­ equality issatisfiedbyalltheterms,1vithatmostafinitenumber 6ofexceptions. In otherwords:anarbitrary positivenumber Ebeingchosen,itisalwayspossible todes~~nate atermx",ofthesequel/ce, beyondwhichthetermsarelessthan Einabsolutevalue.Oranumber nocanalu.!aysbefound,suchthat IXnI<Eforevery Il>no. Remarks andExamples. 1.If,inagivensequence, theseconditions arefulfilled foraparticular ~, theyWillcertainly bcfulfilled foreverygreater £(cf.8,I),butnotnecessanly for anysmaller~. (In6,10,forInstance, theconditions arefulfilled forE=1andthere­ foreforeveryldrger £,ifwcputnocO;for£~~itisnotpossible tosatlsfvthem.) Inthecaseofanullsequence, thecondltlons havetobefulfilled foreveryposItive 8Cf.7,3. 18 ChapterI.Principles ofthetheoryofrealnumbers. £,andinparticular, therefore, foreveryverysmall £>O.Onthisaccount, itis usualtofonnulate thedefimtlOn somewhat moreemphatically asfollows: (xn) isanullsequence if,toevery £>0,however small,therecorresponds anumber "0suchthat Ix"I<£foreveryn>no. Here1,,,neednotbeaninteger. 2.Thesequence 6,1ISclearlyanullsequence; for 1I:\'"I<£,prOVIded n>-, £ 1whatever bethevalueof£.Itisthussuffictent toputno= £ 3.TheplaceInagIvensequence beyond whIchthetermsremainnumeri­ cally<£,WIllnaturally depend ingeneralonthemagmtude of£;speakmg broadly, itWIllliefurtherandfurther tothenght(i.e.n"WIllbelargerandlarger), the smaller thegIven £IS(cf.2).ThISdep('mlence ofthenumber noon£ISoften emphasised bysaymgexpliCitly: "ToeachgIven £corresponds anumber n"~no(t) suchthat..." 4.ThepOSItive number belowwhichIx"Iistohefromsomestageonwards neednotalwaysbedenoted by£.Anypositive number, however deSIgnated, may serve.Inthesequel,where £,()(,K,...,denoting anygIvenpOSItive numbers, we f.d£ ££. ()(may0tenuseInstea2'3'K'c,ex£,£,etc. 5.ThesIgnofXnplaysnoparthere,sinceI-x"I=Ix..I.Accordingly 6,6isalsoanullsequence. 6.Inanullsequence, notermneedbeequaltozero.Butalltenns,whose indexisverylarge,mustbeverysmall.ForIfIchoo~e £=10-6,say,thenforever~ n:>acertainno,IxllImustbe<10-".SimIlarly for£',10-1"andforanyother £. 7.Thesequence (an)spectfied In6,3ISalsoanullsequence prOVIdedIClI<1. Proof.Ifa~0,theassertion istriVial,sincethen,foret'ery £>0,Ix"I<£ 1foreveryn.If0<IaI<I,then(by3,I,4)1'.21>1.Iftherefore Weput 1ra-[=1+p,thenp>O. Butinthatcase,foreveryn;:;;2,wehave (a) (I+p)">1+np. Forwhenn=2,wehave(1+p)"=1+2p-+p2>1+2p;thestatedrelatIOn therefore holdsinthatcase.If,forn-k~2, (1+p)">1+kp, thenby2,Ill,6 (I+p)k+l>(1+kp)(1+p)=1+(h+I)P+kp2>1+(k+1)p, therefore ourrelation, assumed trueforn=h,istrueforn=k+1.By2,V ittherefore holds 7foreveryn;:;;2. 7Theproofshowsmoreover that(a)isvalidforn~2provided only1+P >0,i.e.p>-I,but'*'O.Forp,-00andforn=I,(a)becomes anequalit}. Forp>0,tl,evalidIty of(a)follows immedIately fromtheexpansIon oftheleft­ handsidebythebinomial theorem. -Therelation (a)iscalledBernoulli's Inequality <:lamesBernoulli, Propositiones arithmeticae deseriebus, 1689,Prop.4). §2.Sequences ofrational numbers. l~ Accordmgly, wenowhave la"I_}<1<1 (I+p)n 1+npnp' sothat,however small. :>0maybe,wehave forevery IxnI~IanI<• n>~.pe 8.Inparticular, besides thesequence(I)mentioned in2.,(2~)' (~,). l~n'((:,)n), etc.,arcalsonullsequences.n 9.Asbmlarremarktothatof8,1maybeappended toDefimtlOn 10:no essential modification ISproduced byreading";'S."for"<."there. Infact, If,foreveryn>n",IXnI<"thenafortIOriIXnI;:;.;conversely, if,givenany "nocanbesodetermined thatIXnI;c:;•foreveryn>n3,thenchoosing anypOSI­ tivenumber '1<•thereiscertainly annlsuchthatIxnI~E..foreveryn>n1' andconsequently Ixnl<E forevery n>111; theconditions Intheiroriginal formarcthusalsofulfilled. -Precisely analogous con~lderatlOns showthatinDefimtlon 10">n,,"and"~no"arcpractically inter­ changeable alternatives. Inanyindividual case,however, thedIstinction mustofcoursebetakeninto account. 10.Although Inasequence everytermstandsentirely byItself,Withadefinite fixed ~alue,andISnotnecesqanly Inanyparticular relation Withtheprecedmg orfollOWing terms,yetitISqUitecustomary toascnbe"tothetermsxn",or"to thegeneral term"anypeculiantles Inthesequence "hlchmaybeobserved on running through It.WemIghtsay,formstanee, In6,Ithetermsdlmmlsh; in 6,2thetermsincre.lse; in6,4or6,()thetermsOSCillate; in6,IIthegeneral termcannotbpexpressed byaformul.l, andsoon.-Inthissense,thecharacter­ IsticbehaVIOur ofanullsequence maybedescnbed bysaymgthatthetermsbecome arbitranly small,orlIlfillltely small';bywhichneithermorenorlessismeantthan IScontmned inDefimtlOn 910,viz.thatforevery.>0however smalltheterms areultl/nately (i.e.forallmdlcesn:>aSUitable 110;orfromandafter,orbeyond, acertain 11,,)numericallv lessthan E. 11.Anullsequence ISIpSOfactobounded. ForIfwechoose E~I,thenthere mustbeanmteger 11,suchthat,forevery 1/>n..IxnI<I.Among thefinite number ofvalues I'~II,Ix.l,...,!.\'nll,however, one(cf.8,5)isgreatest, ~M say.ThenforK-M+I,obViouslyI'\·nIisalways<K. 12.Toprovethatagivensequence ISanullsequence, itisindispensable toshowthatforaprc,cnbeu •>0,thecorrespondmg 1/"canactually beprovt:d toeXist(formstance, asmtheexamples thatfollow,byactually deSignating such anumber). Conversely, Ifasequence (xn)isassumed tobeanullsequence, ItIS thereby assumed that,forevery"thecorresponding n"mayreallyberegarded as existent. Ontheotherhand,thestudent shouldmakesurethatheunderstands clearlywhatISmeantbyasequence notbeinganullsequence. Themeaning is this:itisnottruethat,foreverypOSitive number E,beyond acertainpointIXnI 8Thismodeofexpression isduetoA.L.Cauchy (Analyse algebrique, pp.4 and26). 9Thereneedofcoursebenoquestion hereofthesequence beingmonotone. Also,inanycase,someIxnI'sofindex;;;;;nomayalready be<~. 20 Chapter I.Principles ofthetheoryofrealnumbers. ISalwdys<.-;thereeAlsBa,pl'cwlp031tive number '-",suchthatIxnIisnot,beyond ,myIl,ahmys<to;aftereveryIlIIthereisalargermdexn(andtherefore anin­ fimtenumber of~ueh lI1dlcl'~) for~~hlchI""I?:"0' J:l.Finally wemaymdlcate ameansofmterpreting geometncally thespecml character ofanullsequence. Usmgthegraphl~,11 representation 7,5,thesequence isanUlisequence If itstermsultimately (forn>nil)allbelongtotheinterval 10-.....+...Let uscall~uchanlI1ten'al forbreVityan..-neighbourhood oftheongin; thenwemay state(xn)isanullsequence Ifevery..-neIghbourhood oftheongm(however ~mall) cont,lIns allbutafimteIlumber, atmost,oftheternlSofthesequence. Similarly, uSingthegraphical reprc~entatlOn 7,6,wecanstate:(xn)isa nullsequence Ifevery..-striP(however narrow) aboutthe{HISofabscI,.'ae contatn~ theentiregraph,withtheexceptIOn, atmost,ofafinitemitmlportIOn, the..-stnp bemglmllted byparallel~ totheaXIsofabSCIssae through thetwopOints(0,±..). 1-1-.Theconcept ofanullsequence, the"arbltranly smallgiven pO~It1ve number ....,towhIchwesh,lllfromnowonhavecontinually andindispensably to appeal,andwhichmaythusbeSaidtoformanMmsupport forthewholesuper­ structure ofanalysis, appears tohavebeen fir~tusedm1Ii55by:1.IVallls(v.Opera I.,p.:lS2/3). Substantially, however, ItISalreadytobefoundinEuclid,Element, V. Wearealready inabetterposltiun tocomprehend whatisinvolved intheidea,discussed above,ofameaning fory'2or7TorlogG.-In forming ontheonchand(wekeeptotheinstance ofy'~)thenumbers xl=1'4; x2=1"1l; X:,=1·411; x4==1·4H2; ••• ontheother,thenumbers Yl=H:J; Y~=1·42; )'a~1·41fi; )"1=1·411:3;..• weareobviously constructing twosequences of(rational) numbcrs (.x'n) and(Yn)according toaperfectly definite (though possibly verylaborious) method ofprocedure. Thesetwosequences arebothmonotone, (x,,) increasing, (Yn)decreasing. F;,uthermorc Xnis<Ynforcvery 11,butthe differences, i.e.thenumbers 1form,by10,8,anullsequence, sincedn==J(Jn.Theseareclearlythe factswhichconvince usthatwe"know"\1'2,andcan"calculate" it, andsoon,although -aswesaidbefore-noonehasyethadthe value\1'2completely within hisview,sotospeak.--Ifwerefer againtothemoresuggestive representation onthenumber-axis, then, obviously (cf.fig.I,p.25):thepoints XlandY1determine aninterval 10ThewordlJlterval denotes aportIOnofthenumber-axis between adefinite pairofitspoints. According aswereckonthesepointsthemselves asbelonging tothemterval ornot,thisIStermed closedoropen.Unlessotherwise stated,the interval Willalwaysinthe5equclberegarded asclosed.(For10,1:1thisisimmaterial, by10,9.)Supposing atobetheleftendpoint,btherIghtendpoint,ofaninterval, wecallthisforbrevitytheinterval a...b. §2.Sequences ofratIOnalnumbers. 21 ./1oflengthd);thepoints X2andY2'similarly, aninterval.l2 oflength d2•Since thesecondintervallieswhollywithinthefirst.Similarly, thepoints X3 andYadetermine aninterval oflengthda,completely within.12'and generally, thepoints XnandYndetermine aninterval.Incompletely inside.In-I.Thelengthsoftheseintervals formanullsequence; the intervals themselves shrinkup,-onesurmises, -aboutadefinite number, -contract toaquitedefinitepoint. Itonlyremains toexamine hownearthissurmise istotruth.With thispurpose inview,westate,moregenerally, thefollowing: Definition. Toexpressthefactthatamonotone ascending sequence11. (xn)andamonotone descending sequence(y,.)aregiven,whosetermsfor everynsatisfythecondition wndforwhichthedifferences formanullsequence, -1~esay{nrbrevitythatwearegivenanestof intervals (Intervallschachtelung) *.ThenthI1lten:al stretches fromx"toYnandhaslengthdn.Thenestitselfwillbedenotedby(In)or by(xnIYll)· Theconjecture whichwemadeabovenowfindsitsfirstconfirma­ tioninthefollowing: Theoremt.Thereisatmostone(ratinnal) pointsbelonging toall12. theintervals ofagivennest,thatistosaysatisfying, foreveryn,thein­ equality x,,~s<Yn· Proof:Iftherewere,besidess,another number s'differing from it,andalsosatisfying theinequality x,,;Ss':;:;:Yn foreveryn,then,foreveryn,besides Xn<S~Ym • Asetorseriesofsimilarobjectsissaidtoformanestortobenested(inein­ andergeschachtelt) wheneachsmalleroneisenclosed orfitsintothatwhichISnext insizetoit.ThewordnestIShereusedWIththeadditIOnal (ideal)characteristic Implied, thatthesizesdimznish tozero.Whenthisisnotimplied, weshallusethe moreexpliCitphrasethateachiscontained inthepreceding (orVl-emightsaythat theyarenested).tWenotehereforfuturereference thatthistheorem continues toholdun­ alteredwhenthenumbers whichoccurarearbitrary realnumbers. :z (G51) 22 ChapterI.Principles ofthetheoryofrealnumbers. weshouldalsohave(v.3,I,4) -Yn:'S-s'<x,,; by3,I,2and3,11,5,theinequalities - dn=s;;s-s'<d",orIs-s'I~d", wouldtherefore holdforeveryn.Choosing £=Is-s'I,dllwouldnever (afortiorinotforeverynbeyond acertain 110)be<£.Thiscontradicts thehypothesis that(dn)isanullsequence. Theassumption thattwo distinct pointsbelongtoalltheintervals istherefore inadmissible 11. Q.E.D. RemarksandE xampIes. n-In+l. n-In+ld21.Letxn=----,Yn=--;thatIStosaY,./n ~---...---- , 11= -n n n n n Wecanatonceverifythatweactually haveanestof10tervals here,s10ce 2xn<''''n+l<Yn+1<Y"foreveryn,andSince,foreverytl> ,wehavetin<E, E however f£>0bechosen.n-I n+lThenumber s=1herebelongs toalltheIn's,s10ce --<1< -n n foreveryn.Nonumber otherthan1canbelongtherefore toalltheintervals. 2.LetInbedefined asfollows12:./0ISthe10terval 0 . . .1;lithelefthaIf of10;I.therighthalfof)l;.I.thelefthalfof.!.;andsoon.These10tervals are obVIOusly eac",conta1Oed mtheprecedmg; ands10ceInhaslengrhdn_~1 ,'md- ·)n thesenumbers formanullsequence, wehaveane~tofmterval~. Ahttlecons"lera­ tlOnshowsthatthesequence ofthexn'sconsists ofthenumbers 1 I 1 .:;I I 1 21 0,4':4+IQ=TIl'4-+It-+04-(i4'.•• eachtakentu:icerunn1Og; andthatthesequence ofyn'sbeginsWith1andcon­ tinueswith and18eachtakentwicerunnmg. Now ~+1;;-I-u\-+...-+~Ir=~(1-lie)<~ I-~-~-...-~-~F' c~~(1+~)>~. 11Fromagraphical pointofView,whattheproofindicates isthatifsand s'belongtoallthemtervals, theneach10terval hasalengthatleastequaltothe distanceIs-s'Ibetween sands'(v.3,11,G);theselengths cannot, therefore, formanullsequence. l'Herewelettheindexstartfrom0(cf.7,2). 13Foranytwonumbers aandb,andeveryposiuve integerk,theformula ak_bk=(a-b)(ak-I+ak-.b+...+a bk-.+bk-') isknowntohold.Whence, moreparticularly, fora=1=1,theformulae k-I-ak kI-airI +a+...+a1=-1----anda+a'+...-+a=I---.a.-Q -a §3.Irrational numbers. 23 Hence, foreveryn.Xn<!<Y,,;thuss--kisthesln{(lenumber whichbelongs toalltheIntervals. Here,therefore, (.In)"defines" or"determines" thenumberk. or(.In)shnnksuptothenumher!. 3.\:fwearegivenanestofintervals(I..),andanumber shasbeenrecog­ nisedasbelongmg toalltheI.:s,thenbyourtheorem, sISquiteuniquely deter­ mmedby(In)'Wetherefore say,morepointedly, thatthenest(In)"defines" or "encloses" thenumber s.WealsosaythatsIStheinnermost pointofalltheinterval~. 4.IfsISanygivenrational number andweput,forn=I,2,...,xn=s_I I n andYn=s+iJ'then(xnIYn)iseVIdently anestofIntervals determining thenumber sitself.ButthISisalsotheeaseIfweput,foreveryn,xn~sandYn~s.-Mam­ festly,wecan,inthemostvarious ways,formnestsofintervals definmg aJ!lVen number. Thistheorem, however, onlyconfirms whatwemayregardasone halfofourpreviously described impression; namely, thatifanumber sbelongs toalltheintervals ofanest.thenthereisnoneotherbesides withthisproperty, -sisuniquely determined bythenest. Theotherhalfofourimpression, namely, thattheremustalso alwaysbea(rational) number belonging toalltheintervals ofanest. iserroneous, anditisprecisely thisfactwhichwillbecome ourinduce­ mentforextending thesystemofrational numbers. TillSthefollowmg example shows. Asonp.20.letXI=14;x"=c1·41;...; Yl-I,,;Yl=1·.12;...Thenthereisnora/IOna/number s,for\\ll1ch Xn-=-=~.<Yn foreveryn.Infact.Ifweput y..'=Yn2 thenthemtervals In'~xn'•..yn'alsoformanest11.But:I:n'=xn"<2foralln. andyn'---yn2>2forall71(because thiswashowXnandY"werechosen), I.e. xn'<2<Yn'.Ontheotherhand,IfXn::ss~-'-y"weshouldhave,bysquanng (aswemay,by3,I,3),xn'~S2~Y,.'forall11.Byourtheorem 12thISwouldm­ valve S2==2,which IShowever impo,",ble, bytheproofgl\,eninfootnote 17on p.12.Here,therefore, thereIScertamly no(rational) number belongmg toallthe intervals. Inthefollowing paragraphs, wewillinvestigate what,inacasesuch asthis,shouldbedone. §3.Irrational numbers. Wemustcometotermswiththefactthatthereisnorational number whosesquareis2,thatthesystemofrational numbers istoo defective, tooincomplete, toofullofgaps,tofurnish asolution forthe 14ForItfollows fromXn::;:;xnj-I<Yn+l::;:;Yn-smceallthenumbers are pOSItive, sothatsquarmg (cf.3,I,3)isallowed -thatxn'~·~'n+l<Y'''I-':SYn'; further yn'-XII'---(Yn+xn)(Yn-'\·n);therefore, sincexllandY"arecertamly .....2f '-'4. ··d-d1-'.. d h b 10" ,oreveryn,Ynxn<Ion'I.e.<s,prO\1(Ion--4'antIS,Y,0, iscertainly thecaseforeveryn>acertamno. 24 ChapterI.Principles ofthetheoryofrealnumbers. equation x2=2.Indeed, thisisonlyoneofmanyequations forwhose solution thematerial ofthesystemofrational numbers provesinsufficient. Almost allthenumerical valueswhichweareinthehabitofdenoting by{In,logn,sinex,tanexandsoon,arenon-existent inthesystemof rational numbers andcannomorebeimmediately "obtained", or"deter­ mined", orbe"statedinfigures", thancanV2.Thematerial istooeoarse forsuchfinerpurposes. Theconsiderations brought forward inthepreceding paragraphs pointtomeansfmproviding ourselves withmoresuitable material. 'Vcsaw,ontheonchand,that,behind theconviction thatwedo knowv2,therelaynomore,substantially, thanthefactthatwepossess amethod bywhichaperfectly definite nestofintervals maybe obtained; foritsconstruction, thesolution oftheequation x2=2of coursegavetheoccasionI".Wesaw,ontheotherhand,thatifa nest(In)enc1os:.:s anynumber scapable ofs.)~cification atall(thisstill implying thatitisarationalnumber) thenthisnumber sisquiteuniquely defined bythenest(In),--sounambiguously, indeed,thatitiscntirely indifferent, whether Igive(writedown,indicate) thenumber directly, orgive,instead, thenest(In)-withthetacitaddition that,bythelatter, Imeanpr.:cisely thenumber swhichituniquely encloses ordefint:s. In thissense,thetwodata(thetwosymbols) arcequivalent, andmay toacertainextentbeconsidered equall6,sothatwemaywritein­ deed: 15Thekernelofthisprocedure isIIIfactasfollows: Weascertain that 12<2,22::-.2,andaccordIngly putx"=I,Y"--2.WethendIvidethemterval ./0=x"•••YoInto10equalparts,andtakIngthepOIntsofdIvision, 1+tu,for k-"0,1,2,...,9,10,determine bytrialwhether theirsquaresare>2or<2. Wefindthatthesquares corresponding tok=0,I,2,a,4aretoosmall,those correspondmg tok=5,G,...,10toolarge,andaccordIngly weputx,~~1·4and YI==1·5.Next,wed,v,detheInterval /1=XI•..YIInto10equalparts,andgo through asimilartestWithregardtothenewpomtsofdiVision -andsoon.The knownprocess forextractmg thesquarerootof2ISmtended maInlytomakethe successIve trIalsasmechamcal aspOSSible. -ThecorrespondIng treatment of, forinstance, theequatIOn lO'"0=2(I.e.determination ofthecommon logarIthm of2)Involves thefollOWIng nestofintervals: Since10"<2,10'>2,weherepu, Xu=0,Yo=1anddivide./o=Xo•••Yointo10equalparts.Forthepomtsof division, I~'wenexttestwhether IOk/IO<2or>2,thatistosay,whether 10" <21Uor> 210•AsaresultofthIStrial,weshallhavetoputXI~0'3,Y,~0·4. Theinterval./1 =Xl•••Ylisagaindivided into10equalparts,thesamepro- cedureinstituted forthePOIntsofdivision r~+I~Uand,Inconsequence, x.put equalto030andY.to031-andsoon.-ThISobvIOUS procedure isofcourse muchtoolabOrIOUS forpractIcal calculations. 18Thejustification forthisisprovided byTheorems 14to19. §3.Irrational numbers. 25 Consequently, wewillnotsaymerely: "thenest(In)definesthenumber s"butrather''(In)isonlyanothersymbolforthenumber s",orinfine, ''(In)isthenumber s"-exactlyasweareusedtolookuponthedecimal fraction 0·333...asmerelyanother symbolforthenumberl,orasbeing precisdy thenumber1itself. Itnowbecomes extremely natural tointroduce tentatively an analogous modeofexpression withregardtothosenestsofintervals whichcontain norational number. ThusifXnJYndenotethenumbers constructed previously inconnection withtheequation x2=2,one might-seeingthatinthesystem ofrational numbers thereisnot asingleonewhosesquare=2-decidetosaythatthisnest(xnIYn) determines the"true""valueofv2"though oneincapable ofbeing symbolised bymeansofrational numbers, -thatitencloses this f5 -4-X~I--~Xt~J~-JLf'-'--V, ~fYo- '--------J, 11I :'------------.1 0----------..- Fi~.1. valueunambiguously -infine,"itisanewlycrcated symbol forthis number", or,forbrevity,"itisthenumber itself". Andsimilarly inevery othercase.If(In)~(x"Iy,,)isanynestofintervals andnorational number sbelongs toallitsintervals, wemightfinallyresolvetosaythat thisnestencl03es aperfectly definite value,-though oncincapable of beingdirectly symbolised bymeansofrational numbers, -itdeter­ mines 3perfectly definite number,-though oneunfortunately non­ existent inthesystemofrational numbers,.-itisanewlycreatedsymbol forthisnumber, orbriefly: isthenumberitself;andthisnumber, in contradistinction totherational numbers, wouldthenhavetobecalled anirrational number. Herecertainly thequestion arises: CanthisbedoneZl,Jithout furtherjustification? Isitallowable? Maywc,without moreado, designate thesenewsymbols, thenests(x"IYn),asnumbers? Thefol­ lowingconsiderations areintended toshowthattothiscoursethereis noohstacle whatever. Inthefirstinstance, asimplegraphical illustration ofthesefacts onthenumber-axis (seefig.1)giveseveryappearance ofjustification to ourresolution. If,byanyconstruction, wehavemarked apointPon thenumber-axis (e.g.bymarking offtotherightof0thelength 26 ChapterI.Prmciples ofthetheoryofrealnumbers. ofthediagonal ofasquareofside0U)thenwecaninanynumber ofwaysdefineanestofintervals enclosing thepointP.Wemay dosointhisway,forinstance. Firstofallweimagine allintegers ~0marked ontheaxis.Ofthese,therewillbeexactlyone,sayp, suchthatourpointPliesinthestretchfrompinclusive to(p+1) exclusive. Accordingly weputXo-=p,Yo=P+I,anddividethc i:1terval .10=Xo...Yointo10equalparts17.Thepointsofdivision arep+l~(withk=0,I,2,...,10),andamongthem,therewillagain k kbeexactly one,sayp+l~'suchthatPliesbetween Xl=P-I-I~ .I' d+k1+1I'Th' I! mcUSlveanY1=Pjo--excUSlve. eIllterva.1=Xl•••Yl isagaindividedinto10equalparts,andsoon.Ifweimagine thisprocess continued indefinitely, weobtainaperfectly definitenest(in)allofwhose intervals.lncontainthepointP.NootherpointP'besidesPcanlieinall theintervalsin.For,ifthatwereso,alltheintervals wouldhavetocon­ tainthewholestretchPP',whichisimpossible, asthelengthsotthe intervals(.I"haslength l~n)formanullsequence. Foreveryarbitrarily givenpointPonthenumber-axis (rational or not)therearethusnestsofintervals -obviously, indeed, anynumber ofsuchnests-whichcontainthatpointandnoother. Andinthe presentinstance, -i.e.inthegraphical representation onthenumber­ axis-theconverse appears mostplausible; ifweconsider anynest ofintervals, thereseemstobealwaysonepoint(andbythereasoning above,onlythisone)belonging toallitsintervals, whichisthusdeter­ minedbyit.Webelieve,atanyrate,thatwemayinferthisdirectlyfrom ourconception ofthecontinuity, orgaplessness, ofthestraightline18. Thusinthisgeometrical representation weshouldhavecomplete reciprocity: everypointcanbeenclosed inasuitable nestofintervals andeverysuchnestinvariably encloses oneandonlyonepoint. Thisgivesusahighdegreeofconfidence intheadequacy ofour resolvetoconsider nestsofintervals asnumbers, -whichwcnowfor­ mulatemoreprecisely asfollows: 13. Definition. Wewillsayofeverynestofintervals (in)or(xnIYn), thatitdefines or,forbrevity,itis,adeterminate number. Torepresent 17Insteadof10wemayofcoursetakeanyotherinteger ~2.Forfurthel detail,see§5. 18Theproposition, bywhichthe"continuity ofthestraight line"isexpressly postulated -foraproofcannotbehereexpected, sinceitisessentIally adescription oftheformofourconceptofthestraight linewhichisinvolved -iscalledthe Cantor-Dedekind axiom. §3.Irrational numbers. 27 it,'weusethesymboldenoting thenestofintervals itself,andonlyasanab­ breviation replacethisbyasmallGreekletter,writinginthissense19,e.g. (In)or(Xn/Yn) =a. Now,inspiteofallwehavesaid,thiscannotbutseemaveryarbi­ trarystep,-thequestion hastoberepeated mostinsistently: willit passwithoutfurtherjustification? Thesepurelyidealobjects whichwe havejustdefined-thesenestsofintervals (orelsethatstillextremely questionable 'something' whichsuchanestencloses ordetermines) -can wcspeakofthescasnumbers? Aretheyafterallnumbers inthesame senseastherational numbers, ---moreprecisely, inthesenseinwhich thenumber concept wasdefinedbyourconditions 4? Theanswer canonlyconsistindeciding, whether thetotalityor aggregate ofallconceivable nestsofintervals, or ofthesymbols(jn)or (xnIYn)oraintroduced todenotethem,formsasystemofobjectssatis­ fyingtheseconditions 420;asystem therefore--torecapitulate these c('nditions briefly-whoseelements arederivedfromtherationalnumbers, andI.arecapableofbeingordered; 2.arecapableofbeingcombined bythefourprocesses (rules),obeying atthesametimethefundamental laws1and2,I-IV; :3.contain asub-system similarandisomorphous tothesystemofrational numbers; and1.satisfythePostulate ofEud­ oxus. Ifandonlyifthedecision turnsouttobefavourable, allwillbe well;ournewsymbols willthenhavevindicated theirnumerical char­ acter,andweshallhaveestablished thattheyarenUJnbers, whose totalityweshallthendesignate asthesystemorsetofrealnumbers. Nowthedecision inquestion doesnotpresent theslightest diffi­ culty,andwemayaccordingly bebriefinexpounding thedetails: Nestsofintervals -orournewsymbols (xnIYn)-arecertainly constructed bymeansofrational number-symbols alone;wehavethere­ foreonlytosettlethepoints4,1-4.Forthis,weshallgotoworkin thefollowing way:Certain ofthenestsofintervals definearational number 21,something, therefore, forwhichbothmeaning andmodeof combination havebeenpreviously established. Weconsider twosuch rational-valued nests,say(xnIYn)=-=sand(xn'IY•.')=s'.Withthetwo rational number-symbols sands',wecanimmediately distinguish whether thefirstsis<,=or>theseconds';andwecancombine thetwoby thefourprocesses ofarithmetic. Essentially, whatwehavetodoisto endeavour directly torecognise theformerfact,andtocarryoutthebtt~r processes, onthetwonestsofintervals themselves bywhichsands'were 19aisanabbreviated notation forthenestofintervals (In)or(xnI)'11)' 20Thereadershouldherereadtheseconditions throu!1;h l1Rain. 11Wewilldescribe suchnestsforbrevityasratio1lal-valued. 28 Chapter 1.Principles ofthetheoryofrealnumbers. given,andfinallytoextendtheresulttotheaggregate ofallnestsofintervals. Eachprovable proposition (A)relating torational-valued nestswillac­ cordingly giverisetoacorresponding definition (B).Webeginbysetting downconcisely sidebysidethesepairsofpropositions (A)and definitions (B)22. 14. Equality: A.Theorem. If(xnIYn)=sand(xn'IYn')=s'aretwo rational-valued nestsofintervals, thens=s'holdsif,andonlyif, besides wehave23 foreveryn. Onthistheorem wenowbasethefollowing: B.Definition. Twoarbitrary nestsofintervals a=(xnIYn)and (7'-=(xn'IJ'n')aresaidtobeequalifandon~vIf Xn<Yn', xn'<Yn foreveryn. Remarks andExamples. 1.Thenumbers Xnand'l:n'ontheonchand,YnandYr/ontheother,need ofcour~ehavenothing whatever todo\\ithoneanothcr. ThiSisnomoresur­ PrISIng thanthatrational numbers soentirely different 111appcarance asif,iA. and0375shouldbereferred toas"equal". Eqllalily ismdeedsomething which ••TheImportofproposition anddefinitIOn should Incachcasebeinterpreted inrclationtothenumber-axi~. 'dIntotheverySimpleproofsofthepropositions 14to19wedonotpropose toenter,forthegeneral reasons explained onp.2.TheyWillnotpresent the shghtest difficulty tothereader,oncchehasmastcred thecontents ofChapter II, whereas atthiSstagetheywouldappeartohimstrange; moreover theywillserve asexercises inthatchapter. Merely asaspecimen andexample forthesolutIOn ofthoseproblems, wewtllhereproveThcorem 14: a)Ifs=s',thenwehavebothXn~s;;::;Ynandxn'~s~yn',whence at once, Xn:<;Yn'andxn'~Ynforeveryn. b)Ifconversely XnSYn'foreveryn,thens~s'musthold.ForIfwehad s>s',i.e.s-s'>0,then,smce(Yn-xn)ISanullscquence, wecouldsochoose thcmdexp,that Yp-xp<s-s'orXv-s'>Yv-s. Ashowever siscertainly ;<::Yv'thiswouldimplyXv-s'>O.Wecouldtherefore chooseafurtherindexrforwhich yr'-xr'<Xv-s'. Sincexr'~s',thiswouldimplyyr'<x".Choosing anintegermexceed­ ingbothpandr,wecoulddeduce, inviewoftherespective ascending anddescend­ ingmonotony ofoursequences ofnumbers, thatafortioriYrn'<Xm,-whichcon­ tradictsthehypothesis thatXns::Yn'foreveryn.Thuss;<::s'isensured. Byinterchanging throughout theaboveprooftheaccented andnon-accented letters,wededuce inthesamemanner thatlfxn'<Ynforeveryn,thens';;s -Ifthenwehavebothxn';:::;YnandXn~yn'holdmg foreveryn,thens~s necessarily follows. Q.E.D. §3.Irrational numbers. 29 isnotfixedapriori,butnet'd~tobeestabb,hed by~OIl1eformofdefinition, and It"perfectly compatible '\Ithmarked dl,,"nil.lrity 111,Ipurelye,"t"rnoll a'pcet. 2.Thetwonest~(11~-n1 11I;l~1I)and12,2arcequalIIIaccordance WIth ourpre,ent defimtlOn :1.By14,wemaywntee.g.(s-~Is-/-:,)=s-c~(sIs),thelattersymbol denutJn'~ ane,tallofwhosemt,'rval, ha'ebnththeirleftandtheIrnghtendpolllts -"s.Inp.1rt,cul,lr,(-II-/-I)-~(010)=O. 11 11 4.It,tIllremallls toestabll,h -buttheproofISsosllnplethatwewIllnot gomtoItfurther -~that(cf.Footnote :?:l),IIIconsequence ofourdefinitIOn, we 11<1\'ea)a--a(Footnote 24),b)a~~a'alwaysImplies a',~a,andc)a~a',a'~an mvolve a~an. Inequality: A.Theorem. If(xnI)',,)=sand(x,,'IYr.')~s'are15 tworational-valued nests,thenv.:ehm'es<s',~fandonlyIf :X";,:~Y,.'foreveryn,butnotxn'~Y"furevery11, i.e.y",<x",'for{Itleastunem. B.Definition. Gi'venallYtwonestsofintervals a=(x"Iy,.)alld (1'c-(.v,,'Iy,,'),thenweshallsay(1<-a',if x"~-y,.'for£~'£ry 11,blltnotxn'Sy"furevery11, I.e.furatleastonern,y",'_x",'. Remarks andExamples. 1.Itisclearthatby14and15tbetotahtyofallconceivable nestsisordered. ForIfaanda'areanytwoofthem,eItherthereISequality, a~-a',or,foratleast onep,wchave}'p~_-"p',llnplymg a-<a',orfindlly,foratledstoncr,y(<x" Implymg a'<a.ThelasttwocasescannotoccurSImultaneously, smce,form greaterthanrand fJ,weshouldthenhave,afortwn, """<:'·Tt.',whIch lSImpos'lble. Thusbetween aa,-Ja'oneamionlyoneofthethreerelatIOns a<ai,er~--=a',a'<a alwaysholds,andthetut"ht)'ofthesenewsymbols ISthusorderedby14and15. 2.HereagmnItwouldhavetoheestabhshed IIIalldetailthatthelaw,of order1contmue toholdgoodWiththeadopted defimtlol1s ofequahty andIll­ equ,lhty. T.l1-mg asmodeltheproofIIIthefootnote toTheorem 14,thISpresents sofelVessentml dIfficulties thatweWillnotenterintoItfurther: Thelaw'oforder do,effectually, allrell/amvalid. 3.Inconsequence of14and15wenowhave,therefore, foreveryn WhatdoesthismeaniItmeansthateachoftherational numbersx,.is,mac­ cordance with14and15,notgreaterthanthenesta~(x"IYn).Or:Ifwecon- '4HereItmaybeclearlyrecognised thatthiS"law"isbynomeanstnvial: IthasmL~eedtobeprovedthatwiththegivende}imtiotl ofequality everynestof intervals ISeffectually "equal" toIt~e1f,thatistosaythattheconditions ofthat defimtlOn arefulfilled, whenthesamenestistakenforbothofthenestsofintervals whIchwearecompanng. 30 Chapter I.Principles ofthetheoryofrealnumbers. siut'ranyparticular oneofthenumbers xR••ay.\'p'anduenoteItforbrevitybyx, thenwemay'Wnte(see14,Rem.:1) ('1:,,')x~(x-~Ix+~) or~(xlx) anuourstatement takestheform (.'1:I.'1:);5('1:"!Y,,). Wemayproveitasfollows.Ifitwerenottrue,thenforatleastoner, Yr<x, i.e.Yr<x p, andsoafortiori, Ifmi.greaterthanrandp, Y,n<Xm, whichcertainly cannotbethecase.Inthesamewaywescethata:$Yn'Accord­ ingly,aistoberegarded aslYlfIf?betlceen XnalldY"fureach11,l/lotherword;,'"WII­ tamedwit/llllthelIltervalIn' Thefactthatnoothernumber a',beSIdesa,canpossessthesameproperty isnoweastlyproved.IfInfacttherewereaseconunestofInterval. a'--0(\,,'IYr,') suchthatforeveryuefimte IndexfJweal&ohauxp~a':.::::Y,,,thenthelefth.lnd inequalIty means,moreprecIsely (d:3),that(v'pl'-'p)~-;(v',,'IY,,')andso.hy14 and15,xi>":",yn'foreveryn.SincethISmustholdInpdrtlcular for11fJ,we deduce xi>;'-~Y,,'foreveryp,whIchsIgmfies, by14and15,thata:'::0a'.Inthe samemanner thenghthandinequalIty ISseentoImplythata'::;a.Thusneces­ _anlya~a',whichwaswhatwe&ctouttoprove. 4.By15,aIS>0,i.e."po<lIIVe", Ifandonlyif(x"IY,):>(010),thatis tosay,ifforsomeSUItable indexp,xi>:>O.ButinthiScase,asthex,.'sIncrease withn,wehaveafortIOrix""0foreveryn:>p.Wemaytherefon' say:a,­ (\'nIYn)ISpO'I/we If,andonlyIf,alltheendpoll1ts x".y"arcpositIve fromanu afteradefimte Index.-Theexactanalogue holdsofcoursefora<:o. 5.Ifa>0,and,foreveryn~'p,x";..0,letusformanewnest(xn'Iy,,') =a'byputting Xl'~x'2~••• ~~X'i>_lallequaltoXli'huteveryotheroX,,'and yR'equaltothecorresponding x"andYR'By14,obVIOusly a~a';andwemay say:IfaISpOSitive, thentherearealwaysne&tsofintervals equaltoIt,forwhich alltheendpoints ofintervals arepOSItive. Theexactanalogue holdsfora<O. Sofarthen,inrespectofthepossibility ofordering them,ournests ofintervals maybesaidtovindicate theircharacter asnumbers com­ pletely.Itisnomoredifficulttoestablish asimilarconclusion withregard tothepossibilities ofcombining them. 16. Addition: A.Theorem 25.If(x"IYR)and(x,,'IYn')areanytwonests ofintervals, then(xn+xn',Yn+YR')isalsoune,andIftheformerareboth rational-valued andrespectively =sand=s'.thenthelatterisalsorational­ valued,anddetermines thenumbers+s'. B.Definition. If(xnIYn)=aand(xn'IYn')=a'areanytwonests ofintervals anda"denotesthenest(xn+xn',Yn+Yn')deducedfromthem, thenwewrite a"=a-r-a' anda"iscalledthesumofaanda'. 16Withregardtotheproof,cf.footnote 23. §3.Irrational numbers. 31 Subtraction: A.Theorem. If(xnIYn) isanestofintervals, thenso17. is(-YnI-xn);andiftheformerisrational-valued =s,thenthelatter isalsorational-valued, anddetermines thenumber-s. B.Definition. Ifa=(xnIYn)isanynestofintervals anda'de­ notethenestofintervals(-YnI-xn),wewrite,a=-a andsaya'istheopposite ofa.-Bythedifference oftwonestsofinter­ ~'alswethenmeanthesumofthefirstandoftheoppositeofthesecond. Multiplication: A.Theorem. If(xnIYn)and(xn'IYn')areanytwoIS. positive nestsofintervals, -replaced, ifnecessary, (inaccordance with 15,5)bytwonestsofintervals equaltothem,forwhichalltheendpoints ofintervals arepositive(oratleastnon-negative), -then(xnx,,'IYnYn') isalsoanestofintervals.. andiftheformerarerational-valued andrespec­ tively-=sand=s',thenthelatterisalsorational-valued, anddetermines the numberss'. B.Definition. If(x"IYn)~aand(x,.'IYn')=a'areanytwo positivenestsofintervals forwhichalltheendpoints ofintervals arepositive -whichisnorestriction, by15,5--anda"denotethenest(xnxn'IYnYn') derivedfromthem,thenwewrite a"==a·a' andcalla"theproduct ofaanda'. Theslightmodifications whichhavetobemadeinthisdefinition if oncorbothofaanda'arenegative orzero,weleavetothereader,and henceforth consider theproductofanytwonestsofintervals asdefined. Division: A.Theorem. If(xnIY,,)isanypositivenestofintervals19. for7vhichallendpoints ofintervals arepositive,(cf.15,5)thensois(~I~); YnXn andiftheformerisrational-valued, and=s,thelatterisalsorational- valued,anddetermines thenumber!.s B.Definition. If(xnIYn)=aisanypositivenestofintervals for whichallendpoints arepositivf', anda'denotethenest(1/~),thenweYnXn write , 1a=­a andsaya'isthereciprocal ofa.-Bythequotient ofafirstbyasecond positive nestofintervals wethenmeantheproductofthefirstbythereciprocal ofthesecond. Theslightmodifications necessary inthisdefinition, ifa(intheone case)orthesecondofthetwonestsofintervals (intheother)isnegative, 32 ChapterI.Principles ofthetheoryofrealnumbers. wemayagainleavetothereader,andhenceforth consider thequotient ofanytwonestsofintervals ofwhichthesecond isdifferent from0,as defined. -If(XnIYn)=a=0,thentheabovemethod failstoproduce a"reciprocal" nest:di'1.'i"ionby°isherealsoimpossible. Theresultoftheprecedlllg considerations isthusasfollows: By definitions 14to19,thesystemofallnestsofintervals isordered inthe senseof4,1,andadmitsofhavingitselements combined bythefour processes inthesenseof4,2.Inconsequence ofthetheorems 14to19, asstatedineachcase,thissystem possesses further, intheaggregate of allrational-valued nests,asub-system, similar andisomorphous tothe systemofrational numbers, inthesenseof4,3.Itremains toshowthat thesystemalsofulfilsthePostulate ofEudoxus. Butif(xnIy,,)=aand (x,,'IYn')~a'areanytwopositive nestsforwhichallenclpoints ofin­ tervalsarepositive(cf.15,f»,letXmandYm'beadefinite pairofthese endpoints;thetheorem ofEudoxus ensures theexistence ofaninteger p,forwhichPXm>-Ym',andthenestpa,or(px"IpYn),inaccordance with15,istheneffectually>a'. Thenextstepshouldhetoestablish inalldetail(cf.14,4-and15, 2)thatthefourprocesses defined in16to19fornestsofintervals obey thefundamental laws2.Thisagainoffersnottheslightest difficulty and wewillaccordingly spareourselves thetroubleofsettingitforth 26.The Fundamental LawsofArithmetic, andtherebytheentirebodyofrulesvalid incalculations 'l1)ithrational numbers, effectually retaintheirvalzdzty inthe newsystem. Bythis,ournestsofintervals havefinallyproved themselves in everyrespect tobenumbers inthesenseof4:Thesystem ofall nestsofintervals isanumber-system, theneststhemselves arenumbers 27. 28Asregardsaddition, forinstance, itshouldbeshownthat: a)Addition canalwaysbecarriedout.(Thisfollowsatoncefromthedefini­ tion.) b)Theresultisunique; i.e.a~a',T=T'(inthesenseof14)Imply a·1-T=a'1-T',-ifthesumsareformedinaccordance with16andthetest forequalIty earnedoutInaccordance "11th14.Inthecorresponding sense,Itshould beshownfurtherthat c)a+-T=T+-aalways. d)(e+-a)+-T=e+-(a+-T)always. e)a<a'implies a+-T<a'+-Talways.- Andsimilarly fortheotherthreeprocesses ofcombination. 27Whether, asabove,weregardnestsofintervals asthemselves numbers, orimagine somehypothetical entityIntroduced, whichbelongstoalltheintervals In(cf.15,3)andthusappearstobeinaspecialsensethenumber enclosed by thenestofintervals and,consequently, thecommonelement inallequalnests­ thisatbottomisapurematteroftasteandmakesnoessential difference. -The equahty a-.(xnIYn)wemay,atanyrate,fromnowon,(cf.13,footnote 19)read mdlffercntly eitherasHaisanabbreViated notation forthenestofintervals (xnIyn)", orasHaisthenumberdefinedbythenestofIntervals (xnIyn)". §4.Completeness anduniqueness ofthesystemofrealnumbers. 33 Thissystemweshallhenceforth designate asthesystemofrealnumbers. Itisanexte1lSion ofthesystemofrational numbers, -inthesensein whichtheexpression wasusedonp.11,-sincetherearcnotonlyrational­ valuednestsbutalsoothersbesides. Thissystemofrealnumbers isinone-one correspondence with thewholeaggregate ofpointsofthenumber-axis. For,onthestrength oftheconsiderations setforthonpp.24,25,wecanimmediately assert thattoeverynestofintervals acorresponds oneandonlyonepoint, namelythatcommon toalltheintervals In>whichonaccountoftheCantor­ Dedekind axiomisconsidered incachcaseasexisting. Alsotwonestsof intervals aanda'have,corresponding tothem,oneandthesamepoint, ifandonlyiftheyareequal,inthesenseof14.Toeachnumbera(that istosay,toallnestsofintervals equaltoeachother)corresponds exactly onepoint,andtoeachpointexactlyonenumber. Thepointcorresponding inthismanner toaparticular number iscalleditsimage(orrepresentative) point,andwemaynowassertthatthesystemofrealnumberscanbeuniquely andreversibly represented bythepointsofastraightline. §4.Completeness anduniqueness ofthesystemofreal numbers. Twolastdoubtsremaintobedispelled 28:Ourstarting pointin §3wasthefactthatthesystemofrational numbers, byreasonofits "gaps", couldnotsatisfyalldemands whichwouldappearinthecourse oftheelementary processes ofcalculation. Ournev,lycreatednumber­ system-thesystemZaswewillcallitforbrevity-isinthisrespect certainly moreefficient. E.g.itcontains 29anumber aforwhicha2=2. Yetthepossibility isnotexcluded thatthenewsystemmaystillshow gapsliketheold,orthatinsomeotherwayitmaybesusceptible ofstill furtherextension. Accordingly, weraisethefollowing question: Isitconceivable that asystemZ,recognizable asanumber-system inthesenseof4,andcon­ taining alltheelements ofthesystemZ,shouldalsocontainadditional elements distinctfromthese? 30 'sCf.thec10smgwordsoftheIntroduction (p.2). '9ForIfa=(xnIYn)denotethenestofintervals constructed onp.20 inconnection withtheequation x"=2,thenby18wehavea2=(xn''Yn")'Since, however, xn'-..::2andYn'>2,itfollowsthat(J'=2.Q.E.D. 301.e.Zwouldhavetorepresent anextension ofZinthesamesenseasZ Itselfrepresents anextension ofthesystemofratIOnal numbers. 34 ChapterI.Principles ofthetheoryofrealnumbers. Itisnotdifficult toscethatthiscannotbeso,sothatwchavein factthefollowing theorem: 20. Theorem ofcompleteness. Thesystem/.ofallrealnumbers isin- capableoffurtherextension compatible withtheconditions 4. Proof: Let"2beasystemwhichsatisfies theconditions 4and contains alltheelements of/..IfIXdenoteanarbitrary clement ofZ, then4,4 -inwhichwechooseforf3thenumber 1,contained inZ, andalso,therefore, inZ-showsthatthereexistsanintcgap>IX, andsimilarly another p'>-IX.Forthese 31wchave-p'<IX<p. Considering successively the(finitenumber of)integer., between 0-p' andp,starting-with ~p',wcknowthatwemustcometoal.13toncwhich isstill<IX.IftillSbecalledg,then g~IX<g+l. Byapplying tothisintervalg ...g+1themethod, already re­ peatedly used,ofsubdivision intotenparts,aperfectly definite nestof intervals (xnIYn)isobtained. AndarepetitIOn wordforwordofthe proofin15,3showsthatthenumber thusdefincdcanneitherbc>nor <IX.Everyelement of/.istherefore equaltoarealnumber, sothat2 cancontainnoelements otherthanrealnumbers. Afinalobjection mightbethis:Wehavesucceeded informing the systemZinacomparatively natural, butafterallanarbitrary, manner. Othermeasures, obviously, mightbeadopted forfillingupthegdpSin thesystemofrational numbers. (Intheverynextsectionweshallcome acrossother,equallyreadymeanstothisend.)Itisconceivable that adifferent methodwouldleadtoothernumbers, i.e.tonumber-systems differing, inmoreorlessessential particulars, fromtheoneconstructed byus.--Thequestion thusindicated maybegivenapreciseformulation asfollows: Letussuppose thatwehavesomehow, starting withthesystem ofrational numbers, succeeded inconstructing asystem3)ofelements which,besidesstillsatisfying theconditions 4,-asisthecasewithour systemZ,--andtherefore deserving thenameofanumber-system, also fulfilsafurtherreqUIrement, usuallyreferred toasthePostulate of completeness, onaccount ofthetheorem provedabove.-Onthe strength of4,3,g)contains elements, corresponding totherationalnumbers. Let(x"IYn)beanynestandletInandI)nbetheelements of3)associated withx'"Yninaccordance with4,3;thestipulation thenrunsthus:3) shallalwayscontainatleastoneelement"satisfying, foreveryn,thecon­ ditions 1'"<~:s;:\)... Inexactform,ourproblem isnow:Cansuchasystem ~differin ..AtthiSpomt,thePostulate ofEudoxus gainsItsaXIOmatic significance. §4.Completeness anduniqueness ofthesystemofrealnumbers. 35 anyessential particulars fromthesystemZofrealnumbers, ormustthe twosystems beregarded assubstantially identical, intheperfectly definite sensethattheycanbebrought intorelation assimilarandisomorphous tooneanother? Thetheorem statedbelow,bysolving thisproblem inthesense whichweshouldanticipate, closestheconstruction ofthesystemofreal numbers. Theorem ofUniqueness. Everysuchsystem ~isnecessarily similar21. andisomorphous tothesystemZofrealnumbers asconstructed byus.Essen­ tially,onlyonesuchsystemtherefore exists. Proof. By4,3,35contains asuh-system 35',whichissimilarand isomorphous tothesystemofrationalnumbers contained inZ,andwhose elements maytherefore becalled,forshort,therational elements of35. Ifa=(xnIYn)isanyrealnumber, ~must,according toournewstipula­ tion,contain anclement ",whichforeverynsatisfies theconditions In:<;":<;Iln>ifI"and\)"aretheelements of~corresponding tothe rational numbers XnandY1O' Also,theseconditions define ,\uniquely. Forifasecondelement ~',simultaneously with",satIsfied thecondItions In<":s;;\1nforevery 11,thenitwouldfollow,wordforwordasintheproofof12,thatfor everyn I)n-In:--::.:I~-~'I, i.e.?:thenon-negative oneofthetwoelements ,,-~'andl-~. Letrstandforanarbitrary positive ratIOnal number, andIforthecor­ responding clement in65(therefore in65');then,onaccountofthesimilarity andisomorphism of65'withthesystemofrational numbers, \Vemust have,simultaneously withY11-Xp<r,therelation \)1>-'SI<rholding forasuitable indexp.Foreverysuchrtherefore le-0'I<1". Iftherefore 1"1denotes oneparticular such IandifI'mn=1,2,..•, denotestheclement (certainly presentin65',by4,2)which,\vhenrepeated 11times,yieldsthesum1'1'wesce,afterwritingdowntheaboveineyuality for1"=1"nandaddingittoitselfntimes,thatforevery 11=1,2,..•I n.I(j-e'I~1"1 mustalsohold.Since,however, 35satisfies thepostulate 4,J,itfollows that"=~'. Ifweproceed toassociate thisuniquely defined clement "and therealnumber a,itbecomes clear that65contains asub-system ~., similarandisomorphous tothesystem Xofallrealnumbers. That suchasystem (,5*isnotsusceptible offurther extension compatible 36 ChapterI.Principles ofthetheoryofrealnumbers. withtheconditions 4,butmustbeidentical with(,),wastheimpon ofthepreviously established theorem ofcompleteness. Th~reby, itis provedthat(,)andXarcsimilar andisomorphous tooneanother, andtherefore mayberegarded, inallessentials, asidentical: Oursystem Zofallrealnumbers isinallessentials theonlyonepossible satl4ying both theconditio1/S 4andthepostulate ufcompleteness. Afterthesesomewhat abstract considerations, themainresultofour wholeinvestigatIOn maybesummarised asfollows: Besidestherational numbers withwhichwearefamiliar, thereexist others,theso-called irrational numbers. Eachofthemmaybeenclosed (determined, given,...)byasuitable nestofintervals andthisindeed inmanyways. Theseirrational numbers fitinconsistently withthe rational numbers, insuchamanner thattheconditions statedin4are fulfilled bythejointsystemofallrational andirrational numbers, with which,tobebrief,allcalculations maybeeffected, formally, exactlym withtherational numbers alone,hutwithgreatersuccess. Thiswidersystem ismoreover incapable ofanyfurther extension compatible withconditions 4,andisinallessentIals theonlysystemof symbols whichsatisfies theseconditions 4andalsothepostulate ofcom­ pleteness. Wecallitthesystemofrealnumbers. Itiswiththeelements ofthissystem, withtherealnumbers, that wework(atfirstexclusively) inthesequel. Weconsider aparticular realnumber asgiven(known, determined, defined, calculable,...)if eitheritisarational number andsocanbeliterally written downwith thehelpofintegers -inserting ifneedbeafractional baroraminus sign-or(andthisholdsinanycase)wearegiven 32anestofintervals defining thenumber. Weshallverysoonsee,however, thatmanyotherwaysandmeans, besidesthenestsofintervals, exist,fordefining arealnumber. Inpro­ portion assuchwaysbecome knowntous,weshallwidentheabove­ mentioned conditions, underwhichweconsider anumber asgiven. • 2I.e.bythecomplete explICIt specification ofthe(ratIOnal) endpOIntsIn themannerjustdescnbed. §5.Radixfractions andtheDedekind section. §5.Radixfractions andtheDedekind section.37 Afewofthemethods fordefining realnumbers maybementioned atonce,asparticularly important fromthepointsofviewofboththeory andpractice. Inthefirstplace,anestofintervals neednotalwayshegivenin theform(xnIYn)considered byus;itmayoftenhewritten inamore convenient form.Thus,aswehavealready seen,adecimal fraction, e.g.1·4,1421...,maybeimmediately interpreted asanestofmtervals, withtheassumptions ••• ! and,generally, 'Xnequaltothedecimal fractiOn broken offafter th~ n'"digit;Ynbeingderived from XnbyraisingthelastdIgitbyone, i.e.Yn-~Xn+H~n'Practically, wemaythussaythatdecimal fractions represent apeculiarly clearand.convenil.'nt specification ofnestsof intervals :13. Itisobviously quiteanunessential partthatthebaseorradix10 oftheordinary scaleofnotation playsinthisconnection. IfgISany integer :::c:2,wehavetheexactanalogue forfractions inascaleof radiXgorradiXfractions withbaseg.Tobeginwith,givenareal number a,anintegerp(>,=,or--.:::U)isuniquely defined bythe condition p:'Sa<P1-1. Theinterval.10hetweenpandp+1isnextdivided intogequal parts,andeachofthesepartsconsidered -hothhereandsimll.1rly inthefollowing steps-asincludmg itsleftendpoint, butnotits rightone.Then abelongs toonc,andtoonconly,oftheseparts, i.e.among thenumbers 0,1,2,...,g-Ithereisoncand onlyonc-whichweshallcallforbrevity a"digit" anddenoteby Zl-forwhich 33Thedra\\back toitisthatwecanseldom percl'ive t111.'lawof~uccl'~~ion oftilt'digits, i.ethelmuofforll/a/101l ofthe.\·,,'sand)',:s. 38 ChapterI.Principles ofthetheoryofrealnumbers. Theinterval.!l thusdefinedweproceed todivideagainintogequalparts, andeTwill,asbefore,belongtoone,andtooneonly,ofthe~eparts,i.e. adefinite"digit" Z'2willbefoundforwhich P+~1+.::'~-<:;eT<P+-~1+~2t_-.!.gg"-- g 1:" Theinterval./2 thusdefinedweproceed todivideagainintogequalparts, andsoon.Thenestofintervals (In)=(XnIYn)determined bythispro­ cess,forwhich 1 _.+-.::'1Z2 2'n-l Zn+1JYn-Pg+- 1:2+-...+gn-l+---g"(n=1,2,3,...) clearlydefinesthenumber eT,sothat34a=(XllIY,').Butontheanalogy ofdeeim,i1 fractIOns wemaynowwrite -whereofcoursethebasegoftheradixfraction mustbeknownfrom thecontext. Wehavetherefore the 22. Theorem 1.Everyrealnumbercanherepresented inoneandessen- tiallyonlyolle :J~waybyaradiXfractioninthescaleofbaseg. Wemention thefollowing theorem relating furthertothisrepresen­ tatIOn,butshallmakenouseofitinthesequel: Theorem 2.Theradixfractionfurarealnumber CT-whate~)er he 3'Thatwehaveanestofmtervals isimmediately obvious, sincexn_1:< xn<'Yn::::)'n-lthroughout, andYn-xn~~~formsanullsequence, by10,7. .r,ThesltghtalteratiOn inourmethod, reqUIred Ifalltheintervals arecon­ sideredasincludmg theirrightand1/ottheIrleftendpomts, thereaderWilldoubtless beabletoc.uryoutforhimself. Thetwore,ultsdlfTerIf,andonlyIf,thegiven number aISrational, andcanbewritten asafraction having, asdenommator, a powerofg,sothatthepomtaISanendpoint ofoneofourmtervals. -Actually thet\\Onest,ofmterv"i, p+0'.::'1Z••••Z,_1(.::'T-1)(g-1)(g-1)...andp-I-0'.::'1Z2•••Z,_1Zr00..., wherethedigitz,issupposed ~1,areequalby14.Ineveryothercase,tworadix fracUons \\hicharenotIdentical areunequal, by14.-ThereaderWilleaSilyprove forhlm,elf that,exceptmthiscase,therepre,entatlOn ofanyrealnumber aas 11radixfraction withbasegisabsolutely umque. 39 §5.Radixfractions andtheDedekind section. thechosenradixg>2-willproveperiodic (orrecurring)Ifandonlyif aisrational 36. Aparticularly advantageous choicetomakeisofteng=2;thepro­ cessforexpressing thenumber aisthencaller!brieflythemethod of bisection andtheresulting radixfraction, whosedigitscaninthatcase onlybe0or1,iscalledabmaryfraction. Themethod, inasomewhat moregeneral light,isthis:westartfromadefinite interval 10and,in accordance withsomeparticular ruleorpointofview,definitely select oncofitstwohalves,callingit./1;wethenagainmakeadefinite choice ofoncofthetwohalvesof.11'callingit.12;andsoon.Bysodoing,we specify, ineverycase,awell-defined realnumber, determined withab­ soluteuniqueness bythemethod whichregulates ateachstagethechoice between thetwohalf-intervals 37. Inradixfractions, justasindecimal fractions, weaccordingly sec apeculiarly clearandconvenient modeofspecifying nestsofIlltervals. Theyshallaccordingly infutureheadmitted forthedefinitIOn ofreal numbers onthesamefootingasdecimal fractions. Thedistinction liessomewhat deeperbetween nestsofintervals and thefollowing method ofdefinition ofrealnumbers. Wesuppose given,inanyparticular way:l8,twoclassesofnumbers AandB,subjecttothefollowing threeconditions: 1)Eachofthetwoclassescontains atleastonenumber. 2)Everynumber oftheclassAis;::::nlerynumber oftheclassB. 3)Ifanarbitrary positive (small)number EOISprescribed, thentwo numbers canbesochosenfromthetwoclasses,-a',say,fromAand b',say,fromB,-that:l9 b'--a'<€. -Thenthefollowing theorem. holds: S6Hereforsimplicity wcregardtermillat;nl! radixfractions asperiodic with period O.-Thateveryr.ltlOnal number canbl;repre~ented byarecurring deCimal fraction wa~provcdbyy.Wlllhs,DeAlgebra tractatus, p.:~(j+,It!U;l.Thatconversely everyIrratIOnal number canalways, andInoncway onl~',berepresented a~anon­ recurring decimal fraction wasfirstprO\edgenerally byO.Stol::(AlIgcmell1e Anth­ metIkT,p.11!),IHHii). 37Anexample wasgivenin12,2. 38E.g.Acontams allratIonal numbers whosecubeis<5,Ballrational numbers who~ecubeIS>5. 39Wcsayforshort:thenumbers ofthetwoclassestlpproach arbltranly neartooneanother. Intheexample oftheprecedmg footnote, wcseeatoncethat condItIOns 1)and2)aresatisfied; that3)isalso satl~fied \\'erecognise fromthe possibility ofcalculatmg (bythemethod ofpartition mtotenthparts,forImtance) twodeCImal fractIOns ,'l:nandYnWithnplacesofdecimals, differmg onlybyaumt inthelastplace,andsuchthatxn"<5,y,,">0;nbeingsochosenthatl~n<cr. 40 ChapterI.Principles ofthetheoryofrealnumbers. Theorem 3.Thereexistsoneandonlyonerralnumberasuchthat foreverynumberaill.11andeverynumberbillBtherelation a<a~b isalwaystrue. Proof. Itisagainobvious thatnotwodifferent numbers a,a' withthisproperty canexist.ForputtingIa-a'I=c-oE,weshouldhave €>0,yetb-a>Eforeverypairofdements aandbfromAandB respectively, contrary tocondition 3. Thereexiststhenatmostonesuchnumber a.'Vefinditinthe following way:Byhypothesis, thereisatleastonenumber atinAand onenumber b1inB.Ifa1=b1,thenthecommon valueismanifestly thenumber awhichweareinsearchof.Ifa1=f=b1,andtherefore by 2),a1<b1,thenwechoosetworational numbers XlSaI'andY1:::;;b1 andapplythemethod ofbisection totheinterval.l1 whichtheydeter­ mine;wedenotetheleftorrighthalfbY.l2'according asthelefthalf (endpoints included) doesordoesnotstillcontainapointoftheclassB.By thesamerulewenextselectoneofthehalvesof12,callingit.13'and soon. Theintervals .11'./2'...,.In>••.,beingobtained bythemethod of bisection, necessarily formanest (In)=(x"Iy,,)=a. Fromtheirmodeofformation, theypossessmoreover thcproperty that nonumber ofBcanlietotheleftofanyoftheirleftendpoints, andno number ofAtotherightoftheirrightendpoints. Butfromthisitfollowsatoncethatthenumber aenclosed bythcm isthenumber required bytheorem 3.Infact,if,contrary totheassertion inthattheorem, aparticular number iiofAwere> a,sothata-a>O. thenwecouldchoosefromthesucce£~ion ofintervalsInaparticular one, say.1v~-=Xv...Yv'withlength<a-a. Sincexp~a~Yv'thiswould imply Yv-a:'?YP-.'r:']l<_a-a,i.e.Yv<ii, whereas, actually, nopointofAliestotherightoftherightendpoint Yvoflv'Ifontheotherhand,inanyinstance, b<a,itwouldsimilarly followthatforasuitable indexg,b-<Xq,whereas actually nopointof Bliestotheleftoftheleftendpointofaninterval.Iq.Hencewemustin­ variably havea::sa<b.Q.E.D. Asaspecialcorollary, wehavethefollowing theorem, whichsup­ plements Theorem 12,forming anextension ofittothecasewhenthe numbers thereoccurring arearbitrary realnumbers. Intheformulation, weanticipate theobvious definitions 23-25ofnextparagraph. §5.Radixfractions andtheDedekind section. 41 Theorem 4.If(xn)isamonotone ascending, and(Yn)amonotone des­ cending, sequence of(any)realnumbers; If,further, Xnc::::Ynforeveryn, andthedifferences Y"-Xn=dnformanullsequence; thenthereisinvariably oneandonlyonerealnumbera,suchthatforeveryn Xn;::;a-s:;Yn' Wethensay,asbefore(cf.J)(jll1lfioll 11),thatthetwogivnlsequences defhle anestofintervals (x"IYn)andthataisthenumberwhichit(uniquely) deter­ mmes. Proof.Ifwithalltheleftendpoints Xnweconstitute aclassA, andwithalltherightendpoints y"aclassIJ,ofrealnumhers, theseclearly satisfyconditions I)toa)ofTheorem 3,fromwhichthecorrectness of theabovestatement atoncefollows. Remarks andExamples. I,Instead of:l),ItISoftenmoreCOnH'nlent tostipulate thatc,g.e'l:ery ratIOnal number ,hould belong eithertoAortoB(.IS\\.IStheC.lseInthe cxalllple oflastfootnote), Infact,IIIthatca,e,smceratIOnal numbers are dellseonthenumher aXIS,thereqUirement :1)I'fulfilled ofasclf.ToseethiS, wehaveonlytom1.lgme the\\holenU,llher-,lXIS ,ubdl\IlkdIIltoequalportIOns of length<£/2.NowconSIder anyone oftheportIOns contammg anelement from A,and,totherightofIt,t.lke.lnothcr portIOncontdming.ln e1e,n"at frOlllB,together WiththesetwoportIOns, t.lkethefillltenumber ofportions, Ifany,between them. Oneoftheseconsidered portion, must1)<'thefirstofthen}tocontam anclement bfromB.EItherthiSp,lrtlculolr portIOn, ortheprecedmu; one,WIllcontam anelement afromA,andwehaveb-fl--:::Eo 2.ItISoftenstiliInoreconvclllent todiVIdeallrealnumbers mtot\\Oclasses AandB.Inthatcaseofcour,e3)IS,IIforllOlI, alsosatisfied ofItself. 3.IfthetwocI.lssesAandBaregl\'enmoneofthel.lst-mentlOned ways, thenwesayth.ltaDedekind section ISm.lde 1Ilthedom.un ofeitherrational or realnumbers, asthecasemaybe111.Thesome\\hatmoregeneral speCification of twoclasses 11involved inourtheorem :l\\IIIalsoforhre~lty betermed asectian anddenoted by(AIB),Ourtheorem :1e,mthl'nbestatedbrieflyintheform: Asntirm(AIB)",variably defillesadeterl/llllate TealIlIl11lber. AndItsproofeon'lsts simply mpointing outthatthespeCIfication ofasectIOn carriesWithItthespecI­ ficationofanestofintervals, whichfurn"hes anumber a\\Iththeproperties reqUired. 4.Seemgthenthateverysectron immedmtely prOVides adefimte nestof mtcrvals, weshallhenceforth regardsections asperml~slble meansnfdefinmg (determining, speCifying,...)realnumbers; also,wenowwrite,IftheseetW::l (AIB)definesthenumber a, (AIB) a. 00Cf.p.I,footnoten. &1ThiswasgivenIIItheaboveformbyA.CllPelli. Giornale d.MatematiC'l, Vol.35,p.20!l,1807. 42 Chapter I.Principles ofthetheoryofrealnumbers. 5.Theconverse isofcourseequallytrueandevenmoreeasilyproved. Given ane~t(.\'nIYn)=a,wecanconsider allleftendpoints xnasformmg aclassA, andnghtendpoints aclassB,andtheset"oclassesevidently furmsh asection, which definesthl'~amenumber aa~thene,tItself.- Anestcanaccordmgly beregarded asapartIcular kmdofsectIon. H.Byourlastremark, themethod ofsectIOns (forthedefinition ofreal numbers) ISsupenor mgencr,lhty tothatofnests.ItISalsoqUIteasconvenient fromtheintUItIOnal pomtof\lew.ForIfwctake,say,thesectIOn(AIB)inthe somewhat morespecialform,menttoned in2,ofasectton mthedomam ofreal numbers, thenwholtourtheorem nnphes I"thiS.Ifweimagme allpOintsofthe number-axIs "eparated mtotwoclassesAandH,thmking e.g.ofpomt~ofthe onc cla~sasmarked blackandthoseoftheotheraswhite; andIf,whenthiSis done,(I)thereISatlea~toncpomtofeachkmd,(2)everyblackpomtliestothe leftofeverywhitepomt,and(:l)e7'erypomtonthenumber-axIs iseffectually coloured eitherblackor\\l11te,thenthetw0cla~sesmustcomeIntocontact ata perfectly defimte place,andtotheleftofthiSplaceallisblack,totherightofItall ISwhite. 7.Wemusttakecare,however, nottoaccepttheIllustration justgivenas aproof.Hadwenotalready Withthehelpofnestsofmterval~ invented theclass ofrealnumber" ourtheorem couldnotbeprovedatall-anymorethanItcould beprm'edthate\('ryne,t dcfine~ anumber. \VeSImplyagreed-andwereamply Ju~ttfied bytheresult-toregarde\eryne"tasanumber. Inexactly thesame wayweColnagree-andthl'ISactually thecoursefollowed byR.Dedelllnd 42 inh,scon~tructJon ofthesy"temofrealnumbers _.toregardeverysectIOn mthe domam ofratulf/al numbers a,a"realmll1lher", andweshouldthen,exactly as inourmve~tlg,ltlOns In§:l,onlyhavetoeX,nnIne whether thl"ISpernllsslble; Le, weshouldhavetomakesurewhether tbetotalItyofallsuchsectIOns(AIH)forms anumber s}steminthesen,cofconclItlOns 4 -whIch ISnotmoredifficult than theanalogous investigatIOns carnet!outIn§3. Henceforward -andforthepresent exclusively -realnumbers formourworking material. Wemayeven,ifweplease,droptheword "real": Forthepresent, "number" shallinvariably meanarealnumber. Exercises onChapter I. 1.Fromthefundamental laws1and2deduce themostimportant ofthe furtheranthmetlcal rules,e.g.(a)theproduct oftwonegative numbers ISpositive; (b)Cl,+c<b+cinvariably imphes Cl<b;(c)forevery Clwehavea'()~(); etc. 2.WhenIn3,II,4arethesignsofequality correct? 3.Express thefollOWing numbers asbmaryandasternary fractions (I.e. inscalesofnotation ofwhichthebasesarerespectively 2and3): 1 3 1 10 2'8'3'7'17' findthefirstfewfiguresofthebinaryandternary fractions forv'2,v'3, 1Tande. ••Stetigkelt undIrrationale Zahlen, Bru1"\swick 1872, §6.Arbitrary sequences andarbitrary nullsequences. 43 exn,--fJn4.Inthesequence 6,7provexn--",,__fJ'where exandfJ<iretheroots ofthequadratic equatIOn x·--~x+1.(Hint: the~equence~ (exn)and(fJn)have thesamelawofformatIOn asthe~equence 6,7.) 5.Formthesequence (\"n)ofnumbers given,forn:_:I,bytheformula whereaandharegiven PO~ltlve nun~bers andtheInttIaIterm,x",Xl0,I;I,0; -~I,ex;I,fJ;orarcarhltrary. (Here exandfJdenote re~pectlvely thePc)~ltlve andthenegative rootoftheequatIOn x·-ax+b)Ineachofthefourcases J!lveanexpliCit formula forxn. 6.If./0'./1'I.,...isasequence ofnested intervals (i.e.eachcontamed Intheprecedmg) aboutwhoselengths nothing further isknown, thenthere ISat leastonepomtwhichbelong, toalltheIn's. 7.Arealnumber aI~IrratIOnal, ifwecanfindanascendmg ,equence of Integers ('In)'suchth.ltqn'JI'not.InInteger forany".butIf,"henPn't<indsfor theinteger nearesttoqna.«(/na-Pn)ISanullsequence. 8.Provethat(\"nIy,,)ISane,tmeachofthefollo\\mgexample,: j(X'n1-.1',),\:'nl-t_~\'n•J'n )'11+13(\'n-+-.J.',),)'n+la)I'I2"[-I(11-1)'1"+2'+I11'x7I-- J}'n--- 11311' b)0·-XI<J'I.1Ildforevery 11~J,'"nf-l--,IYnJ'n-'-- l'IlH c)0·-"--.1'1 ,"'ftll - d)()·-XI<:YI ,Ynt-l e)0..::::\'1<YI"J'n+l f)()---::XI<:YI",)'n+l g)()..---::\"1<:")'1.. ,J'n+-t(n=1.2•...); 1(\",d-I+.:I'n); 1(vnI)'nll); Evaluate thenumbers dcfllled 111e".unples (.1)<ind(g).(CLproblems 91 <i11l1\)2.) Chaptern. Sequences ofrealnumbers. §6.Arbitrary sequences andarbitrary nullsequences. Wenowresumeourconsiderations of§2,-andgeneralise them byallowing allthenumbers whichthereoccurtobearbitrary realnumhers. Since,withthese,wcmayoperate precisely aswithrational numhers, boththedefinitions andthetheorems of§2will,inallessentials, remain unchanged. Wemayaccordingly bebrief. 44 Chapter 11.Sequences ofrealnumbers. 23. CDefinition 1.Iftoeachposith,e integerI,2,3,•••,corresponds adl'jiml('realmmlber .\:,!!thenthenumbers aresaidtoformasequence. Examples 6,1-12,may,ofcourse, alsoservehere.Similarly, theRemarks 7,I~liretamfullvahdlty. \Vegiveafewmore<examples, mwlllchItISnotun­ meJllIte1y apparent \\hether thenumbers InquestIOn arcratIOnal ornut. Examples. 1.Leta---O.:lOI0•••,i.e.etlu,,1tothedecimal fr.lctian who_efirstfew digitswereobta1l1eJ 111afuotnote (p.24)fromtheequation lOt.-2;andput xnO~anfornc_-I,2,:1,..• 12.\Vlththesamemeanmg fora,letXn= .a+n 3.Applythemethod of'UCCC"""C bisection totheinterv.!! 10~0..•1, takingfi"tthelefth.llf,thentWIcerunnmg thenghthalf,tl1l'nforthenextthree stepsagmnthelefthalf,thenfourtimesrunnmg thenghth.llf,,mdsoon.])enote tht'number' sodefined byb(\\hltI'It,v.llue,approxlmlte1y'), amiputfOlxn' &ucce,"1 vely, -I-b,--b,+~,-~,+b2 ,-b2 ,-1-:2'b"-Ib3 ,••• 4.WIththes.lmemeanmg forb,putforXII'successIvely, 1 -b,1+b,I--b',llb', 1--b3,1I-b3,••• 5.WIththesamemeanmg foraandb,let\:,b~themlddl~pomtofthestretch between them, I.e.XI--~(a1-b);X,themIddle POllltbetween X,andb,X3, thatbetween x,anda,x"thatbetween x,andb;--I.e.generally, x1l+1,themiddle pomtbetween x"andeIthera,orb,accordmg asnISevenorodd. 24. Definitions: cl.Asequence (x,.)issaidtobebounded Ifaconstant Kexists,suchthattheinequality issatisfiedforeveryn. 2.Asequence (xn)issaidtobemonotone increasing IfXn<X1l+lfor everyn;monotone decreasing,ifXn?:Xn+1foreveryn. Allremarks madeIn8and9retamtheirfullvahdity. 1Forthemeaning ofthemark 0cf.thepreface, asalsolaterthebeginning of§52. ,Wrttten asabWlIryfractIOn,b--~001100011110 ••• §6.Arbitrary sequences andarbl7rary nusequences. Examples.45 2. 1 '1>I.Thesequences 23,1,2,4and5areevidently bounded. Sequence 3is notbounded, andmfactneitherontheleftnoronthenght;forwecertamly have 1 1 1()<b<.2andtherefore bin> 2m>m.andaccordmgly -b'"< -m.Terms ofthesequence maytherefore alwaysbefound,whichare>Kor< -K,how­ everlargetheconstant Kischosen.-For5,theboundedness followsfromthe factthatallthetermsliebetween aandb. 2.Thesequences 23,1and2aremonotone decreasing': theothersarenot monotone. Thedefinition 10ofanullsequence andtheappended remarks­ whichthestudent shouldreadthrough againcarefully -alsoremain unchanged. oDefinition. Asequence (xn)shallbetermedanullsequence If,25. subsequently tothechoiceofanarbitrary positivenumbere,anumberno=no(e) mayalwaysbeassigned, suchthattheinequality 3 Ix,,1<e: isfulfilledforevery n>no. Examples. I.Thesequence 23,1ISanullsequence, fortheproof10,7isvalidforany reala,forwhichIaI<I. 123,2ISalsoanullsequence, forhereIx"I<..,thcrcfore<e:.providedn e: Fornullsequences -thesewilllateronplayadominating part­ anumber ofquitesimpletheorems, whichwillbecontinually appliedin thesequel,willalsobeprovedhere.Thefollowing two,inthefirstplace, areobvious enough: oTheorem 1.If(xn)isanullsequence andthetermsofthesequence26. (xn'),forevery nbeyondacertainvaluem,satishtheconditionIx,,'I::sIXnI. or,moregenerally, thecondition Ix,,'I<K.Ix"I, inwhichKisanarbitrary (fixed)positivenumber,-thenx,,'isalsoanull sequence. (Comparison test.) 3GivenanypOSitive realnumber e:,apositive rational number e:'<e:canbe designated; infact,bythefundamental law2,VI,wecanfi:1danaturalnumber n>;,ande:'~.~~satisfies therequirements. Fromthisitfollowsthat,forrational sequences, theabovedefiOltlon isequivalent tothedcfiOltion 10,insPiteoftht: factthatonlyrational e:wereallowed there. 46 Chapter H.Sequences ofrealnumbers. Proof.IftheconditionIx,:I<K.IXnIissatisfied forn>m ande:>0isgiven,thenbytheassumptions wecanassignno>m,so thatforeveryn>no,IXnI<k'Sinceforthesevaluesofnwethcn alsohaveIx,:I<e,(xn')isthereforc anullscquence. Thefollowing theorem isonlyaspecialcaseofthepreceding: oTheorem 2.If(xn)isanullsequence, and(an)anyboundedsequence, thenthenumbers alsoformanullsequence. Onaccountofthisthcorem wesayforshort:Anullsequence "may" bemultiplied byabounded factor. Examples. 1.If(xn)isanullsequence, isalsoanullsequence. 2.If(Xn)ISanullsequence, sois(IXnI). 3.Asequence, allofwho.,etermshavethesamevalue,sayc,iscertainly hounded. If(x,,)ISanullsequence, (cxn)istherefore alsoanullsequence. In partIcular, (~),(ca")forlal<1,ctc.arcnullsequences. Thenextpropositions arelessobvious: 27. 0Theorem 1.If(xn)isanullsequence, theneverysub-sequence (xn') of(xn)isanullsequence 4. Proof. If,foreveryn>no,IXnI<e,thenwehave,ipsofacto, foranysuchn, sinceknisccrtainly>no,whennis. oTheorem 2.Letanarbitrary sequence (xn)beseparated intotwo sub-sequences (xn')and(xn"),-sothat,therefore, everytermof(xn)belongs tooneandonlyoneofthesesub-sequences. If(xn')and(xn")arebothnull sequences, thensois(x,.)itself. 4IfkI<k.<k,<...<kn<...isanysequence ofpositive integers, then thenumbers (n,-,=I,2,3,••.) aresaidtoformasub-sequence ofthegivensequence. §6.Arbitrary sequences andarbItrary nullsequences. 47 Proof.Ifanumbere>0bechosen, thenbyhypothesis anum­ bern'exists,suchthatforeveryn:-..n',IXII'I<B,andalsoanum­ bern",suchthatforeveryn>n",IXII"I<e.ThetermsXII'with index<n'andthetermsXII"withindex<n",havedefinite places, i.e.definite indices, intheoriginal sequence (XII)'Ifnoisthehigher oftheseindices, thenforeveryn>no'obviouslyIXIII<c,q.e.d. °Theorem 3.II(XII)isanullsequence and(x,,')anarbitrary rearrangement~ 01it,then(XII')isalsoanullsequence. Proof. Foreveryn:>no'IXIII<e.Among theindIcesbelong. ingtothefinitenumber ofplaceswhichtheterms Xl'Xli'•••,XlIo occupy inthesequence (XII')'letn'bethelargest. Thenobviously, foreveryn>n',IXII'I<c;hence(x,,')isalsoanullsequence. 0Theorem 4.IIIx,.)isanullsequence and(x,,')isobtained Irom itbyanylimtenumber 01alterations H,then(x..')isalsoanullse­ quence 7. Theprooffollows immediately fromthefact,thatforasuitable lOtegcrp~0,from,omenonwards wemusthaveXII'=x..+p'For ifevery:rllforn:::'::1lthasremained unchanged, andx,,)hasrecehTd theindexn'inthesequence (x,,'),theninpomtoffactforevery n>n', ifweputp=n1-n'. Theorem 5.II(x,,')and(x..")aretwonullsequences andilthe sequence (XII)issorelatedtothemthatfromacertammunwards X'<X<X"11== 71=n (n>m) then(x,,)isalsoanullsequence. ProofHaving chosen e>0,wecanchoseno>111sothat,for everyn>no'-e<XII'andXII"<+c.Forthesen'swethenhave, ipsolacto,-e<x..<+B,thatisIXIII<c;q.e.d. •IfkI>k2,•••,km...isasequence ofpositive integers suchthateveryin­ tegeroccursonceandonlyonceinthesequence, thenthesequence formed by xn'=-xkn issaidtobearearrangenzent ofthegivensequence. •Wewilldescribe thisconcept asfollows:Ifwealteranysequence, by omittm~, orinsertmg, orchanging, afinitenumber ofterms(orbydomgallthree thmgsatonce),andthenrC!1umbt,r thealtered sequence, \\ithout changmg the orderofthetermsleftuntouched, soastoexhtbititasasequence (x,.'),then",e shallsay,(xn')isobtained orhasresulted from(xn)by{/fimtenllmberofalteratIOns. 7Itisprecisely because ofthistheorem thatonemaysayofasequence that theproperty ofbemganullsequence concerns onlytheultll1lute behuvwur ofItsterms (ef.p.16). 48 Chapter H.Sequences otrealnumbers. Calculations withnullsequences, finally,arefounded onthe following theorems: 28. °Theorem 1.11(x,,)and(x,,')aretwonullsequences, then (y,,)=(x"+x,,~, i.e.thesequence whosetermsarethenumbers y"=x"+x,,',isalso anullsequence. -Briefly. Twonullsequences "may" beaddedterm byterm. Proof.IfE:>0hasbeenchosenarbitrarily, thenbyhypothesis (cf.10,4and12)anumber 1/1andanumber 112existsuchthatforevery n>111,IX"I<~,andforevery 11>112,IX,,'I<;.If110isanumber ?:;1/1and>112,thenfor11>110 Iy"I=IX"+-xn'I~IX"I+-Ix,,'I<~+-~=e:. (y,,)istherefore anullsequence 8, Since,by26,3(or10,5),(-x,,1isanullsequence if(x,,')is, (y,,')=(x"-x,,')isthenbytheabovealsoanullsequence, I.e.we havethetheorem' °Theorem 2. (y,,')==(x"-x,,'). bytem111(x,,)and(x,,')arenullsequences, thensois Orbriefly:nullseqltences "may" besubtracted term 1 , 1 hstantly=1.Ifwetakex"=11'x.=n~'tenthe videabounded sequence.Remarks. 1.Sincewemayaddtwonullsequences termbyterm,wcmayalsodo sowiththreeoranydeflntle number ofnullsequences. Forsupposing thisprov- edfor(p-1)nullseqlIences(x,,'),(x.."),....(x~P-1».i.e.supposing the sequence tobealready recognised asanullsequence, Theorem 1ensures thatthe sequence lX..),forwhich xn=(xn'+...+x~p-1»)+x~Pl, Isalsoanullsequence. Thetheorem thusholdsforeveryfixednumber ofnull sequences. 2.Thattwonullsequences "may" alsobemultlplted termbyterm,is immediately clearfrom26,1,sincenullsequences, by10,11,arenecessarily bounded. 3.TermbytermdLvision, onthecontrary, isingeneral notallowed, as isalready obvious, forinstance, fromthefactthatwhenx"=1=0,Xniscon­ x" .x"dratIos-,0notevenprox. 8Forthelastmequahty 3,11,4isused. §7.Powers, rootsandlogarithms. Special nullsequences. 49 4.Inthecnseofothersequences (X,.)also,littlecanbesaidInthefirst instance aboutthesequence (~)ofthereciprocal values. Thefollowing is anobvious, butoftenusefultheorem: oTheorem 3.1/thesequence(Ixn!)0/absolute values0/theterms0/(x,,) haveapOSitIVe lowerbound.-1/.therefore. anumber">0eXists,suchthat/0' everyn. Ix"I>,,>O, thenthesequence(;J0/reciprocal valuesisbounded. 1Infact,fromIx,.I:2:">0itatoncefollows thatfor](=- '1wehave foreveryn. Inordertoincrease thescopebothoftheapplication ofourcon· ceptsandoftheconstruction andsolution ofexamples, weinsert ~ paragraph onpowers, roots,logarithms andcircular functions. §7.Powers, rootsandlogarithms. Specialnullsequences. As,inthediscussion ofthesystemofrealnumbers, itwasnot ourintention togiveanexhaustive treatment ofalldetails, butlather toputfundamental Ideasaloneinaclearlight,assuming asknown, thereafter, thebodyofarithmetical rulesandconcepts, with\\hich afteralleveryone isthoroughly conversant, sohere,inthediscussion ofpowers, rootsandlogarithms, wewillrestrictourselves toanexact elucidation ofthebasicfacts,andthenassumeknownthedetailsoftheir application. I.Powers withintegral exponents. Ifxisanarbitrary number, weknowthatthesymbolxkforpositive integralexponents k;'::2isdefinedastheproduct ofkfactors,allequal tox.Herewehavetherefore onlyanothernotationforsomething weknow already. ByXlwcmeanthenumber xitsclf,andifx=4=0,itisconvenient toagree,besides,that XOreprescnts thenumber 1,x-kthenumber -1k-(k=1,2,3,..'j' sothatxPisdefined foreveryintegralp~O.ForthcsepOllers'" withintegral exponents, wemerelyemphasize thefollowing facts: 1.Forarbitrary integral exponentspandq(~O)thethree29. funtlantental 'ruleshold: xP·xq=xP+rl; xlt.yP= (xy)1J;(x1J)q=xPIl, IIIx1Jisapowerofbasexandexponent p.Thiscontinental useofthe wordpowercannotbeheredispensed with,inspiteoftheslightambigully resulting frombyfarthemostfrequent useofthewordinEnglish todesignate theexponent. Thissenseshouldbeentirely discarded fromthereader's mind, notably for§35,2aandothers. (Tr.) 50 Chapter H.Sequencesofrealnumbers. fromwhichallfurtherrulesmaybededuced, whichregulate calcu­ lationswithpowers 11. 2.Since,inapowerwithintegral exponent, merely arepeated multiplication ordivision isinvolved, itscalculation hasofcourseto beeffected byISand19.Iftherefore xispositive anddefined forinstance bythenest(x,,!y"),withallItsendpoints ~0(cf.15,5), thenwehavesimu1talH'Ously with --(x"IY,,),x"=(x~Iy~)atonce, forallposlt1ve integral exponents: andsimilarly -withappropriate restrictions -forx<0ork.s::o. 3.Forapositive xwehavefurthermore Xk+1~x"accordmg asx~1 asweatoncededuce fromX~l,ifwemultiply (v.3,T,:3)byx".-. AndquiteasSImply wefind: IfXl'x2andtheintegral exponent karepositive, then according as 4.Forpositive integral exponents nandarbitrary aandbwe havetheformula (a+b)"=a"+(~)a,,-lb+(~)a"-2b2+... +(~)an-kbk+.,.+(:)b", where (~),for1<k<n,hasthemeaning (~)=n(n-l)(n---2) ..•(n-k+l)1·2.3 ... k and(~)willbeput=1.(Binomial Theorem.) 11.Roots. Ifabeanypositive realnumber, andkapositive integer, then Va shalldenoteanumber whosekthpower =a.Whatinterests ushere issolcJytheexistence question: Istheresuchanumber, andtowhat extentisitdetermined bytheproblem thusset? Thisisdealtwithinthe 9Inthis.thevalue0forthebasezor:yisonlyadmissible ifther.or responding exponent ispOSItive. §7.Powers, rootsandlog,lnlhms. Special nullsequences. 51 (a>0). ;k=(JTheorem. Thereis,invariably, oneandonlyonepositivenumber; 30. satisfying theequation l<a<(g+1)le• IheInterval10determined bygand(g+1)wedivideinto10equal partsandobtain, inthemanner nowrepeatedly worked out,adefi· niteoneofthedigits0,1,2,...,9,-whIchwemaydenote,say,byzl' -andforwhich~,-- WewriteS=Vetandcall;thektorootofa, Proof. Onesuchnumbt:r mayimmcdi,ltdy bedetermined bya nestofintervals, anditsexistence thereby established Weusethe decimal-section method. SinceOk=0<a,but,pdenotmg anypositive integer>a,pk~P>a,-thereisoneandonlyoneintegerg~0 forwhichtu (Z,)k(ZI+l)kg+16<a<g+-10 andsoon,andsoon.Wetherefore obtaina (JJ=(xnIYn)whoseendpoints havedefinite values +ZI+ZB++Z"-1+Z"x..=~g1010.-•••10.-::-1-Ionnestofintervals oftheform (n=1,2,3,...) and --tz,+z,+ + Zn_, LZn+1Y..=g-iD109...10,.-1T-IOn' If~=(XnIy,.)bethenumber thereby determined, thensincehereall endpomts ofintervals are~0,itatoncefollowsby29,2that ;k=(x~IY~), But,byconstruction, x~<a<Y~fore\'cry n~hence,by§5,Theorem 4, wemusthave Thatthisnumber ~is,moreover, theonlypositivc solution ofthe problem, follows directly from29,3,SInceitwastherepointed out tbatforaPOSitive;1=F;,necessarily ;1=F;k,ie,=Fa. Ifkisanevennumber, then-~isalsoasolution ofthe problem. Weshallnot,however, takethism\oaccount inthefollow­ ingpages, butinterpret thekthrootofapOSitive number aas meaning onlythepositive number ~,completely anduniquely dete1' ,kJ-minedby3011.-Fora=0,wemayalsoputv a=012 10gisthelastofthenumbers0,1,2,.",pwhosekthpoweris:;;;:;... '11naccordance withthiswehave,forinstance,.jx·'notalways =x, bUlalways=IxI· k._ nFornegative a'swewillnotdefinevaatalljwecan,however, if 1:_t-- 11isodd,writeifa= -ifIaI. 52Chaptern.Sequences ofreal nU~)Jber•. givesWewillnotenterfurtherintotherule~forcalculation'> withroots, outcomider themasfamihar toevery OIlC,andwillonlyprovethe folloWIng siIIIpIetheorems: 29,3givesatoncethe :U. Theorem 1.Ita>0andal>0,thenV-a-;; -V~~,according as a~al'-Further wehavethe Theorem 2.Ifa>0,then(y-;)isamonotone sequence,. and wehave,moreprecisely, _3_ a>,Ia>-Va>...>1,ila>1, but _3_ a<Ja<)/a<...<1,11a<1. (Fora=l, thesequence 1S01course_1.) Proof By29,3,a>1involves an+!>an>1,andthere!ore bythepreceding theurem, takingn(n+1)throots, Ya>n+V-a>1. Sincefora<1alltheirequality signsarereversed, thisprovesthe \\holestatement, -.Hencefmallywededucethe Theorem 3.Ifa>0,thenthenumbers n__ x=Va-ln formanullsequence (monotone bythepreceding theorem). Proof.Fora=1,theassertIOn istnvial,asthenxn=O.If n- n-a>1,andtherefore Va>1,i.e.xn=Ya-1>0,thenweredson n-asfollows: Bytheiuequalny ofBcrnoulli (1'.10,7), Va=1+xn a=(1+xnf>1+nxn>nxn. aConsequently xn=IxnI<'n'therefore (x,,),by26,1or2,ISanull sequence. 1If0<a<1,then->1,andso,bythele~ultobtained,a (V~--1) n_ ISanullsequence. ]fwemultiply thistermbytermbythefactors\1a, n_ -whichcertainly formabounded sequence, asa<Va<1,-then itatoncefoHows, by26,2,that andtherefore also(x,,), ~sanullsequence, -q.e.d. §7.Powers, rootsandlogarithms. Special nullsequences. 53 Ill.Powers withrational exponents. Weagainregardassubstantially known, inwhatmanner onemay passfromrootswithintegral exponents topowers withanyrational 71 exponent: Byaq,withintegral p~0,q>0,wemean,loranyposi- tivea,thepositive numberuniquely delinedby '!'.- IfP>0,thenamayalsobe=0;a'lmustthenbetakentohave thevalueO. Withthesedefinitions, thethreefundamental rules29,1, i.e.the formulae remainunaltered, foranyrational exponents, andtherefore calculations withthesepowers areformally thesameaswhentheexponents are integers. Theseformulae contain, atthesametime,alltherulesforworking withroots,sinceeveryrootmaynowbewrittenasapowerwitha rational exponent. -Ofthelessknownresultswemayprove,as theyareparticularly important forthesequel,thesetheorems: Theorem 1.Whena>1,-thenaT>1,il,andonlyil,r>O.32. Similarly, whena<1(butpositive), thenaTis<1il,andonlyil, , >o. q-Proof. By31,2,aandVaareeitherbothgreaterorbothless ('1-)71istrueofaandVa=aTifandonlyif lhan1;by29thesame p>O. Theorem la.IItherational number r>0,andbothbasesare PositivethenaT::::;a Taccording asa<a, > l' >l' Theproofisatonceobtained from31,1and29,3. Theorem 2.Ifa>0,andtherational number rliesbetweenthe rational numbers r'andr",thenaTalsoalwaysliesbetween aT'and aT"13,andconversely, -whether abe<,=or>1,andr'< • or>r". Proof. If,firstly,a>1andr'<r",then I ,ar"a'=a'·a,-r=~. a 13Theterm"between" maybetaken,asweplease,eithertoinclude orexcludeequality onbothsides,-excepting whena=1,andtherefore all thepowers aTalso=1. s (051) Chapter 11.Sequences ofrealnumbers. (a>0) thensois(xn).Xn=aT"-I, alsoformanullsequence. 11(rn)ismonotone, Proof. By31,3,(V~-1)and(vI-1)arenullsequences. Iftherefore t:>0begiven,wccansochoosen1andn.thatByTheorem 1,thisalreadyprovesthevalidityofourstatement forthis case,andintheotherpossihle casestheproofisquiteaseasy.-From thisproofwededuce, indeed,moreprecisely, the Theorem 2a.11a>1,thentothelarger(rational) exponent also corresponds thelargervalue01thepower. 11a<1(butpositive) thenthelargerexponent givesthesmaller power.-Inparticular: 11the(positive) basea9=1,thendifferent exponents givedtllerent powers.-Hencewcdeduce, further, Theorem 3.If(rJisany(rational) nullsequence, thenthe numbers lIence, for Ia'"-11<e, proving that(aT-1)isanullsequence. -Thatitismonotone, If (r,,)is,follows immediately fromTheorem 2a. Thesetheorems formthebasisforthedefinition offorn>no' thesen's,forn>ni'iV~-1I<e, andforn>n~,IV~-1I<e. Ifmisaninteger largerthanbothnJandnJ'thenthenumbers (a~-1)and(a-~-1)bothliebetween -eand+e,i.e 1 1 amanda-;nliebetween 1 -t:and1+e. ByTheorem 2,a'thenliesbetween thesamebounds, ifI'IJesbe· tween-~and+~.Byhypothesis wecan,however, sochooseno'm m thatforeveryn>no'111I""I<mor-m<""<+m; a'"IStherefore between 1-eand1 -f-l!• a=(aZ "Ia"n) a=Ca"flIaZn)IV.Powerswitharbitrary realexponents. Forthiswefirststatethe 33. Theorem. If(x"IY,,)isanynestofintervals (withrational end­ points)andaispositive, then 101'a~1, andfora~1, §7.Powers, rootsandlogarithms. SpecIal nullsequences. 55 isalsoanestofintervals. Andif(xnIYn)isrational valuedand=r, thena=aT, Proof. Thatineithercasetheleftendpoints formamonotone ascending sequence, thenghtendpoints amonotone descending se- quence, followsatoncefrom32,2a.Bythesametheorem, aXn<a"" intheonccase(a::?::1)andalln<aXnintheother(a<1),forcveryn. Flllally,thatinbothcasesthelengths oftheIntervals formanull sequence, follows, withtheaidof26,from Ialln-aXnI=\a",,-xft -l\.axn; forherethefirstfactor,by32,3,isanullseqnence, because(y"-x,,) isbyhypothesis anullsequence withrational terms;andthesecond factorisbounded, becausc forcveryn 0<aX"<a'" intheonecase(a~1), intheother(a~1). Nowif(x"IY,,)=r,thcnrliesbetwccn x"andYn'forcveryn, andsoby32,2,aTliesbetween aX "anda"",foreveryn;henceby §5,Theorem 4,necessarily (J=aT. Incomequence ofthIStheorem, wemayagreetothefollowing Definition le.Ifa>0,and(!=(xnIYn)zsanarbitrary real number, then: a(!=6,i.e.{=(aX "Ialln) =(a"nIaXn)it if ThisdefinitIOn canofcourseonlyberegarded asappropriate, iftheconcept ofageneral powerthereby determined obeyssubstan­ tiallythesamelawsasthetypeofpowersofarconsidered, that withrational exponents. Thatthisisso,inthefullestsense,isshewn bythefollowing considerations. 1.ForratIOnal exponents, thenewdefinition givesthesameresult34. astheold. 2.Ife~e',then15ae~a?'. HThiscomblllatlOn 33oftheorem anddefinition is,fromthepoint ofviewofmethod, ofexactly thesamekindasthosesetforthin14-19: \Vhatisdem0nstrabII'inthecaseofrational exponents ISraised, 111the caseofarbItrary eJlponents. totherankofadefinition, -whoseappropriate­ nesshasthentobeveriflCd. 16Thisassertion, formally rathertrivialinappearance, whenputsome· whatmoreexplicitly, TUnsthus:If(x"Iy,,)=eand(x,,'IY..')=(/aretwonests ofmtervals, whichmayberegarded asequal 111thesenseof14,thensoare thosenestsofintervals equal(again Jl1thesenseof1-1),wInchbyDefinition :13 givethepowers allandal/. 56Chapter 11.Sequences ofrealnumbers. 3.Fortwoarbitrary realnumberseande',andpositive aandb. thethreefundamental rules ae.ar!'=ae+e'; hold,sothatwiththegeneral powers nowintroduced wemaycal· culateformally inprecisely thesamewayaswiththespecialtypes hitherto used. Intotheextremely simple proofsofthesefactswewill,as emphasized onp.49,notenterfurther16;wewillalso,sofaras concerns theextension oftheorems 32,1-3togeneral powers, now immediately possible, content ourselves withthestatement andafew indications oftheproof.Wehavetherefore thetheorems, generalized from32,1-3: 3a. Theorem 1.Whena>1,wehaveae>1it,andonlyit,f!>O. Similarly, whena<1,(butpositive), wehaveae<1it.andonly it,e>O. Forby32,1,wehavee.g.fora>1,a:rn>1if,andonlyif, x,.>O. Theorem 1a.Ittherealnumbereis>0,andbothbasesare positive, thenat!~af,according asa~a1• Proofby32,laand15. Theorem 2.Ita>0andeisbetweene'andr/',thenaI!isal­ waysbetween aI!'andae".-TheproofispreCIsely thesameas 32,2.Ityields,moreexactly, the Theorem 2a.Ita>1,thentothelargerexponent corresponds thelargervalueofthepower;ifa<1(butpositive), thenthelarger exponent givesthesmallerpower.Inparticular: Ita9=1,thendifferent exponents give d~tterent powers.-Andfromthistheorem, exactlyas in32,3,follows thefinal 16Asamodelwemaysketchtheproofofthefirstofthethreefundamental rules:Ife=(xnIYn)ande'=(xn'IYn'),thenby16,e+e'=(xn+xn'IYn+Yn') andtherefore -weassumea2:;1-: ae'=(a"'n'lalln'), Sinceallendpoints (aspowers withrational exponents) arepositive, we have,byIS. Since,however, forratIOnal exponents, thefirstofthethreefundamental rules hasalready beenseentohold,thislastnestofintervals isnotonlyequal,in thesenseof14,tothatdefining al!+e',butevencoincides withItterm byterm. (:a)foreverya>0isa§7.Powers, rootsandlogarithms. Specialnullsequences. 57 Theorem 3.It(!?,,)isanynullsequence, thenthenumbers x"=at'"-1 (a>0) formanullsequence. It(e,,)ismonotone, then sois(x,,). Asaspecialapplication, wemaymention the Theorem 4.It(x,,)isanullsequence withallitstermspositive, thentoreverypositive et, isalsothetermotamtllsequence. -Thus nullsequence. 1 Proof.If€>0begivenarbitrarily, €'iisalsoapositivenumber.By hypothesis, wecanchoosenosothat,foreveryn>no(cf.10,1and12), 1 Ixi=x<ea.UI tJ Forn>no'by35,la,wethenalsohave,however, x"Q=Ix,,'I<e whichatonceprovesthewholestatement. Theabovetheorems comprise themainprinciples usedincal­ culations withgeneralized powers. V.Logarithms. Thefoundation forthedefinition oflogarithms liesinthe Theorem. Ifa>0andb>1aretworeal,andinallfurther36. respects quitearbitrary numbers, thenoneandonlyonerealnumber ~ alwaysexists,forwhich Proof. Thatatmostonesuchnumber canexist,alreadyfollows from35,2a,because thebasebwithdifferent exponents cannotgive thesamevaluea.That!:>uchanumber doesexist,weshowcon· structIvely, byassigning anestofintervals whichdetermines it,­ thusforinstance bythemethod ofdecimal sections: Sinceb>1, (b-")=(in)isanullsequence, by10,7,andthereexists,conse- IquentIy, sinceaand-arcpositive, naturalnumberspandqforwhicha andb-q<~aor If,now,weconsider thevarious integers between -pand+qin succession, asexponents ofb,theremustbeone,andcanbeonly one-callitg-forwhich bU;:;;;a,butbU\-1>a. 58 Chapter H.Sequences ofrealnumbers. Theinterval10=-g...(g+1)thereby deternuned wedivIdeintu10 equalpartsandobtain,justasonp.51,a"digit"zl'forwhich but ByrepetitIOn oftheprocess ofsubdivision wefindaperfectly definite nestofintervals l:.exIy)with ..=nn' forwhich bZ "<a<bV" foreveryn,-forwhich,therefore, IIIaccordance with33, be=a. Thistheorem justifies usinthefollowing Definition. Ita>0andb>1arearbitrarily gwcn,thenthereal number ~,uniquely determined by be=a iscalledthelogarithm ofatothebaseb,'and,symbolically, S=log ba. (gisalsocalledthecharacteristic, andthesetofthedigitszl'z~,za'" themantissa, ofthelogarithm.) Wespeakofasystem oflogarithms, whenthebasebisassum· edfixedonceforallandthelogarithms ofallpOSSible numbers are takentothisbaseb.Thesuffixbin10gbisthenusually omitted assuperfluous. Verysoonaparticular realnumber, usuallydenoted bye,appears quitenaturally asthemostconvement foralltheo­ retical considerations; thesystem oflogarithms builtuponthis baseisusually calledthesystemofnaturallogarithms. Forpractical purposes, however, thebase10is,asweknow,themostconvenient; logarithms tothisbasearecalledcommon orBriggs'logarithms. These arethelogarithms foundinalltheordinary tables!? Therulesforworking withlogarithms weassume, aswedid withpowers, tobealready known, andcontent ourselves withamere mention ofthemostimportant ofthem.Ifthebaseb>1isarbitrary, 17A~amatterofcourse, asystemoflogarithms mayalsobebuiltupona positive baselessthan1.This,however, isnotusual.Thefirstlogarithms cal­ culated byNapierinIIH4were,however, builtuponabaseb<1,whichpresents somesmalladvantages, particularly forlogarithms oftrigonometrical functions. NeitherNapiernorBriggs,however, reallyusedanybase.TheIdeaofloganthms astheinverseofpowersonlydeveloped inthecourseofthe H~t,hcentury. §7.Powers, rootsandlogarithms. Special nullsequences. 59 butassumed fixedinwhatfollows, andifa,a',a".•.denoteany positive numbers, then 1.log(a'a")=loga'+loga". 37. I2.log1=O; log-=-loga; logb=1.a 3.logal.!=eloga(earbitrary, real). 4.loga~loga',accord1I1g asa~a';mparticular, I::>:0 d' ::> 5.oga<:'accorll1gasa<:1. 6.Ifbandbiaretwodifferent bases(>1),and~and ~ithe logarithms ofthesamenumber atothesetwoba~es,i.e. then 1n=2,3,4,...isaIlullsequence. Infact---<E,logn~=~l·logbbl' - asfollowsatoncetrom(a=)be=b/',bytakinglogarithms onboth sidestothebasebandtakingaccount of37,2andCl 7(_2_).log-n' 1 1 provIded logn>-,thatis,n>b-;. 6 VI.Circular functions. Tointroduce theso-called circular functions (thesineofagiven angleiM,withthecosine,tangent, cotangent etc.)inanequally strict manner, ie.avoiding onprinciple allreference togeometncal in­ tuitionaselement 01proalandfounding solelyontheconcept ot therealnumber, isatthisstagenotyetpossible. Thisquestion will beresumed later(§24).Inspiteofthis,wewillunhesitatingly enlist themtoenrichourapplicatIons andenliven ourexamples (butof coursenevertoprovegeneral propositions), insofarastheirknow­ ledgemaybepresupposed fromelementary work. Thu!>e.g.thefollowmg twoSImplefactscanatoncebeascertained: 37a. 1.Ifai'alll.••,an'...arcanyangles(thatistosay,anynumbers), then (sinan)and(cosan) Ilreb0unded sequences iand 18Angleswillingeneral bemeasured inradiansIfinacircleofradius umtyweImagine theradiustoturnfromadefinIte initialposition, thenwe measure theangleofturning bythelengthofthepathwhichtheextremity ofthemoving radiushastraversed -takingitasposll1vewhenthesenseof turmng iscOltnterclockwlse, otherwise asnegatIVe. Anangleisaccordingly a purenumber jastn!19htanglehasthemeasure+nor-n,arightanglethe n n m~asure+2-or--2-'Toeverydehll1tcly placedangletherebelongs an infinitenumber ofIllea!>ures which,however, differfromoneanother onlyby integrnl multiples of2n,i.e.bywholeturns.Themeasure 1belongs tothe angle,thearccorresponding towhichisequaltotheradius,andwhichthere­ fereindegrees is57017'44"·8nearly. 60Chaptern.Sequences ofrealnumbers. 2.thesequences (Sinnan)and(co:an) IanI= ---]---- ---. 1+(7)e+.,.+(:)e"38.are(by26)nullsequences, fortheirtermsarederived fromthoseofthe nullsequence (~)bymultiphcation bybounded factors. VII.Specialnullsequences. Asafurtherappliration oftheconcepts nowdefined, wewill examll1c anumber ofspecialsequences: 01.ItIa1<1,thenbesides(a")even(nan)isanullsequence. Proof. Ourreasoning isanalogous tothatof10,711':For a=0,thcassertion istrivial;for0<IaI<1,wemaywrIte, with(!>0, IIaI=Cf(;'andtherefore Sincehereinthedenominator eachtermofthesumispositive, we haveforeveryn>1, lanl«~)/ Thuswchavethcrefore1nl 1·2na<(U=-I)e~' In anI<E,assoonas i.e.forevery1·2 (n-l)/?~<e 2n>1+-,.e·/?- Theresultthusproved isveryremarkable: itasserts, infact, thatforalargenthefraction (1-:/?)"isverysmall,anditsdenominator therefore verymuchgreater thanitsnumerator. Thisdenominator is however constant(=1)forQ=0,andwheneisverysmall(and positive), itonlyincreases veryslowlywithn.Nevertheless, ourresult showsthatprovided onlynbetakensufficiently large,thedeno­ minator isverymuchlargerthanthenumerator 20.Thepointno,from whichinanl=(-l~)-liesbelowagivene-wefoundno=1+~-+/?" B'l! doesindeedlieveryfartotheright,notonlywhen E,butalsowhen e=r~I-1,isverysmall(i.e.IaIverynearto1).Substantially this 19Exceptthat 11and/?neednolongerberational. §7.Powers, rootsandlogarithms. Specialnullsequences. 61 andonlythisistrue:HoweverIaI<1ande:>°maybegiven,we havealways, fromareadilyassignable pointonwards,InanI<e. Fromthisresultmanyothersmaybededuced, e.g.thestillmore paradoxical fact: 02.IfIaI<1andIXrealandarbitrary, then(n"an)isalsoanull sequence. Proof.IfIX-S0,thenthisisevident from10,7,because of26, 1 2;ifIX>0,writeIaI;X=aI'sothatby35,1a,thepositive number atisalso<1.Bythepreceding result,(natn)isa'nullsequence. By 35,4 therefore, finally,(by10,5),n"anitselfisalsothetermofanullsequence 21. 3.Ifa>0,theneO~"n) isanullsequence 2~,towhatever baseb>1 thelogarithms aretal~en. Proof. Sinceb>],a>0,\Vchave(by35,la),b">1.There- fore(b~)n)isanullsequence, by1.Given e:>0,wchaveconsequently fromacertainpointonwards, -sayforeveryn>m- n,e: (b')"<e=ba' But,inanycase, ~ogn<g+1=b".g+1 n"(bY)" (b,,)q-}-l' ifgdenotethecharacteristic oflogn(sothatg<logn<g+1).If, therefore, wetaken>bm,logn,andafortiurig+1,is>m.Hencethe lastvalueabove,withourchoiceofm,is <b"· e:=e:I.e.b'],logn<e:forevery Il>1'0=bm. n" 20Writing asaboveIaI=-_!_--InanI~--n_.wemayalsosay'1+e' (1+e)n' • (1+e)"becomes -forapO~ltivee-morepronollncedly lart~e,or,alsomorepro­ nOllncedly infinite, thannitself,-bywhichweagain(cf.7,3)meannothing more andnothing lessthanthatoursequence isprecisely anullsequence. -Forfuture reference weremark herethattheresultsprovedin1and2arcalsovalidfora complex a,provided onlyIaI<1. 21Withthesamechangeofnotation asabove,wemaysayhere:"(1+e)n becomes morepronouncedly i.ifinitethanevery(fixed)powerhowever largeofn Itself". 22Or,inwords,"lognbecomes lesspronouncedly larf!ethaneverypower,how­ eversmall(butdett.rminate andpositive), ofnitself". 3. (G51) fi2Chaptern.Sequences ofrealnumbers. 4.11IXandfJarcarliitrary positive numbers, then JSanullsequence-,however large IXandhowever smallfJmaybe23• Proof. By3.,C:fl:) isanullsequence, because ~>0;by Sil,4,therefore, soisthegivensequence. 5.(xn)-(-ii1t-1)isanullsequence. (Thisresultisalsovery remarkable. Forwhennislarge,wehavealargenumber under the11-;theexponent ofthe11is,itistrue,alsolarge;butitis notatallevident aprioriwhichofthetwo-radicand orexponent ­ will,sotospeak,provethestronger.),,-Proof. Forn>1,wecertainly have11n>1,therefore n_ X=--=11n-1certainly>' O.Hence m ,. allthetermsofthesumarepositive. Consequently wehave,in particular,n>(n)X~=_~(Ilnxi2" I 2 nor24 Hence 2IX,.I<-\'n'n_ sothat(xJ=Yn-115111factby26,1and35,4anunsequence. 6.11(X,.)isanullsequence whosetermsareall> -1,then10,. every(fixed)integerk,thenumbers kx:.=yC+ ~',,-1 alsolormanullsequence 25. _3"Every power oflogn,however large,(butfixed)becomes less pronouncedly largethaneverypowerofnitself,however small(butfixed). UThesubstitution, whenn>1,ofthevaluen-;for(n-1)which itcannotexceed, isanartifice oftenusefulInsimplifying calculations. 2.Bytheassumption thatallx,,'s>-1,wemerely wishtoensurethat thenumbers xn'aredefined foreveryn.Fromadefinite pomtonwards thisisautomatically thecase,since(XII)isassumed tobeanullsequence and therefore fromsomepointcertainly Jx"I<1,andhenceXII>-1. §7.Powers, rootsandlogarithms. Special nullsequences. 63 Proof. Fromtheformulae setforthonp.22"Footnote 13,where k wcputa=Vl-~x"andb=1,itfollows t11at28 therefore, sincethetcrmsinthedenominator areallpositive and thelas~is1, whence, by26,thestatement atoncefollows. 7.11(xtl)isanullsequence 01thesameIdndasm{j.,then then'~mbers Y..=log(1+:x-,,) alsolormanullsequence. Proof. Ifb>1isthebasetowhichthelogarithms aretaken,and E>Uisgiven,wewritc I)'-1"-=81>1 -b'-=82 sothatwehave81~-b'.<:2:><:2:>O.Wcthenchoose nosolarge,that forevery II>110,IXnI<<:2'ForthoscIl'Swehave,afurtiori, therefore (by35,2or37,4) Iy"I=,Ilog(1+x,,)I<E:j withwhiehthestatement isproved. H.It(xtl)isagainanullsequence 01thesamekindasin6., thrnthenumbers alsolormanullsequence,itedenoteanyrealnumber. Proof. By7.and26,3,thenumbers en=e.Iog(1+xn) formanullsequence. By35,3and37,3thesameistrueofthenumbers bun-1=(1-t-:r."r-'-1=zn' 18Wcassume k~2,sincefork=1theassertion istrivial,q.e.d. 64 Chaptern.Sequences ofrealnumbers. §8.Convergent sequences. Definitions. Sofar,whenconsidering thebehaviour ofagivensequence, wehave beenchieflyconcerned todiscover whether itwasanullsequence ornot. Byextending thispointofviewsomewhat, inamanner whichreadily suggests itself,wereachthemostimportant concept ofallwithwhich weshallhavetodeal,namely, thatoftheconvergence ofasequence. Wehavealready(cf.10,10)described theproperty whichasequence (:en)mayhave,ofbeinganullsequence, bysayingthatitsmemhers becomesmall,becomearbitrarily small,withincreasing n.Wemayalso say:Itsterms,asnincreases, approach thevalue0,-without, ingeneral, everreaching it,itistrue;buttheyapproach arbitrarily neartothis valueinthesensethatthevaluesofitsterms(thatistosay,theirdifferences fromO)sinkbeloweverynumber 0:(>O),however small.Ifwesubstitute forthevalue0inthisconception anyotherrealnumberg,weshallbe concerned withasequence (xn)forwhichthedifferences ofthevarious termsfromthedefinitenumber ~-thatistosay,by3,Il,6,thevalues IXn-gI,-sink,withincreasing n,beloweverynumber 0:>0,how­ eversmall. \Vestatethemattermoreprecisely inthefollowing: 39. 0Definition. If(xn)isagivensequence, andIfitISrelatedtoa definitenumberginsuchawaythat (xn-t) formsanullsequence 1,thenwesaythatthesequence(xn)converges tot,orthatitisconvergent. Thenumber ~iscalledthelimitingvalue orli1nitofthissequence; thesequenceisalsosaidtoconverge to;,and zeesaythatitstermsapproach the(limiting) value ~,tendtog,havethe limit~. Thisfactisexpressed bythesymbols Xn~;orlimXn=;. Tomakeitplainerthattheapproach togiseffectedbytakingtheindexn largerandlarger,wealsofrequently write 2 Xn~;forn~00orlimxn=;. n~rr, Including thedefinition ofanullsequence inthenewdefinition, wemayalsosay: Xn~~forn~00(orlimXn=g)ifforeverychosen 0:>0,wecan 1l-70-"1'J alwaysassignanumberno=no(o:),sothatforeveryn>no,wehave IXn-~I<0:. 1Ora-Xn)orIXn-~I;by10,5theresultisexactlythesame. 2Read:"xn(tends)towards ~forntending toinfinity" intheonecase,and "Limit Xnforntendingtoinfinityequalsf'intheother.Inviewofthedefinitions 40,2and3,itwouldbemorecorrecttowritehere"n......+00";butforsimplicity the+signisu~uallyomitted. §8.Convergent sequences. tm Remarks andExamples. 1.Instead ofsaying"(xn)isanullsequencc", wemaynow,moreshortly, write"xn~0".Nullsequences areconvergent sequences Withthespeciallimiting valueO. 2.Substantially, allremarks madein10thercfore holdhere,sinceweare concerned onlywithaveryobvious generah~ation oftheconceptofanullsequence. 3.By31,3and38,5,wehavefora>0 Va-~1amiV" >-1. 4.If(xnIy,,)--17,then .~'Il->-17andYn--';17.Forboth Ixn-171andalsoIYn-171are;SIYn-x"I, sothatboth,by26,I,formnullsequences together with(y,.-xn). ~' __(-l)n'. . 143 6 5 D.Forxn-1 --n- ,thatIS,forthesequence 2,2'3'4'5'G'.••,x"-+1, 1forIxn-1Ic=nformsanullsequence. G.Ingeometrical language, XII->-gmean~thatalltermswithsuffiCiently largeindiceshemtheneighbourhood ofthefixedpomtg.Ormoreprecisely (cf. 10,13),meveryE-neighbourhood ofg,thewholeoftheterms,WIthatmostafillite /lIanberofexceptions, aretobefound 3.-Inapplymg themodeofrepresentation of7,G,wedrawparallel, totheaXIsofabSCIssae, through thetwopoints(0,g±E) andmaysay: Xn->-g,ifthewholegraphofthesequence (xn),Withtheexception ofafimtemitialportIOn, liesmeveryE-strip(however narrow). 7.Thelaxmodeofexpression: "for 11=00,xn=f'insteadofxn->g, shouldbemostemphatically rejected. -Forallintegern=00doesnotexil'lalld \:nneedneverbe~g.Weareconcerned merelyWithaprocessofapproXImation, sufficiently clearfromallthatprecedes, whichthereisnoground whatever for Imagining completed manyform.(Inoldertextbooksandwntmgs wefrequently find,however, thesymbohcal modeofwnting: "Iimxn~C,towhich,smceIt n-"'G() ISafterallmeantonlysymbohcally, noobjection canbetaken,-exceptmg that Itisclumsy, andthatwntmg"n.~00"mustnecessanly createsomeconfUSIOn regardmg theconceptoftheinfimteinmathematics. 8.Ifxn-+g,thentheisolated termsofthesequence (.\:,,)arealsocalled approximations tog,andthedifference g-xniscalledtheerrorcorrespondmg to theapproximation .~'1l" D.Thename"convergent" appears tohavebeenfirstusedbyJ.Gregory (Veracirculiethyperbolae quadratura, PaduaHi(7),and"dIvergent" (40)byBemoulli (LettertoLelblll:::of7.4.1713).Itwa~through thepubhcatlOns ofA.L.Callchy (~eep.7'2,footnote 18)thatalimiting valuecametobedenoted generally bythe prefixed symbol"hm".Thearrowsign(-->-),\\hichissoparticularly appropriate, camemtocommon useafter1906,through theworksofG.H.IIardy,whohimself referred itbacktoJ.G.Leatham (1905). Tothedefinition ofconvergence weatonceappendthatofdiver­ gence: oDefinition 1.Everysequence whichis1Iotconvergent inthesense40. of39iscalleddivergent. 3Frequently thisisexpressed morebriefly: Ineverye:-neighbourhood of ~"almost all"termsofthesequence aresituated. Theexpression "almost all" has,however, othermeanings, e.g.intheTheoryofSetsofPoints. 66 Chapter 11.Sequences ofrealnumbers. Withthisdefinition, thesequences ti,2,4,7,8,11arecert<linly divergent. Among divergent sequences, onctypeisdistinguished byits particularly simpleandtramparentbehaviour, e.g.thesequences (n'l), (n),(an)fora>1,(logn),andothers. Theircommon property is evidently thatthetermsincrease with incrca~ing nbeyond everybound, however high.Forthisreason, wcmayalsosaythattheytendto-j00, orthatthey(ortheirterms)become infimtely large.Thisweput moreprecisely inthefollowing Definition 2.Itthesequence (xn)hastheproperty that,givenan arbitrary (large)positive numberG,another number nocanalwaysbe assigned suchthatforeveryn>no x",,'G, then4weshallsaythat(xn)diverges to1-00,tendsto+00,orisdefinitely divergent 57viththelimit+00;and1VPthenwrite x..--++00(forn-;..00)orlimXn=+00orlimXn=+00. 11----> ~£ Wearemerelyinterchanging rightandlefthydefining further: Definition 3.Ifthesequence (xn)hastheproperty that,givenall arbitrary negative number-G(largeinabsolute value),anothernumber nocanahvaysbeassigned suchthatforeveryn>110 Xn< -G, thent0eshallsaythat(xn)diverges to-00,tendsto-00or1Sdefillite~'Y divergent 5withthelimit-00,andwewrite x..--+-00(for11--+00)orlimx..= -00orlimXn-=-00. n--+~ Remarks andExamples 1Thesequences (n),(n2),(nn)fora>0,(logn),(logn)"fora>0, lendto+00;thosewhosetermshavethcsevalueswiththenegative sign tendto-00. 2.Ingeneral' Ifxn-+ro, thenx,,'=-x,,--CO, andconversely.­ Iti~thcrefore sufficient, suhstantIally, toconSIder divergence to+COIIIwhat follows. 3.Ingeometrical language, Xn-+00means, ofcourse, thathowev('r a pointG(veryfartotheright)maybechosen, allpointsx"'exceptatmosta finitenumber ofthem,remam beyond itontheright.-Withthemodeof •Noticethatherenotmerelytheabsolute valuesIx"I,butthenumbers x" themselves, arerequired tobe>G. 5Itissometimes evensaId,-withapparent distortion offacts,-that thesequence converges to+CO.Thereasonforthisisthatthebehaviour described inDefmition 2resembles inmanyrespects thatofconvergence (39). Wewillnot,however, subSCribe tothismodeofexpression, although amis­ understanding wouldneverhavetobefeared. -Similarly {or-00. §8.(;onvergtl1t sequences. 67 representation in7,6,itmeansthat·however farabovetheaxisofabscissae wemayhavedrawntheparallel toit,thewholegraphofthescqucnc(' (x,,)­ excepting ahniteinitialportion, liesstIllfurtheraboveIt. 4.Thedivergence to±!Xlneednotbemonotone; thusforinstance the sequence1,2\2,223,23,4,2',...,k,2",alsodiverges to+00. 5.ThGsuccession 1,-2,+3,-4,,(-l)"-l n,...doesnotdiverge to+00orto-00. This lead~ustothefurther Definition 4.Asequence (x,,),whicheitherconverges inthesense ofdefinition :-19,01'divergesdefinitely inthesenseofthedefini­ tions40,2and3,willbesaidtobehavedefinitely (forn-+oo). Allothersequences, whichtherefore ne/the1'converge, nordivergedefini­ tely,willbecalledilulejinitely di"eruent or,torshort,indefinite 6. Remarks andExamples. 1.Thesequences (-1)"],[(-2)"],(an)fora;;::;-1,andlikewise these­ quences 0,1,0,2,0,3,0,4,...and0,-1,0,-2,0,-3,"',asalsothese­ quences6,4,8areobVIously indefillltcly divergent. 2.Onthecontrary, thesequence(Ia"I)forarbitrary a,and,inspiteof allirregularities indetaIl, thesequences (3"+(_2)"), (n+(-l)"logll), (nS+(-1)"n),showde/lnltebehaviour. 3.Thegeometrical mterpretation ofindefinite behaviour follows imme­ diatelyfromthefactthatthereisneither convergence (v.39,6)nordefinite divergence (v,40,3,rem3). 4.Bothfrom:l'n-.+00andfromx"-.-00itfollows, provided every term+07,that-.!..-.0;forIx,.I>G=~evidently imphesI-.!..I<•._..On ~ 6 ~ theotherhand, Xn-.°10nowayinvolves deflllite behaviour of(.~-). x" Example: Forx"=(-=-.!l~, wehavex,.-+0,but(.!.)indefinitely diver-n XII gent.-Wehavehowever, asISeaSilyproved, the Theorem: 1/(x,,)ISanullsequeme whosetermsallhavethesamesign. thenthesequenre (-~)isdef'nltely dIvergent; -andofcourseto+00or J'1l -00,accordmg asthexn'sareallposlllve orallnegatIVe. 8"Vchavetherefore tocon~ider threetypical modesofbehaviour ofa eqnence, namely: a)Convergence toanumher ~,inaccordance with39; )dIvergence to±00,IIIaccordance w;th40,2and3;c)neither ofthe wo.-Sincethebehaviour b)shows50111eanalogy WItha)andsomewithc), \nodesofexpressions inuseforitvary.Usually, Itistrue.b)isreckoned as fivergencc (themodeofexpre••ionmentlOned IIIthelastfootnote cannot reconsistently maintallled) but"lilmtll1g values" -+-00and-00areatthe ametimespoken of.-Wetherefore speak,inthecases11)andb),ofade­ inite,inthecasec)ofanIndefInite, behaviour; incasea),andonlyin thiscase,wespeakofconvergence, inthecasesb)andc)ofdivergence. ­ Instead of"defmilely andmdefinitely dlvergent", thewords"properly andim­ properly divergent" areahousedSince,howl'ver, asremarked, defilllte di­ vergence 5tillshowsmanyanalogies toconvergence andalimitisstillspoken ofinthiscase,itdoesnotseemadvisable todt'signate thiscaseprecisely as thatofproper divergence. 7Fromsomeplaceonwards thisiscertainly thecase 68Chapter Jr.Sequences otrealnumbers. Tofacilitate theunderstanding ofcertaincaseswhichfrequently occur,wefinallyintroduce thefollowing furthermodeofexpression: oDefinition 5.Ittwosequences (xn)and(Yn)'notnecessarily con­ vergent. aresorelatedtooneanother thatthequotient Xn y" tends,torn---I-00.toadefinite finitelimitdifferent from zer08•thenweshallsaythatthetwosequences areUl,ymploti cally prop01'lional andwritebriefly Itinparticular thislimitis1.t/zenwesaythatthetwosequences are asymptotically equalandwrite.moreexpressively x..~y... Thusforinstance V;~2+i.'Vn,log-(5nil+2:3)""logn,v/n+1-Vu""-~,Vn 1+2+.,.+ n""n",11+22+'.,+n"~_!n3• Thesedesignations areduesubstantially toP.duBOls-Reymolld (Annali dimatematica puraedapp!.(2)IV,p.338,1870/71). Tothesedefinitions wenowattachaseriesofsimple,butquite fundamental Theorems onconvergent sequences. 41. °Theorem 1.Aconvergent sequence determines itslimitquite uniquelyD, Proof.Ifx.._~,andsimultaneously xn-t. then(xn-~)and (xn-f)arenullsequences. By28,2, ((Xn-~)-(x"-$'))=W-~) ISthenalsoanullsequence, i.e.~=f,q.e.d.10 8x"andYnmustthennecessarily be=F°tramsomeplaceonwards. This isnotreqUIred foreverynintheabovedefinition. SAconvergent sequence therefore defInes (determines, gIves...)its limitquiteasuniquely asanynestofintervals orDedelilnd sectiondefinesthe number towhichitcorresponds. Thusfromthispointwemayconsider areal number asgIVenifweknowasequence convergmg toit.Andasformerly we saidforbrevitythatanestofintervals(x,,!Y,,)oraDedekind section (AIB) oraradIXfraction isarealnumber, sowemaynowwithequalrightsaythat asequence (x,,)converging toEistherealnumber;, orsymbolically: (x,,)=;. Forfurtherdetailsofthisconception, whichwasusedbyG.Cantortoconstruct histheoryofrealnumbers, seepp.71)and95. 10Thelaststepinour rea~oninl!, bywhichtherel\dermayatfirstsight betakp.naback,amounts simplytothIS:Ifwithrespecttoadefinite numerical value" weknowthat,foreveryE>0,wealwayshaveI"1<B,thenWfl ~M.Convergent sequences. oTheorem 2.Aconvergent sequence (xn)isinvariably bounded. And ifIXnI<K,thenforthelimitgwehave11IgI~K. Proof.IfXn-+g,thenwecan,given E>0,assignanumberm, suchthatforeveryn>m g-E<Xn<g+E. Iftherefore K}isanumber greaterthanthemvaluesIXlI,Ix21,•.., IXmI,andgreaterthanIgI+E,thenobviously IXn1<KI foreveryn.NowletKbeanyboundofthenumbersIX..I.Ifwehad IgI>K,thenIgI-K>0andtherefore, fromsomeplaceonwards inthcsequence,IgI-IXnI:-:;;IXn-gI<IgI-K andthereforeIXnI>K,whichiscontrary tothemeaning ofK. oTheorem 2a.Xn-+gimpliesIXnI->IgI. Proof. Wehave(v.3,H,4) 11xnI-IgII~Ixn-gI; therefore (IxnI-I~I)isby26,2 anullsequence when(xn-~)is. oTheorem 3.Ifaconvergent sequence (xn)hasallitstermsdifferent fromzero,andifitslimitgisalso=1=0,thenthesequenceG)isbounded; n orinotherwords,anumbery>0exists,suchthatIXnI>y>0forevery n..thenumbersIXnIpossessapositivelowerbound. Proof. Byhypothesis, 11~I=E>0,andthereexistsaninteger m,suchthatforeveryn>m,IXn-gI<EandthereforeIXnI>!I~112• Ifthesmallest ofthe(m+1)positive numbersIXlj,Ix21,...,IXmI andtIgIbedenoted byy,theny>0,andforeveryn,IXnI>y. I}I;;:::K=!,q.e.d. Xn Y If,givenasequence (xn)converging tog,weapplytothenullse­ quence(xn-g)thetheorems 27,1to5,thenweimmediately obtain thetheorems: necessarily haveex=O.For0istheonlynumber whoseabsolute valueislessthan everypositIve E.(InfactI0I<Eistrueforevery E>O.ButIfex'4=0,sothat 1exI>0,thenIexIiscertainly notlessthanthepOSItive number E=iIexI.)Simi­ larly,ifweknowofadefinite numerical value exthat,foreverye:>0,wealways have Cl<~K+E,thenwemusthavefurther Cl<~K.Themethod ofreasoning involved here:"IfforeveryE>0,wealwayshaveIexI<e:,thennecessarzly Cl<=0" isprecisely thesameaswasconstantly applted bytheGreekmathematicians (cf. Euclid,Elements X)andlatercalledthemethodofexhaustion 11Herethesignofequality in"I~I~K"mustnotbeomitted, evenwhen, foreveryn,IXnI<K. 12Forn:---rn,allthe:lCn'saretherefore necessarily '4=O. 70 Chapter H.Sequences ofrealnumbers. 0Theorem 4.If(x,,')isasub-sequence ot(x,,),then implies x,,'-~. °Theorem 5.Itthesequence (x,,)canbedivided intotwosub· sequences 1.1otwhicheachconverges to~,then(x,.)itseltconverges to~. oTheorem 6.It(x,,')isanarbitrary rearrangement otx"'then implies oTheorem 7.Itxn-~and(x,,')resultstrom(x,,)byatinite number otalterations. thenxn'-~. °Theorem 8.Itx,,'-~andx,,"-~,anditthesequence (x,.)is sorelatedtothesequences (x,,')and(X,,")thatfromsomeplaceonwards, (i.e.toreveryn::::::m,say.) x,,'Sx"~x,,", thenx"-I;. Calculations withconvergent sequences arebasedonthefollowing fourtheorems: oTheorem 9.x"-~andy"-r;alwaysimplies (x"+y,,),""" ~+r;, andthecorresponding statement holdstortermbytermaddition otany tixednumber -S(lYP-atconvergent sequences. Proof.If(x"-~)and(Yn-r;)arenullsequences, thenso,by 28,1,is(x"+Y,.)-(~+'7))'Inthesameway,28,2givesthe °Theorem9a. x,,_~andY,,-r;, alwaysimplies(x"-y,.)-~- r;. oTheorem 10.x"-~andY"-'7,alwaysimplies x"y"-~r;. andthecorresponding statement holdslortermbytermmultiplication 01anylixednumber -sayp.-ofconvergent sequences. Inparticular: x,,_~ implies cx,,-c~, whatever numberp denote. Proof. Wehave x"y"-I;r;=(x"-~)Yn+(y"-r;H; andsincehereonthefighthandsidetwonullsequences aremulti­ pliedtermbytermbybounded factorsandthenadded, thewhole expression isitselfthetermofanullsequence, q.e.d. 0Theorem 11.x,,-~andy,,-'Yjalwaysimplies, ifeveryx,,=!=O andalsoI;=l=0, Y.. 1'/---+-. X" ~Proof. Wchave toOrthree.oranydefmite number. §8.Convergent sequences, 71 Herethenumerator, forthesamereasons asabove,represents anull 1sequence, andthefactors-/:-are,bytheorem 3,bounded. Therefore ".i-Xn thewholeexpression isagamthetermofanullsequence. -Only aparticular caseofthisisthe oTheorem 11a14.x,.-rgalwaysimplies,ifeveryx,.andalsoeare =t=0, Thesefundamental theorems 8-11lead,byrepeated application, tothefollowing morecomprehensIVe oTheorem 12.LetR=R(X(l),X(2),X(31,•••,x(P))denoteanex­ pression builtup,byafinitenumber ofadditions, subtractions, multi­ plications, anddivisions, fromtheletters X(l),X(2),, , "xIP),andarbttrary numerical coefjicients F,..andlet (X~I»),(xi:»),••.•(x;:'») bepgivensequences, converging respectively to~(l).~(2),•••,.;(p).Then thesequence ofthenumbers R..=R(x~), X~2),•••,x;;I)-+R(~(l), ~(2),•••,~(p») provided neitherintheevaluation ofthetermsR",norinthatofthe number R(';(1),.;'2),.•",;(p)),division by0isanywhere required. Thesetheorems giveusallthat Ii'>required fortheformalmani· pulation ofconvergent sequences: \Vegiveafewmore Examples. 1.XII--~implics,ifa>0,nlvanably, For :r tt(T_t )a"-a'=a""an""_1 (sanullsequcnce by3:-i,3 2,XII-+i;Implies, Ifcvery ,ellandalsoI;lIrc>0,that logXII--logS. Proof. Wehave 1XII (Xn-~)logX..-logI;=og1-=log'1+--r- h'hb3S7' 11'>0' I'xn-/;-... 1 wICY , ISanusequence, since XII Imples---r./-. 14Intheorems :1,11andlIa,itissufficient topostulate thatthelimitof thedenominators is'*0,forthenthedenommators arc,fromsomeindex t1lon­ wards,necessarily'*0,andonly"afimtenumber ofalteratIOns" needbemade, orthenewsequence needonlybeconsidered forn>m,toensurethiSbemgthe caseforall, 15Moreshortly: arational function ofthepvariable~ :e(l"\:C'",•••,XCp)with arbitrary numerical coeffi,'\,·nts.42. 72 Chapter H.Sequences ofrealnumbers. 3.Underthesamehypotheses asin2.,wealsohave,forarbitrary reale. Proof. Wehave whichby38.8isanullsequence 18,since~n_~~> -1andtcndsto0asn-+00. (ThiS IStoacertamextentfurthercompleted by35,4.) Cauc1ly's theorem oflimitsanditsgeneralisations. Thereisagroupoftheorems onlimits1?essentially morepro­ foundthantheabove,andofgreatsignificance forlaterwork,which originated intheirsimplest formwithCauchy18andhaveinrecent timesbeenextended indifferent directions. Wehavefirstthesimple 43. °Theorem 1.It(xo'xl'...)isanullsequence, thenthearith- meticmeans IXo+X,+...+XnxR= n+1 ' "=0,1,2,..., alsoformanullsequence. Proof.If8isgiven>0,thenmcanbesochosen, thatfor everyn>mwehaveIxnI<;.Forthesen's,wethenhave Ix'I~LXI)+Xl+...+XmI+-!....n-m. ..- n+l 2n+l Sincethenumerator ofthefirstfraction ontherighthandsidenow contains afixednumber, wecanfurther determine "0'sothatfor n>nothatfraction remains<;.Butthen,foreveryn>no'we haveIxR'I<E,-andourtheorem isproved. -Somewhat more general, butnevertheless animmediate corollary ofthis,isthe 0Theorem 2.IfXR-+~,thensodothearithmetic means x'==0+X,+...+Xn-+.;. R n+l I~Examples 1.to3.mean-inthelanguage ofthetheoryoffunctions ­ thatthefunction aZiscontinuous ateverypoint,thefunctions logxandxl! ateverypositive point. UThereadermaydeferthestudyofthesetheorems until,inthelater chapters, theycomeintouse. 18Augustin LouISCauchy, born1789inParis,died1857inSceaux. In hisworkAnalyse alg~bl'lque. Paris1821(German edition, Berlin1885,Julius Springer) thefoundations ofhigheranalysis areforthefirsttimedeveloped Withfullrigour,andamongthemthetheoryofinfiniteseries. Inwhatfollows weshallfrequently havetorefertoit;theabovetheorem 2maybefound OD p.59ofthattreatise. §8.Convergent sequences. Proof. Bytheorem 1,73 ISanullsequence when(xn-~)is,q.ed. Fromthistheorem, thecorresponding oneforgeometric means nowfollows quiteeasily. Theorem 3.Letthesequence (Yl'Y2'.•.)-'fJ,andhaveallits members anditslimit'fJpositive. Thenalsothesequence0/geo­ metricmeans Proof. FromYn-fj,sinceallthenumbers arepositive, we deduce, by42,2,that xn=logYn-...~=log"I' Bytheorem 2,itfollows that ,Xl+Xg+...+XnI-:'1I' Ixn=----,.-=ogVYlY2'"Yn=ogYn-ogfJ· By42,1,thisatonceprovesthetruthofourstatement. Examples. I.1because --.O.n s._3_ ft_l+y2+V 3+...+yn----------- -.1, ,."_becauseVn-.1. 4.Because (l+~)n_e(v.46ain thenext§),wehavebytheorem S, V(~)~~r. (~y...(n~\.lr =V~~llt= :+1also_e ynl or,therefore, 1"- 1_,In!--nV e ' "- narelation whichmayalsobenotedintheform"ynI'"-".-" 74 ChapterIt,sequences ofrealnumbers. Let(Xo'Xl'...)beanullsequence andsuppose ofthesystemEssentially morefar-reaching, and following generalisation ofCauchy's p.Toeplitz1ll: ~°Theorem 4. thecoefficients a""yetaseasilyproved, isthe theorems 1and2,dueto aou aiOall (A)a~o a~l a~';! \~":'anian2 ann.... satisfythetwoconditions: (a)Everycolumn contains anullsequence, i.e.forfixedP>° anp-O whenn-+oo. (b)Thereexistsaconstant K,wchthatthesumoftheabsolute valuesofthetermsinanyone row,i.e.,foreveryn,thesum Ia"oI+Ia"lI+...+IannIrematns<K.. -Thenthesequence formedbythenumbers +annx" tSalsoanullsequence. Proof. If815given>0,determine tit~othatforeveryn>m IxnI<2~'"Ihenforthosen's, Ixn'I<IanoXo+...+a"".x",I+-~-. Bythehypothesis (a),wemaynow(asmisfixed)chooseno>m, 50thatforeveryn>no'wehave1£1,,0Xo+...+an",X.;.\<~.Smce forthesen'sIx,,'Iisthen<8,ourtheorem isproved. Inapplications itisusefultohavethefollowing °Complement. If,forthecoefficients a"",aresubstituted other numbers a~"=a""'''"",obtained fromthenumbers a".tbymultiplication 19Cl/Itchy's Theorem 1hasbeengenerahsed inseveral ways,inparticular byJ.L.W.V.Jensen(OmenSiitmng afCauchy, Tldsknft forMathematlk, (5) Vo!.2,pp.81-84. 1884)andO.Stolz(GberemeVerallgemeinerung einesSatzes vonCauchy, Mathemat. Annalen, Vo!.33,p.2:n.1889).Theaboveformulation, duetoO.Toeplttz (Gberhneare Mlttelblldungen, Pracematematycznofizyczne, Vo\.22,p.11:l-119. 1911),isinacertamsenseafinalgeneralisation, forthisreason thatItshows(I.c.)theconditions, recogmsed mTheorem 5assuffiCient, tobe alsolIece\\ary, for'"n•ttlJImply."0:,,'.,tInallcases(cf.221,andtheworkofI. Sthur:OberlmeareTransformatlOnen mderTheone derunendlichen Reihen, Jour. f.d.reineu.angew.Math.,Vo!.151,pp.70-111. 1920). §8.Convergent sequences. 75 byfactors cc,,!.-allinabsolutevaluelessthanafixedconstant a,­ thenthenumbers x::=a;oXo+a~lXl+..,+a~nX" alsoformanullsequence. Proof The a~!.'salsosatisfytheconditionS' (a)and(b)of thp.orem 4;for,Ifpisfixed, a~p--0by26,1,andthesums Ia~oI+Ia;l!+...+Ia~nfremain<K'=,aK. FromTheorem 4wemaynowdeducethe...../l...u:.. 0Theorem 5.If:1.:,,_.;,andthecoefficients al,,,satisfy,besides theconditions (a)and(b)ofTheorem 4,thefurthercondition (c) a,,0·+an1+...+a"n=A,,-l,'o thenalsothesequence formedbythenumbers Proof. \Vcnowhave whence ourstatement atoncefollows, inconsequence ofcondltIon (c), bytheorem 4. Beforegivingexamples andapplications oftheseimportant theorems, wemayprovethefollOWing furthergenerahsation, whichPOlIltsina newdirection. °Theorem 6.Ifthecoefficients al,,,ofthesystem(A)satisfy, ,I' besidestheconditions (a),(b)and(c)mentioned inTheorems 4and0, thefurthercondition, that (d)thenumbers ineachofthe"diagonals" ofAformanull sequence, i.e.forfixedp,ann-p-0whenn-+00, thenitfollowsfromXn~.;andYn-r;thatthenumbers Proof. Since X"Yn-"=(x"-~)Yn-,'+~.Yn-", wehave " "21,.=2Jan,'Yn-.(x"-~)+$.2,'an,-Y,,-,.. 'J'::::f) ...=0 10IntheapplicatIOns. weshallgenerally haveAll=1. 11Forpositiveal,,,,thistheorem maybefoundinapaperbytheauthor "OberSummen derFormaobN+a1bll_1+...+aNbo"(l<cnd.delcircolomat. diPalermo, Vol.32,p.95-110. 1911). 76 Chaptern.Sequences ofrealnumbers. Herethefirstsumtendstozero,byTheorem 4anditscomplement, for(xp-~)isanullsequence andthefactors Yn-parebounded. Andifthesecondsumbewrittenintheform n n ~.2)ann-pYp==~.~'a~"Yp,,=0 ~ ,.=0 wesee,bytheorem 5,thatthis,andthereby alsozn'tends -+-~17; forthenumbers a~p=ann-psatisfy,inconsequence of(d),precisely thecondition (a)therest;pulated. 44. Remarks, applications andexamples. 1.Theorem 1isaparticular caseofTheorem 4;weneedonlyput,in thelatter, (n=0,1,2,...) Theorem 2isderived inthesamewayfromTheorem 5.Theconditions (a),(b),(c)arefulfilled. 2.If"0'".,...areanyposlhve numbers, forwhichtheSlIms "0+"1+...+a"=Cl.-+00, itfollows 22fromx"~~that ,aDXo+a,Xl+...+a"Xn Xn= also-E. aO+"I+'''+'''' Infact,weneedonlyput,intheorem 5I {n=0,1,2,. v=0,1, ,11 (b),(c)areflllfJlled.apanv=­ 0" toseethatthestatement iscorrect. Theconditions (a), -Fora.==1,weagainobtamTheorem 2. 2a.Thetheorem ofno.2.remains truefor;=+00or;=-00.The sameremark holdsforthegeneral theorem 5,provided alltheapp'sare~0 there.ForifXn_+00and,asintheproofofTheorem 4,mbesocho&en, givenG>0,thatforeveryn>mwehavex.> G+I,thenforthosen's wehave x,,'>(G+I)(a'm+1+...+a".)-anoIXo1-'"-a"mIx,.I. Inconsequence oftheconditions (a)and(c)inTheorems 4and5,wemay therefore sochoose nothatforeveryn>nowehavex,,'>G.Hence x,,'-++00. . U:l.Insteadofassuming the"',,'spositive anda"~+0Ci,itsuffices[hy(h)] toreqUIreonlvthatI"'0I+I""I+...+I"'"I~+0Ci,withtheproviso, however, thataconstant Kexists,suchthat23foreveryn I"'0I+I"'1I+...+I"'",;SK·'"'0+"'1+...+"'"I. (Forpositivea.,K=1givesallthatishererequired.) 2~O.Stolz,lococit.-Ofcourseitalsosuffices, thatthea,,'sbetram somePOintonwards ~0,provided only0"-++00.Thex,,"smllstthenbecon­ sidered fromthatpointonwards, afterwhich Cl"is>O. 23]ensen,lococit.-Ifa",isthefirstofthe,,'stobeof0,thenthex.'·s aredefmed onlyforn~m. §8.Convergent sequences. 4.Ifin2.or3weput,forbrevity, c(..x"=3'..,thenweobtain:77 3'0+Y1+...+Y.. ~ '-"--'--..:~'-----,.--'--..: -s-, eto+et1+",+et..provided andprovided theetn'Ssatisfytheconditions givenin2.or3. 5.Ifwewritefurther Yo+Yl+...+Yn=Yn,andeto+et1+...+et"=A.., thenthelastresulttakestheform: Y"_~,provided An andprovided thenumbers etn=A"-An-1(n>I,eto=Ao)satisfytheconditions givenin2.or3. 6.Thuswehave,forinstance, by5.: !im1+2+.,.+n=lim n=lim__n_=_.':.. n'9 n2-(n-l)" 2n-12' Similarly wehave limP+22±--=-:..:..+n2=lim ..n2 ..1 n3 n3-(n-l)3=3' andgenerally I"IP+2P+, ..+nPl' nP un =Im-,--;-----...,-,,-nP+1 nP+1_(n_l)p+l . nP1=hm =--, (p+l)n P-(Ptl)nP-1+ ...p+l ifPdenotes apositive integer. 7.Similarly wefind,ifweanticipate theproofin46aoftheconvergence ofthesequence ofnumbers(1+~)"1.+1! log1+log2+..,+logn=lognI_1 nlogn lognn ' 8,Thenumbersi.e.lognI""lognn. {n=0,1,2,... v=O,I,...,n fulfiltheconditions (a),(b)and(c)ofthetheorems 4and5iforifpbefixed, 2"p_0,seeingthatitis la"ol+···+la" ..I=a"o+···+a",,=I,whileandtherefore (v.3S,2) f01everyn.Therefore Xn_~always implies Xo+(7)Xl+(;)x2+..,+ (:)X" " -E·2 78 Chapter If.Sequences ofrealnumbers. 9.ThesamespeciallsatJons asweregivenin1.,2.,3.and8.fortheorem I) mayofcourse alsobeapphed totheorem 6.Wemerely mention thetwo follOWIng theorems: (a)Fromx"-~andY"-+YJitalways follows that Xoy,,+X'Y"-1+X2Y"-2+ ...+x"Yu ~ -----~- --n+l- ----:;'1' (b)If(x.)and(y,,)arctwonullsequences. thesecond ofwhIchfulfils theextracondItion thatforeveryn IYoI+IY,I+...+IY"I remains le~sthanafixednumberK.thenthenumbers formanullsequence. (Fortheproofwcputanv=Yn-vintheorem 4.) 10.Thereader WIllhavenoticed thatItISInnowiseessential thatthe rowsofthesystem(A)oftheorem 4shouldbreakoffexactly atthenthterm. Onthecontrary, these lOWSmaycontaIn anynumber ofterms. Indeed, after wehavemastered thefIrstprinCIples ofthetheoryofinfmite series,weshall secthattheserowsmaycontam evenanmfmity ofterms(a"0'a"".."an.'I•••), provided onlytheothercondItions Impo!>ed allthesystem befulfilled. The theorem hereby indIcated WIllbeformulated andproved in221. §9.Thetwomaincriteria. Wearenowsufficiently prepared toattacktheactualproblems of convergence. Therearctwomampointsofviewfromwhichwe propose, inwhatfollows, toexamine thesequences whichcomebefore us.Wehaveabovealltoconsider the Prob Ie mA.Isagivensequence (x,,)convergent, ordefinitely orindefmitely dtvergent? (Briefly: Howdoesthesequence behave withrespect toconvergence?) -Andifasequence hasploved to beconvergent, sothattheexistence ofalinlltmg valueisensured, wehavefurther toconSIder the Problem n.Towhatlimit; doesthesequence (xn),recognized tobeconvergent, tend? AfewexampIesmaymakethesignificance oftheseproblems clearer:Iffol'instance wearegiventhesequences (1+22+ 33+...+n") n2 J(1+1_+...+-.!_)2 n 1'etc.ogn examination oftheirconstruction showsthattherearealwaystwo(01 more)forceswhir.hhere,sotospeak,opposeoneanotherandthereby callforththevariation oftheterms. Oneforcetendstoincrease, §9.Thetwomaincriteria. 79 theothertodiminish them,anditISnotc1e.lfataglancewhichof thetwo"illgettheupperhandorinwhatdegreethiswillhappen. Everymeanswhichenables ustodecidethequestion ofconvergence ordivergence ofagivensequence, wecallacriterion ofconvergence orofdivergence; the~eserve,therefore, tosolvetheproblem A. Theproblem 13isingeneral muchmoreclIfficult. Infact,we mightalmostsaythatitisinsoluble, -orelse1<;trivial.Thelatter, because aconvergent sequence (xJ,bytheorem41,1,entlrely deter­ minesitslimit ~,whichmaytherefore beregarded as"given" bythe sequence itself(cf.footnote to41,1).Onaccount, however, ofthe boundless complexity andmultiplIcity offormwhichsequences show, thisconclu~ion doesnotseemverysatisfactory. Weshallwish,rather, nottoconsider thelImit ~as"known", untilwehavebeforeusa Dedekind section, orstillbetteranestofmtervals, forinstance aradix fraction, inparticular adecimal fraction. The<;elatterespecially arethe methods ofrepresenting arealnumber withwhichwehavealwaysbeen mostfamiliar. Ifwcregardtheproblem inthis!lght,wcmaycaB itthequestion otnumerical calculation otthelimit!. Thisquestion, oneofgreatpractical significance, isusually in theoretical considerations ofverysecond-rate imparlance, forfroma theoretical pointofview,allmodesofrepresentatIOn forarealnumber (nests,sections, sequences, ...)arepreCIsely equivalent. Ifweobserve further, thattherepresentation ofarealnumber byasequence may beconsidered asthemostgeneralmodeofrepresentation, ourproblemB maybestatedinthefollowing form. Problem B'.Twoconvergent sequences (xll)and(x~)aregiven,­ howmaywedetermine whether ornotbothdefinethesamelimit,or whether ornotthetwolImitsstandinasimplerelation tooneanother? Afewexamples WIllservetoillustrate thekindofque~tion referred to: 1.Let(1)"'(rx)" X,,=1+-;; andXn=1+-;; .4:>. Bothsequences arequiteeasIlyCv.46aand111)5eentobeconvergent. Butitisnotsoapparent thatifI;denotes thelimitofthefirstsequence, that ofthe sccond is=gx. 2.Giventhesequence 1 3 7 1741T'-2-'5'-12-'2\l' inwhichthenumerator ofeachfraction isformed byaddingtwicethenume­ ratorofthelastfraction preceding tothenumerator ofthelastfraction but one(e.g.41=2·17+7),andsimilarly forthedenominators. -Thequestion of 1Numerical calculation ofareal nl1mb~r=representation ofthatnum­ berbyudecimal fraction. Forfurther details,scechapter VII!. 80 Chaptern.Sequences ofrealnumbers. convergence againgivesnotrouble, nordoesthenumerical evaluation ofthe limit,-buthowarewetorecognise thatthislimit=l'2? S.LetZn=(l-~+~-~+' ..+~_~n-l)3 5 7 2n-l(11=1,2,... andletx~betheperimeter oftheregular polygon withnsidesInscribed in thecircleofradius1.Herealsobothsequences areeasilyseentobecon­ vergent.If;and;'aretheirlimits,-howdoesoneseethathere;'=8.~? Theseexamples makeitseemsufficiently probable, thatProblem B orB'isconsiderably hardertoattackthanProblem A.Wetherefore confine ourattention inthefirstinstance entirely tothelatter,andto beginwithmakeourselves acquainted withtwocriteria, fromwhich allothersmaybededuced. Firstmaincriterion (formonotone sequences). 46. Amonotone bounded sequence isinvariably convergent,' amono- tonesequence whichisnotbounded isalways definitely divergent. (Or,therefore: Amonotone sequence alwaysbehaves definitely, and ISthenandonlythenconvergent, whenitisbounded, andthenand onlythendivergent, whenitisnotbounded. Inthelattercasethediver­ genceistowards+00or-00according asthemonotone sequence is ascending ordescending.) Proof. a)Letthesequence(xn)bemonotone ascending andnot bounded. Sinceitisthen(because xn~Xl)certainly bounded on theleft,itcannotbebounded ontheright;givenanyarbitrary (large) positive numberG,thereisthenalwaysanindexno'forwhich xno>G. Butthen,sincethe.sequence ismonotone increasing, wehavefor everyn>no'afortiori, xn>G,andso,byDefinition 40,2,actually xn-.+00.Interchanging rightandleft,weseeinthesameway thatamonotone descending sequence whichisnotbounded must diverge to-00.Thusthesecondpartoftheproposition isalsoproved. b)Nowlet(xn)beamonotone ascending, butbounded sequence. ThereisthenanumberK,suchthatIxnI~Kforeveryn,sothat Xl<xn~K foreveryn.Theinterval11=Xl.•.Ktherefore contains alltheterms of(x•.);tothisinterval weapplythemethod ofsuccessive bisection: Wedenotetherightorthelefthalfof11byJ'J'according asthe righthalfdoesordoesnotstillcontain pointsof(xn).From1'Jwe selectonehalfbythesamerule.andcallthis J~;andsoon.The intervals ofthenestsoconstructed havetheproperly'J, thatnopoint •Thereadershouldillustrate thecircumstances onthenumber-al[i». §9.ThetwomaIncriteria. 81 ofthesequence liestotherightofthem,butatleastoneliesinside eachofthem.Orinotherwords: thepointsofthesequence (while monotonely progressing towards theright)penetrate intoeachinterval, butdonotemerge fromitagain;ineachoftheseintervals, therefore, allpointsfromacertain indexonwards cometolie.Wemaythere. fore,ifwesUIJpose thenumbers nI'n2,•••properly chosen, saythat: Inlklieallx,,'swithn>nk,buttotheright01lklieno morex,,'s. If~isnowthenumber determined bythenest(I,.),itcanat oncebeshewnthatx"_~.ForIf8isgiven>0,choosetheindexp sothatthelengthoflpislessthan8.Forn>np'allthex.:slie, together with1;,inJ:,sothatforthesen'swemusthavep Ix"-1;1<8. (x"-1;)istherefore anullsequence, andx"-1;,q.e.d. Byasuitable interchange ofrIghtandleft,weseethatmonotone descending bounded sequences mustalsobeconvergent. Thusevery partofthetheorem isproved. Remarks andExamples. 1.Wefirstdrawattention againtothefactthat(cf.41,1)evenwhen Ix"I<J(,wemayhaveforthelimiting value ~theequalityI~I=](. 2.Let (n=I,2, ...). As 1 1 1 1 1 x"+1 -X..=2n+f+2n+2--n+1=2n+1- 2 n+2>0, thesequence' ismonotone increasing, andasx"<n._1_<I,itisalsobound­-n+l ed.Itistherefore convergent. Ofitslimit ~weknownomore,sofar,thanthat x,,<~-:;:; 1 37foreveryn,which t'.g.forn=3becomes 60<:~<1.Whether ithasara- tionalvalue,orwhether, bearsacloserelationtoanumber appearing inanyother connectIon -inshort:ananswertoproblem B -cannotherebeperceIved at once.LateronweshaHseethat,isequaltot~enaturallogarithm of2.I.e.the loganthm of2whosebaseisthenumber eintroduced in46abelow. 3.Letx"=(1+{+}+...+~),sothatthesequence (x..)ismonotone increasing (cf.6,12).Isitbounded ornot?-IfGisgivenarbitrarily> 0, chosem>2Gjthenforn>2m x>(1+..!..)+(~+..!..)+(1+...+..!..)+...+(_1_+ ...+~)" 2a4 5 :::; ·2m-1+1 2m 1 1 4 1 8 1 2 m-11m >-+2.-+ ._+._+...+.---=->G.24!:l16 2m2 Thesequence istherefore notbounded andconsequently diverges .....+00. 82 Chapter 11.Sequences ofrealnumbers. 4.If0=(XnIy,,)isanarbitrary nestofintervals, theleftandrightcnd· pointsofthemtervals respectively formtwomonotone, bounded andtherefore convergent sequences. 'Vethenhave limT"=hmy"=(x"IYn)=a• !lOa. Asaparticularly important example, wewillconsider thetwc' sequences whosetermsare -7~'\"'"1 )'\.•\kML' .,.)i". L. 1-", -\.,. xn=(1+!_)nantiY..=(1+.!.)n+l. (n=1,2,3,...)n ~I;I. n _, t- o 'r",',- ..J.JJ;"~' "- Wehavenomeansofperceiving immediately (cf.thegeneralremark onp.78)howthesequences behaveasnincreases. Wcproceed toshowfirstthatthesecondsequence ISmonotone descending, thatistosaythatforn:2::! Thisinequality isIIIfd.ctequivalent 3to (l)nl+n_11--->1+- 11 n+-n orto (n9)n 1-->1+-n2-In'i.e.to(l)n 11+---,->1+-.n'-1 n Butthetruthofthisinequality iseVIdent, since,byBernoulli's m· equality10,7wehave,fora>-1,a+0andeveryn>1. (1+at>1+na, orinpartIcular (l)nn n 11+-.,-1>1+\i-I>1+--;;=1+-.n"- n- n" n As,moreover, Yn>1foreveryn,thesequence (Yn)ismonotone des­ cending andbounded, andtherefore convergent ItslimItWIllofter occurlateron;itis,sinceEuler'stIme,denoted bythespecial4lettere. Asregards thisnumber, wecanonlydeduceforthepresent that 1<e<Y.. whichfore.g.n=5becomes •Thatistosay,eachinequaltty followsfromalltheothers. 4Rl/ler use~ thi~lettertodeSignate theaboveItmltmalettertoGoldbach (:!:i.!\IO\-.Ii:H)andm17:itimhiSwork:Mechanica SIvemotusscientm analyllce eXposlta, 1I.p.~51. §9.Thetwomaincriteria. 83 Thefirstofourtwosequences, onthecontrary, ismU1/otone asctnding. Infact,xn-1<x"heremeans G or(1)"--1(I)" 1+---- <1+n-J n (1+1 )" 1-1 11 (1+n-----1)<1--'--1- , 11-I I.e. But,againby10,1_1..--(112-I)n=(1__1)" 11-n2 112• 7,wehaveactually foreveryn>I, (1-IQ)n>1-~,=1_I . 11" 11" n Thesequence (x,,)istherefore monotone increasing. As,inanycase, (1+~r<(1+~r+l,I.e.x"<)'no wehave,foreveryn,x"<Yl'i.e.(x,,)isalsobounded andhencecon­ vergent. As,finally,thenumbers Yn-xn=(1-I-!)11•(1+1-1)=-01.Xn 11 11 11 areallpositive and(by26,1)formanullsequence, weconclude atonce that(xn)hasthesamelimitas(y,.).Thus limXn=limYn=e. Andforthisnumberewchavefurthermore, ashasappeared intheproof,in e=(x"IY,,)=-.((I+~r1(1+~rll) anestofintervals defining it.(Itprovides, forinstance takingn~3, theinequality -~}<e<2H5t";weshallhowever become acquainted later on(§23)withothersequences converging toe,whicharcmoreconvenient fornumerical calculation.) Thisisthenumher ethat(cf.p.fJ8)formsthebaseofthenatural logarithms. Weshallaccordingly agreetousethesymbol logtomean thisnaturallogarithm tothebasee,unlcssthecontrary isexpressly stated. Thefruitfulness ofthefirstmaincriterion isdueabovealltothe factthatitallowsustodeduce theconvergence ofasequence of numbers fromveryfewhypotheses, andthesesuchasareusuallyvery easytoverify-namely, frommonotony andhoundedness alone.On theotherhand,however, itstilIrelatcsonlytoaspecial, eventhough particularly frequent andimportant kindofsequence, andtherefore &Cf.footnote 3. 84 Chaptern.Sequences 01realnumbers. appears theoretically insufficient. Weshalltherefore a"kforacriterioll whichenables ustodecidequitegenerally astotheconvergence or divergence ofanysequence. ThIsisaccomplished bythefollowing 47. 0Secondmaincriterion (18tform). Anarbitrary sequence (xn)isconvergent itandonlyit,given 8>0,anumber no=no(8)canalwaysbeassigned, suchthattorany twoindicesnandn'bothgreaterthanno'W3haveineverycase Ixn-x...1<E.- Wefirstgiveafew Explanations andExamples. 1.Theremarks 10,1,3,4and9arealsosubstantially applicable here; andthereaderisrecommended toreadthemthrough oncemoreinthiscon­ nection. 2.Thecriterion states-toputitinintuitive language: allx,.'swith veryhighindices must!leveryclosetogether. 3.LetXo=0,x,=1,andleteverytermalterthesebethearithmetic meanbetween thetwotermswhichprecede it,i.e.forn~2 X,._l+xo_.x.=2- sothatx.=t,X3=t,x.=~-,....InthiSevidently notmonotone sequence it isclcar,ontheonehand,thatthedifferences bctween consecutive termsform Rnullsequence; foritmaybeverificd quiteeasilybyinductIOn thatU andsotendstoO.Ontheotherhand,between thesetwoconsecutive numbers allthefollowing oneshe.Iftherefore, after BhasbeenassIgned> 0Iwe 1choosepsolargethat--<BIwehave 2P Ixn-xn'l<. provided onlynandn'are>p.Bythe2ndmamcriterion thesequence (xn) istherefore convergent. Thelimit ~alsohappens tobeeasilyobtainable. Alittle reflection infactleadstothesurmisethat;=j-.Inpomtoffact,theformula 2 2(_1)0+1 Xn-3=lr-~ canimmediately beprovedbyinduction andshowsthatxn-iisactually a nullsequence. Beforetryingtofathom themeaning ofthe2ndmaincriterion further, weproceed togiveits Proof.a)Thatthecondition ofthetheorem -letuscallitfor brevityits8-condition -isnecessary, i.e.thatitisalwaysfulfilled 8 0 Xk+1 -XkXk-Xk-1Thisistrueforn=and1.FromXk+s-Xk+1=--2--+-"2--- itfollowsthatifprovedforeveryn:;;;;k,itistrueforn=k+1. §9.ThetwomalOcriterIa. 85 12k'•••thenweget:it(x,,)isCOll\ergent, is!oeenthus:If::rll--"~' then(x"-~)i"anull !ocquence jgivene>0,wecan!oochoose nothatforeveryn>no' Ix"-~Iis<-~.Ifbesidesn,wealsohaven'>no'thenI::rn'-$\ isalso<i.andso Ix"-Xn'I=I(xn-;)-(xn'-;)I< Ix"-;I+IXn'-;I<i+i=e, whichprovesthispartofthetheorem. b)Thatthee·condition isalsosufficient isnotsoeasytosee Weagamproveitconstructively, bydeducing fromthe~equence (x,,) anestoflIltervals(IJandthenshowing thatthenumber determined thereby isthelimitofthesequence. Thisisdoneasfollows: Anye>0beingchosen,Ix"-3'",1mustalwaysbe<eprovided onlytheindicesnandn'bothexceed somesufficiently largevalue. Ifwesupp0'ie theonefixedanddenote itbyp,thenwemayalso say:Givrnanye>0,wecanalwaysassignanmdexP(actually, as fartotherightaswcplease) sothatforeveryn>P Ix"-a:p!<e. 1 1 e=~,4'..., Ifwechoose successively 1)ThereisanindexPIsuchthat foreveryn>P1'wehave suchthat >Pk-l'suchthat 1Ix"-xPkl<2k'2)Thereisanindex P~,\\hichwemayassume> PI' foreveryn>P'J'wehaveIx"-xp,l<{" andsoon.A kthstepofthiskindgives: k)ThereisanindexPI,'whichwemayassume foreveryn>Pk•wehave Accordingly wcformtheintervalslk: 1.Theinterval xp,-~•••xp,+~call11;itcontains allthex,,'s forn>PI'inparticular, therefore, thepointxp,'Ittherefore contains inwholeorpartthelI1terval xp.-i...xp.+hinwhichallx,,'s withn>P2lie.Asthesepointsalsoliein11,theyheinthecommon partatthetwointervals. ThIScommon partwcdenote 2)by12andmaystate:12liesin11andcontains allpointsx" withn>p'J'IfinthisresultwereplacePIandp'JbyPk-landPk' anddenote therefore k)bylktheportion of liesin1,,-1'wemaythen pointsx"withn>Pk' 4h· I 1 +1.t emterva xPk-2"...xPk2kwhich state:]"liesin],'-1andcontains all (G51) 86 Chapter 11.Sequences ofrealnumbers. But(lkIisthenanestofmtervab; foreachlIltervdl besIIIthe .) precedmg andthelengthoflkis<27.. Nowif~isthenumber thusdetermined, weassert,finally,that x,,-~. Infact,ifanarbitrary B>0benowgiven,wechooseanindex"so 2largethat-<e.Wcthenhave2' foreveryn>Pr'Ix"-~Iis'<e, since ~,together withallx,,'sforn>Pr'liesinlrandthelength oflrIS<e.Thisprovesallthatwasrequired7• Further examples andremarks. 4S. 1.Thesequence 45,3caneasIlynowbeseentobeconvergent For wehavehere,ifn'>n: (1 1 (-1)"'-"-1) X,,'-Xn=±2n+1-2n+3+...+2//'+I. Ifinsidethehracket, wetakethesuccessive termsinpairs,wesee(cf.later SIc,3)thatthevalueofthebracket ispositive, sothat 1 1 (_1)"',,-1 Ix"'-x,,I=2n+1-2n+3+'" +-21/'-+1-· Itwenowletthefirsttermstandbyitselfandtlkethefollowing termsin pairs,weseefurtherthat 1ThereforeIXn'-X"Iis<ll,provided nandn'areboth> 211'Thesequence istherefore convergent. 2.Ifx,,=(I+~+"'+~)' wehavealready seenin46,3that(xn)IS notconvergent. Withtheaidofthe2ndmaincriterion, thisISdeducible fI0111 thefactthatherethe6-condition isnotsatisfied for6<~.Forhowever nu maybechosen, wehaveforn>noandn'=2n(alsotherefore> no) 1 1 1 1 1 X'-xn=--+--+···+->n-=-nn+ln+2 2n2n2' nottherefore<8.Theseqnenee istherefore divergent, andinfactdefinitely divergent, sinceitisevidently monotone ascending. 3.Theprevious example showsatthesametimethatthecontrary ofthe fulfilment ofthe6-condltion isthefollowing (cf.also10,12)":Notforevery choiceof6>0cannubesoassIgned thattheE-condition isthenfulftlledi thereeXistsonthecontrary (atleast)oneparticular number Eo>0suchthat, 7'Neshallbecome acquainted withotherproofsofthiSfundamental cri­ terion. Theproofgivenaboveleadsimmediately tothedefinition ofthelimit bytheaidofanestofintervals. - Acritical account ofearlierproofsofthe criterion maybefonndinA.Prmgshelm (Sitzungsber. d.Akad.MUnchen, Vol.27, p.303.1897). §9.Thetwomamcriteria. 87 aboveeverynumberno,however large(therefore ir.1inite1y often)twopositive in­ tegersnandn'maybefoundfo!"which Ixn'-:enI2:EO>0 . 4.The2ndmaincriterion isnowusually, afterP.duBoisReymond (Allge. meincFunl<tl0nentheorie, Tubingen 1882),calledthegeneral prmc~ple ofconver­ gence.Insubstance, itorigmated withB.Bolzano (1817,cf.O.Stolz,Mathem. Ann.Vol.18,p.25H,1881)butwashrstmadeastartmg point,asancxpre~sly formulated princIple, byA.L.Cauchy (Analyse algebrique, p.12.5). Ourmaincriterion mayalsobegivensomewhat different forms, whicharesometimes moreconvenient inapplication". Wesuppose thenotation forthenumbers nandn'sochosen thatn'>n,and therefore wemaywnten'=n+k,wherekisagainapositive integer. Wethenformulate thusthe oSecondmaincriterion (Formla). 49. Thcnecessary andsufficient condition fortheconvetgence ofthe sequence(x,,)isthat,givenanyE>0,anumbct no=no(e)canalways bcassigned sothatfatevetyn>noandevetyk~1wealwayshave Ix••+k-;K:nI<l?• Fromthisstatement ofthecriterion wecandrawfurthercon­ clusions. Ifwesuppose quitearbitrary natlJralnumbers kl'k~,...,k",... chosen, thenwemusthave,IIIviewoftheabove,foreveryn>no Ixn+kn-XIII<e. ButthISimplies thatthesequence ofdifferences formsanullsequence. -Inordertomakeourselves morereadily understood, wewillcallthesequence (d,,)forshortadifference·sequence of(x).Init,distherefore thedifference between Xandsomede-" " "finitelaterterm.Ourcritenon maythenbeformulated thus: 50. one01itsoSecondmaincriterion (2ndform). Thesequence (x")isconvergent ifandonlyifevery d~ff/:'rence·sequences isanullsequence. Proof. Thenecessity ofthiscondition wehavejustproved; we havestilltoshowthatitissufficient. Weaccordingly assume that everydifference-sequence tendsto0,andhavetoshowthat(x,,)con· verges. Butif(x,.)weredivergent, therewould,by48,3,existapar· ticularnumber eosuchthataboveeverynumber no'however large, twonumbers nandn'=n+kwouldalwayshe,forwhichthe difference 51.Chapter 11.Sequences ofrealnumbcrs. Sinccthismustbcthecaseinfinitely often,"therewould-incontradic­ tiontothehypothesis -existdifference-sequences 8whichdidnottend to0;(xn)musttherefore converge, q.e.d. Remark. If(xn)ISconvergent, and\\cchooseapart/CIllar difference-sequence (d,.),wetherefore certamly havedn--'>O.ButItshouldbeexpressly emphasized thatfromdn-->-0alonetheconvergence of(x,.)Ileednotfollow.qnthecontrary, forthIS,ItISonly ~ufficlent thate1'eryarbitrary dIfference-sequence (notmcrely aparticular onc)shouldprovetobeanullsequence. Ifform,tance thesequence0,n,I,0,I,...)isconsidered, everydlfferencc­ sequence forwhIchallkn's(fronlsomcpomtonwards) areevennumber, ISanull sequence. Kevcrthele,s thesequcnce mquestion isnotconvergent. SII111larly III thedl\'ergent sequence (x,,)\\ithxn"I+t+...+Ie1'elYdIfference-sequence 11 forwhIchtheindIcesk"arcboullded formsanullsequence. Extcnding somewhat furtherthelastobtaincd formulation ofthc criterion, wemayfinallyformulate itthus: oSecond maincriterion (3rdform). IfVI'1'2'•••,I'n'•••isallYsequcncc ofpositivc integers 9which diverges to-1-00,andkl,h2,•••,km...areanypositivc integers (with­ outanyrestriction), andifweagaincallthesequence ofdifferences forshortadifference-sequence of(xn),thcnfortheconvcrgcnce of(x,,) itisagainnecessary andsufficient that(dn)isilleverycaseanullsequence. Proof. Thatthiscondition issufficiellt isobvious fromthepre­ ccdingformofthecriterion, since(dn)must,inthepresentcasealso, alwaysbeanullsequence when Vnischosen=11.Andthatitisnecessary mayatoncebescen.ForifE:ISchosen> 0,therecertainly eXIsts,if (xn)isconvcrgent (v.Formla),anumber m,suchthatforeveryn>m andevery l~:2:1,wehave AsVndiverges -+-I-00,theremustbeanumber "0suchthat forn>"0'wehavealways vn:>m. Butthen,bythepreceding, wehave,forn>11o,always IxVnHn-x"nI-'--'-IdnI<E, 1.e.(dn)isanullsequence, q.c.d. HForIfwedenotebyn""2,1IJ'•••themfinitenumber ofvaluesof11for \\hichthatmequahty (eachtImeWItha'ultable choiceofk)isa5sumed tobeposs­ Ible,adIfference-sequence ....ouldeXI't....hoseIll''''nib,n}",...termswereallin absolute \aJue2'E"/O.ThIScouldnotthenbeanullsequenCe. •Equalorunequal, monotone ornotmonotone. §10.Limiting pointsandupperandlowerlimits. 89 (a)§10.Limiting pointsandupperandlowerlimits. Theconcept oftheconvergence ofa5equence ofnumbers as defined inthetwoprecedlllg paragraphs admits ofanother, some· whatmoregener.tI modeoftreatment, bywhichweshallatthesame timebecome acquainted withsomeotherconcepts, oftheutmost importance forallthatcomesafter. In:J9,6,wch.wealrc.ldy Illustrated thetlctofagivensequence (xn)beingconvergent bysaymgthateverye·neighbourhood (however small)of~mustcontainallthetermsofthe,;equence -withthepossible exception ofafUlltenumber atmost.-Thereistherefore ine\'elY neighbourhood of~,however small,certaInly aninfinite number of termsofthesequence. Forthisreason, ~maybecalledalimiting pointorpointofaccumulation ofthegivensequence. Suchpoints may,asweshallatoncesce,occuralsointhecaseofdivergent sequences, andwedefinetherefore quitegenerally: oDtfinition. Anumber ~shallbeculledalimiting point*at52. ugivensequence (x,,)ifeveryncighboltrhood uf~,however small,contains aninfinite number ofthetermsofthesequence,. or,therefore, If,for anychosen B>0,thereisaIway,;anmfinitenumber ofindicesn forwhich Ix"-~I<8. Remarksandexamples. 1.ThedistinctIOn between tIllSdefImt,on andthedebmtion oflimitgiven53. manlIes,asalready Il1dllaled,inthefactthathneI,'"-i;I.-:::::Fneedstobeful· fillcunotforevery 11afteracertain poil1t,butonlyforanyinflllite number oflI'S,andtherefore 1llpartieuI.\l· foratlca~tulle1Ibeyondc\eryno'Onthe otherhand,inaceOIdanceWIth:19,thelinl1t ~ofacom'clg-ent,equence (.r,,) ISalways ahmiting pOIntofthesequence. 2.Thesequence ti,1hasthellllllting point0;6,4,thelimitlllg points oand1.(Every number whichoccurs anIllfil11tC' number oftunesina sequence (.r,,)IStp'OfaCIOallI1l1tlllg" rain!.)6,2,7and11havenolimiting pomt;6,9and10havcthelImiting l'0lllt1. 3.\'\'enowformanexamplc ofmorethanJllustratlve SIgnificance: Ifp ISanmtcger ~2,thereisobVIOusly onlyafmltenumber ofrositi\'e fractions forwhichthe~umofnumerator anddCllollunator =p,namely thefractIOns 1'-1p--2 1-1-'-2--,...,p"_-i'Ofthe~ewesuppose leftOlltallthosewhIcharenot intheirlowestterms,andnowconSIder insucces5ioll allthefractions thus formed forp=2,3,4,....Thisgivesthesequence, beginmng with 1132 I1,2'2,3'-3,4, 2.-3-'4'···. whichcontains allposilwe rational numbers. Ifaftereachofthesenumbers weinserttheSamenumber withsignchanged andstartwith0asfirstterm, wehaveinthesequence •German: }-fiiuflltlgs1IJet't, IIcillfl11lgspllnl~t orl/iiltfllngsslellt. (Tr.) 90 (b)Chapter H.Sequences ofrealnumbers. -2, 3,-3,114,-4. 0,1,-1,2,2~,--2'-i'3 ' 3 3 221 2'--2-'off'-:f'4' thusformed obviously allratzonal numbers occurnng, eachexactly once. Forthisremarkable sequence everyrealnumber isalimitIng point;for everyneighbourhood ofeveryrealnumber contains aninfinity ofrational numbers (cf.p.l:!l. 4.Weshallfrequently makeuseoftheprinciple ofarrangement inorder applied inthisexample Wetherefore formulate itsomewhat moregenerally: Suppose thatforeverykoftheserie~k=0,1,2,.••asequence X(k)x(k) x 2(k),••,oJ1J (k=0,l,2,...) isgiven. Wecanthen,inmanydifferent ways.formase<juence (xn)wluchcon­ tamseverytermofeachofthesesequences andcontams ztexactlyonce. Theproofconsists simply IIIassigning asequence (xn)whichfulfilswhat isrequired. Forthispurpose wewritethegivensequences inrowsonebe· lowtheother: jx(O)reo):reo) x(O) o ' 1 '.J,...,'n' X(I)X(I)X(I);l:~I),• 0 ' 1 '2,..., Ix(k) (k)x(k) x(l) o 'Xl'2'..-,'n' The"diagonal" ofthissystemwhichjoinstheelement x6P)totheelement x~o, thencontains allelements x~k)forwhichk+n=p,andnoothers. Theyare p+1innumber. Thesetermswewritedowninsuccession, takingp=0,1,2,..., anddescribe eachofthediagonals sayfrombottom totop.Thusweobtain thesequence ~(O) X(l) x(O) x(2):lP) x(O) x(8) x 1(2',••0, ....0'0' 1I0Jl'2' 0I whichevidently fulfilstherequirements. (Arrangement bydlagonals*). Another arrangement frequently usedisthat"bysquares". Herewe firstwritetheelements X&P),xfP),•, " x~P)ofthept.brow,thentheelements standing vertically abovexii)intheabovesystem:x:-1),••"x~o).These groupsof2p+1termsarethenwritten downinsuccession forp=0,]I2,.", andthisgives,beginning with x(O) 1IX(21 1 'x~o), x~8), thearrangement bysquares**. Ifsomeoralloftherowsintheabovesystem consist ofonlyafinite number ofterm'l,orifthesystem consists ofonlyafinitenumber ofrows, thenthearrangements described aboveundergo slightanditnmediately ob viousmodifIcations. *German: Anordnung nachSchraglinien. (Tr.) ....German: Anvrdnung na~hQ"adraten. (fr.) §10.Limiting pointsandupperandlowerlimits. 91 5.Anexample similar to3.isthefollowing: Foreveryp~2thereare 11. fh..exactlyp-1numbers oftheform-+-forwhichthesum0tepOSIt!vekm integers kaudmisequaltop.Ifwesuppose thesewrittendowninsuccession, forp=2,3,4,...,weobtainthesequence andnoothers. 6.Asinthecaseofthelimitofaconverg-ent sequence, thelimiting pointsofanarbitrary sequence mayverywellnotbelong-tothesequence ItSelf.Thusin3.theirrational numbers, andm5.thevalue0,certamly do notbelongtothesequence concerned. Ontheotherhand,inbothcasesthe valllet,forinstance, isbothaIUlllting pointandatermofthesequence. Weproceed togiveatheorem which ISfundamental forour purpose, dueoriginally toB.Bolzano 10,thoughitssignificance wasfirst fullyrecognised byK.Weierstrass H. °Theorem. Everybounded sequence possesses atleastonelimit-54. ingpoint. Proof.Weagaindetermine thenumber inquestion byasuitable nestofinter\'al~. ByhypotheSIS thereexistsaninterval10which contains allthetermsofthegivensequence (x,,)TothisIllterval weapplythemethod ofsuccessive bisection anddesignate as11its leftorrighthalfaccording asthelefthalfcontains aninfinite nItmber01thetermsofthesequence ornot.Bythesamerulewe designate adefimte halfofI1asI~,andsoon.Thentheintervals ofthenest(JJsoformed allhavetheproperty thataninfinite number oftermsiscontained ineach,whIlsttotheleftoftheirleft endpoint thereisalwaysatmostafinitenumber ofpomtsofthe sequence. Thepoint ~thusdefined isobVIOusly alImiting point; forife>0isgivenarbitrarily, choosefromthesuccession ofinter· valsI..one,say11"whoselengthis<e.Thetermsof(x,J,in number infinite, whichbelong totheintenalIpthenlieipsofacto inthee·neighbourhood of~,-whichprovesallthatwerequire. Thesimilanty ofthedefinitions oflimiting pointandlimit(or limitingvalue)inspiteofthedifference emphasized in53,1("every limitisalsoalimiting point,butnotconversely") naturally creates acertain relationship between them.Thisiselucidated bythe following 10Reinanalytischer BewelsdesLehrsatzes, daDzwischcn jezwey\Verthen, dieelnentgegengesetztcs Resultat gewllhren, wenigstens einereelleWurzel derGlelrhung liege,Prag1817. 11InIllSlectures. 92 Chapter H.Sequences ofrealnumbers. 55. °Theorem. Everylimitingpointgofasequence (x,,)maybert; gardedasthelimit0/asuitablesub·sequence 0/(xn). Proof. SInceforevery Ii>0,wehave,foraninfinitenumber ofindices,Ix"-~I<e,wehave,inparticular, forasuitable n=k1, IXL,-~1<1;forasuitablen=ll~>k1,wehavesimilarlyIXk.-~I<~, andingeneral, forasuitablen=kv>kv-1 Ixk"-~I<~ (v=2,3,...). Forthesubsequence (x"')=(XL,.)thuspickedout,wehavex,,'-.~, as(Xkn-~),by26,2,formsanullsequence. Tileproofofthetheorem ofBolzano-Weierstrass givesoccasion forafurther mostimportant remark: TheintervalsInofthenest thereconstructed notonlyhadtheproperty thatwithinthemlayan infinite number oftermsofthesequence (x,,),butaswenotlCcd, theyhadthefurtherproperty thattotheldtoftheleftendpointof anydefinite oneoftheintervLlls therelayalwaysafinitenumber onlyofthetermsofthesequence. FromthiS,however, itfollows atoncethatnofurtherlimiting pointcanlie totheleftofthelimiting point ~alreadydetermined. Forjfwechooseanyrealnumbere<~, wehavee=~(~-e)<0;choosing anintervalJqoflength<8,wc havethcwholeofthee·ncighbourhood ofthePOlJ1telyingtothc leftoftheleftendpointof],andtherefore containing onlyafinite number oftermsofthesequincc. Thl'fefore nopoint ~'totheleft of~canbealimlting pointofthesequence (x,J,andwehavethe 56. Theorem. Everybounded sequence hasawell-defined leastlimit- ,ingpoint(i.e.onefarthesttotheleft). Ifweinterchangc rightandleftintheseconsiderations, weobtain 12 quitesimilarly the 57. Theorem. Everyboundedsequencehasawell-defined greatestlimiting point 13(i.e.onefarthesttotheright). Thesetwospeciallimiting pointswewilldesignate byaspeCial name. Theleastlimiting pointofa(bounded) sequence will lowerlimitorlimcsinterior. Denoting itby'"Definition. called* its wewrite5S. be orhminfx"=",,-.'" 12Orbyreflection attheorigin. 13Thesetheorems areagainobvious exceptinthecaseinwhichthesequence (x,,)hasaninfinitenumberoflimiting points,Ekee.g.thesequence 53,5.For amongafinitenumber ofvaluestheremustalwaysbebothagreatest andaleast. •TheGerman texthas"unterellaufungsgrenze, U1ltererLimes,Limesinferior", (Tr.) §10.Limiting pointsRndupperandlowerlimits. 93 (possibly omitting thesubscript n-oo). Itpisthegreatest li· mitingpointotthesequence, wewrite limx"=/Lor n-)oet:lEmsupx..=/L ",··)000 andcall!~.theupperlimit ~rlimessupet'im' atthesequence (x"). Wehavenecessarily always ~Sp. Smceeverye·neighbourhoocl ofthepoint ~contains aninfinite number oftermsofthesequence (x,,),andsinceontheotherhand onlyafimtenumber oftcrmsofthesequence canlietotheleftof theleftendpoint ofanysuchneighbourhood, ~(orsimilarly !~)isalso characterised bythefollowing conditions: Theorem. Thenumber x(orp)isthelower(orupper)limitat59. thesequence (x,,)ifandonlyif,givenanarbitrary e>0,wehave stilltoraninfinite number ofn's, x"<x+e(or>/L-e), butforatmostafinitenumber 14ofn's, .\'"<X-e:(or>I~+5:). Beforewegivcafewexamples andexplanations ofthistheorem, letuscomplete ourdefinitions forthecaseofunbounded sequences. Definitions. 1.Ifasequence isunbounded ontheleft,thenwe60. willsaythat-00isalimiting pointotthesequence;' andifitis unbounded ontheright,wewillsaythat+00isahmiting point ofthesequence. Inthesecases,however largewechoosethenumber G>0,thesequence hasaninfinityofterms 15below- Gorabove+G. 2.Iftherefore thesequence (xn)isunbounded ontheleft,then-00 istheleastlimitingpoint,sothatwehavetowrite x=~x,,= -00. "~~+'" Similarly wehavetowrite p=limx"=+00 n~+'" Ifthesequence isunbounded ontheright.Inthesecases,nowever largewechoosethenumber G>0,wehave,foraninfinityofindices, xn< -Gorxn>+G. •TheGerman texthas"obereHli'ujrmgsgrenze, oberellLimes,Limessuperior". (Tr.) UOr:Thereisanindex"0fromandafterwhichweneverhavexn<le-e (>P.+e)butbeyondeveryindexn,thereisalwaysanothernforwhichxn<le+E: (>P.-e). 15Heretherefore -andsimilarly inthefollowing definitions -theportion ofthestraight linetotherightof+Gplaysthepartofans:-nelghbourhood of+00,theportiontotheleftof- Gthatofans:-neighbourhood of-00. 94Chn.ptern.Sequences ofrealnumbers. 3.If,finally,thesequence isbounded ontheleft,butnotunthe rightand(besides+00)hasnootherlimiting pOlllt,then+00is notonlyitsgreatest, butatthesametimeitsleastlimltmg point,and Wcshalltherefore equatethelowerlimitalsoto+00: x=limxn=+00; ft--J-+CX) Andcorrespondingly weshallhavetoequatetheupperlimitto-00, fJ.=lunxn= -00 n~+oo ifthesequence isboundedontheright,butnotontheleft,and(besides -00) hasnootherlimiting point.Theformer(latter)caseoccursifandonly if,givenanyG>0,theinequality Xn>G(xn< -G) holdsforaninfinitenumberofn's,buttheinequality Xn<G(xn:>-G) foratmostafinitenumberofn's,thatistosaytherefore whenXn-++00 (-00),Cf.63,Theorem 2. Examp Iesandexplana tions. 61. 1.Inconsequence ofthepreceding definitions, everysequence ofnumber~ nowofItselfdefIDes,absolutely uniquely, twodeterminate symbols ><and(-t, (whichmaynow,itistrue,standfor+00or-00,andwhichbearthere­ lationxSJLtooncanother 16.Andthefollowmg examples showthatY.and I~ mayactually assume allfiniteorinfmite values compallble withthein equalityY.;£(-t. Infact,forthe!>cqucnce 1.(n)==I,2,3,4,... ,(-I)' _1 12.(arn)=a+, a+2,a+--,a+4,...3 3.a,b,a,b,a,b,. . .(a<b) 4.(a+(-~)n)=="a-],a+;,a_{,a+-}, 5.«(-1)'.'1)==-1, +2,-3,+4,... I6.(a-nl-I)')=a-I, a-2,a-:r'a-4,... 7.(-n)=-I, -2,-3,...wehave l<=I~ +00+00 a+00 ab aa -00+00 -00 a -00-00 2.Thereadershouldnoteparticularly thatitisnotcontradictory to theorem liDthatanmbmtenumber oftermsofthcsequence shouldlietothe leftof><ortotherightof(-t.Thusforinstancc wehavc,forthesequence ((l)nn+1). f 23456 .--n-' I.e.orthesequence -'+2'-3'+4'-5"" eVidently 10\Yesayofeveryrealnumber thatitis<+00and>-00,andfor thisreason weoccasionally de~ignate itexpressly as"finite". §10,Limiting pointsandupperandlowerlimits, 95 "= -1,1£=+I,andbothtotheleftof"andtotherightof1£liesan infinitenumber oftermsofthesequence (andbetween" andf'liesnotermof thesequence I).Itistherefore 'notatallneces~ary thatthereshouldbeonlya finitenumber oftermsofthesequence outside theinterval x...!~.Theorem ISDonlyasserts infactthatatmostafimtenumber oftermsofthesequence canbetotheleftofx-eortotherightoff'+e. 3.HAfmitenumber ofalteration~" hasnoeffectonthelimiting points ofasequence -none,inpartIcular, onitsupperandlowerlimits. These therefore represent anulltnzate property ofthesequence. 4.Sinceasequence (xn)determines boththenumbers "andf'with complete uniqueness, andsincetheirvalue,inconnection withourdefmition, wa~ alsoenclosed byawelldefined nestofIntervals, wchaveheremanewlegi. timatemeansofdefining (determining, giving) realnumbers: arealnumber shallhenceforth alsoberegarded as"gIVen". if~tistheupperorlowerltmztofa gIVensequence. Thismeansofdetermining realnumbers isevidently stillmore general thantheonementioned in41,150lncenowthesequence utilisl!d need notevenbeconvergent, orbesubject toanyrestriction whatever17• Asmaybeseen,inthelightof5;i,wehavealsothefollowing Theorem. Theupperlimit/1ofthesequence (x,,),p,=limx"'is62. ulso,inthecaseIt=F±00,characterised bythetwofollowing conditions: a)thelimiteateveryconvergent sub'sequence (x,,')of(x,,)is invariably :cs;/1,'butthereexists b)atleastonesuchsub-sequence, whoselimitisequaltop;­ andcorrespondingly torthelowerlimit. Aconcept relatedtoth.!toftheupperandlowerlimits,though onewhichmustbesharply distmguished fromit,istheconcept of lIpperandlowerbounds ofasequence (x,,),whichisderived from thefollowing consideration: Ifnotermofthesequence liestothe rightofp=limx",sothatforeveryn,X,,:CS;,It,thenItisabound above(8,4)ofthesequence, -butonewhIChcannotbereplaced byanysmaller one;/1istherefore inthiscasetheleastboundabove. Butsuchaleastboundalsoexistsifthereisatermofthesequence >p.For1fforinstance xpis>Il,thenby50thereiscertainly onlyafinitenumber oftermsinthesequence whichare~x,and-p amongthesethereisnecessarily (8,5)alargestone,sayxq'\Ve thenhave,foreveryn,x"<xq'i.e.xqisaboundaboveofthese­ quence, -butagainone,whichcannotbereplaced byanysmaller one.Everysequence bounded ontherightthere/ore possesses adefinite leastboundabove.Since,inthesameway,everysequence bounded 17Whereas therefore anestofintervals (withrational endpoints) wasat firsttocountastheonlymeansofdefining arealnumber, wehavenow deduct"d quiteaseriesofothermeanswhichwcnowadmitasequally legI­ timate: Radixfractions, Dedekind sections, nestsofintervals witharbitrary realendpoints, convergent sequences, upperandlowerlimitsofasequence In allthe~ecases,however, wesawhowatoncetoassign l\n('st01intervals (withrational endpoints) whichencloses thegivennumber. 96 Chaptern.Sequences ofrealnumbers. ontheleftmusthaveadefinitegreatest boundbelow,wearejustified inthefollowing Definition. Wedefineastheupperbound" ofasequencebounded ontherighttheleastofitsboundsabove(invariably determinate byourpre­ liminary remarks), andsimilarly asthelowerbound" ofasequence bounded ontheleftthegreatestofitsboundsbelow.Asequence unbounded ontherightissaidtopossesstheupperbound+00,oneunbounded onthe left,topossessthelowerbound-00. Theconcepts ofupperandlowerlimitsareduetoA.L.Callchy (Analyse algebnque, p.1:12.ParisIS21)butwerefirstmadegenerally knownbyP.dllBOI<­ Reymond (Allgemeinc FunktlOnentheoric, Tubmgen IS82). Bothnomenclature andnotatIOn haveremamed vanable uptothepresent day.ThepartIcularly con­ venient notation hmandhmusedinthetextwasintroduced byA.Przngsheun (Sltzungsber. d.Akad.zuMunchen, vo!'28,p.62.lR!l8),towhomthedeSIgnatIOns ofupperandlowerhmltsarealsodue"". Itshouldbcexpressly pomted outagainthattheupper(andsimilarly the lower)bound 1&notnecessarily determmed bythetall-end ofthesequence. Thus theupperboundofthesequence (~)is1,andISobviously alteredIfthefirsttermof thesequence ISaltered. Theprevious investigations ofthisparagraph werecarriedoutquite independently oftheconsiderations onconvergence of§§RandH,and giveus,forthisveryreason,anewmeansofattacking theproblem of convergence Aof§H.Itmaybeshewnthattheknowledge ofthelower andupperlimitsxandiLofasequence -theknowledge, therefore, of twonumbers whoseexistence isaprioriensured-entirely sufficesto decidewhether orhowthesequence converges ordiverges. Wchave infactthetheorems 63. Theorem 1.Thesequence(xn)isconvergentifandonly ~fitslowerand upperlimitsxandiLareequalandfinite.IfAisthecommon value(different, therefore, from+00or-00)ofxandiL,thenXn~A. Proof. a)Letx=iLandtheircommon value~A.Then,by59, given E,thereisatmostafinitenumberofn'sforwhich Xn<K -E=A-E, "German: Obere,untereGrenze(frontier). Theword"frontier" isnotusual inEnglish wntmgs, thoughsometlmes foundmFrench. Thedlstmction between anyboundsandthenarrowest boundsisemphaSIzed chieflybythearticlethemthe lattercase;theupperboundandthelowerboundalwaysdenoting thelatter.For fearofambIguity, however, theword"bound" inthegeneral senseisavoided as muehaspossIble inEnglt'h text-books. (Tr.) "..Wehaveomitted reference heretotheuntranslated term"HilUfungsgrenze" ofdIeGerman text:"Die101Textebenutzte ausfuhrliehere Bezeichnung Hdufun~s­ gI;tnzesolinurdenUnter,ehled zudersoebendefimerten unteren undoberen Grenzestarkerbetoncn". (Tr.) §10.Limiting pointsandupperandlowerlimits. 97 andsimilarly atmostafinitenumberofn'sforwhich Xn~f.L-I-e="-I-e. Foreveryn>someno,wetherefore have "-e<Xn<"-I-e,orIXn-"I<e, i.e.thesequence isconvergent and"isitslimit. b)If,conversely, limXn=",then,givene>0,wehave,forevery n>no(e),,,-e<Xn<"-I-e.Therefore theinequality X n<"-I-e(>"-:::) issatisfied foraninfinitenumberofn's,buttheinequality Xn<"-e(>"+e) foratmostafinitenumberofn's.Theformerinequalities (with<)imply It=",thelatterf.L,,-".Thisprovesallthatwcrequired. Theorem 2.Thesequence (xn)isdefinitely divergent if,andonlyzf, itsupperandlowerlimitsareequal,buthavethecommon value18+00or -00.Intheformercaseitdiverges to+00,inthelatterto-00. Proof. a)Ifx=f.L=-I-00(or-00),thenthissignifies, by 60,2and3,that,givenG>0,wehavefromandafteracertainno Xn>+G«--G); wctherefore thenhavelimXn=+00(-00). b)If,conversely, limXn~-I-00,then,givenG>0,wehavefor everynafteracertainno,."<:n>-I-G;therefore theinequality Xn<+Gissatisfied foratmostafinitenumber of n's,whereas theinequality Xn>+Gissatisfied foraninfinitenumber ofn's. Butthisimplies, by60,thatIt=-I-00andipsofactoalsofL=+00. Therefore x=f.L=+00.Andinprecisely thesamewayweshowthat iflimXn= -00,thenIt=f.L= -00. Fromthesetwotheorems weatoncededucefurther: Theorem 3.Thesequence (x,,)isindefinitely divergent ifandonlyif itsupperandlowerlimitsaredistinct. Thecontentofthesethreetheorems provides uswiththefollowing Thirdmaincriterion fortheconvergence ordivergence ofasequence:64. Thesequence (x",)behaves definitely orindefinitely, according asits upperandlowerlimitsareequalordistinct. Inthecaseofdefinitebehaviour, itisconvergent ordivergent, according asthecommon valueoftheupper andlowerlimitsisfiniteorinfinite. 18Inoccasionally speaking ofthesymbols+00and-00(whicharecer­ tainlynotnumbers) as"values", wemakeuseofamereverbal!tcence, towhich noimportance shouldbeattached. 98 Chaptern.Sequences ofrealnumbers. Thefollowing tablegivesasummary ofpossibilities asregardsthe convergence ordivergence ofasequence andofthedesignations used inthisconnection. ;<=~/-l,both=),+±oo ;<=,t=+00or-00 ;«/-l convergent (wIthlImit),)divergent (orpossibly: con- lim:rn=.l.verg-ent) towards (or:with Illlllt)-I-00or-00;inboth mdefmitely(t1-~+(0)cases:deflmtely ulvcrg-cnt. divergentxn-), bmxn=-I-00or-00 (forn-+oo) x,,_+00or-00 convergent dIVergent defllllte behaviourindefInIte behavlOl1r §11.Infiniteseries,infiniteproducts, andinfinite continued fractions. Anumerical sequence canbespecified inthemostdiverseways; thisissufficiently evident fromtheexamples whichhavebeengiven. Inthese,however, forthemostpart,thenthterm Xnwasforconveni­ encegivenbyanexplicit formula, enabling ustocalculate itatonce. Thisisbynomeanstherule,however, intheapplications ofsequences inallpartsofmathematics. Onthecontrary, thesequences tobeexamined generally present themselves indirectly. Besides severallessimportant kinds,threetypesespecially comeintoconsideration; ofthesewewill nowgiveabriefdiscussion. 66. 1.Infinite series. Thesearesequences giveninthefollowing way.Asequence isatfirstassigned inanymanner (usually bydirect indication ofitsterms),butwithout beingintended itselftoformthe objectofdiscussion. Fromitanewsequence istobededuced, whose termswenowdenotebySmwriting so=oo; Sl=OO+Ol; S2=00+01+02; andgenerally Sn=ao-I-a1-I-a2-I-•••-I-an(n=0,1,2,...). Itisthesequence (sn)ofthesenumbers whichthenformstheobjectof investigation. Forthissequence (sn)weusethesymbolical expression 67. a) ao-I-a1-I-a2-I-•••-I-an+ ormoreshortly b) ao+a1+a2+... orstillmoreshortlyandmoreexpressively: QC §11.Infinite series,infiniteproducts, andinfinitecontinued fractions. 99 andthisnewsymbolwecallaninfinite series; thenumbers Snare calledthepartialsumsorsections «<oftheseries.-Wemaytherefore statethe aDefinition. Aninfiniteseriesisasymboloftheform Of: };anoraoI-at'+a2-+-... n--U or ao+at-+-a2-+-...+an+... bywhich ISmeantthesequence (sn)ofthepartialsums Sn-ao+at-+-...-+-an (n=0,1,2,...). Remarks andExamples. 1.Thesymhols68. '"all+.Elln;an 11tif) il,-IEa,,; u-2'"all+a,-I•••+am+Ean "m1-1 '" shallbeentirely equivalent toEll",Theindex 11IScalledtheindexofSIImmallOll. 11·~~0 Ofcour.eanyotherlettermaytakeItSplace 00 en Ea.;all+a,+a.+Ea,;etc. 11--0 e-'Jl: Thenumhers anarethetemHoftheseries.Theyneednotbeindexed from0on­ wards.Thusthesymbol 00 EaAdenotes thesequence (a"a,+a.,a,+{l.+(l3'•••) A~l andmoregenerally, denotesthesequence ofnumbers $p,sP-I-"sP-I-"•••givenby Sn=lip+ap-I-'+...+anforn'~p,p+I, Herepmaybeanyinteger ;;:;O.FmallywealsowriteqUIteshortly ~ Ea. whenthereisnoambiguity astothevalueswhichtheindexofsummatIOn hasto assume, -orwhenth,s,samatterofind,fference. 2.Forn=0,1,2,...letanhe 1 1 ~~2n: b)(n+l)(n+2); c)=1;d)=11; e)=~-:);j,0=(-W;g)=(-1)"(211+1); h)~(otf-n)(1X1+n+I) IX=arealnumber4:0,-I,-2,... «<German: TeilslImmm oderAbsclmitte. 100 Chapter H.'Sequences ofrealnumbers. Wearethenconcerned withtheinfinite series 001 1 1 1a).2 2.;==1+~+4+8-+···; n=ll ex> 1 1 1 1b).2------=--+--+--+ ...;n=o(n+l)(n+2)-1.2 2·33·4 c)1+ 1-+1+...;d)0+1+ 2+3+...; ";(-1)"1]1 e)2,-;;-+-1="'-1--2+:r-"4+-...; 1,=0 t)l(-I)A=:=I-I+I-I+- ...; A=Og)1 - 3+5 - 7+9 -+...; 00 1 1 1 1 h)k~(a:+k)(a:+k+1)==a:(a:+1)+(a:-+1)(a:+2)+(a:+2)(a:+3)+.... Andwehaveinthesesimply anew-andaswillbescen,verycon venient -symbol forthesequences (S0'51>s~.•.•)forwhich 5"is 1 1 1l'a)=1+-2+4+"'+2,,=2-211;• 111 1 b)=r:2+2.g+g.4+···+(n+l)(n-+2) =(1_.!-)+(.!-_i)+...+(_11_)=1__1_. 223 n+ln+2 n+2' 1 d)=n(n+-..!2"C)=n+ ; 2 e)=1-{-+~-+"'+~~)i (cf.45,3and48,1); f)=t[I-(-1)"+1] (seefootnote lU); g)~~(-I)"(n+I); 1 1 1 h)=~'"+I)+('"-+I)('"+2)+...+('"-+n)(",-+-:--n---:-+-l::-7) =(~-;-~-l)+(",+1-"'-+2)+...+(~~- 111-;+~-+-1) 1 1 =;X-'"+1;-+i' 3.WeemphaSIse aboveallthatthenewsymbols havenosignificance inthem­ selves.Addition, ItIStrue,ISawell-defined operation, alwayspossible, withregard totwooranypartIcular number ofvalues,inoneandonlyoneway.Thepartial sumssntherefore, however thetennsanmaybegiven,haveunderallcircumstances definite values. Butthesymbolfa"hasinitselfnomeaning whatever, -not 11-0 eveninacaseastransparent, seemingly, as2a;fortheaddition ofaninfinitenumber oftermsissomething quiteundefined, something perfectly meaningless. Itmust beconsidered substantially asaconvention thatwearetotakethenewsymbol tomcanthesequence ofItspartialsums. 1"EqualtoIor0,accordmg asnisevenorodd. §11.Infiniteseries,infiniteproducts, andinfinitecontinued fractions. 101 4.ThereadershouldtakepartIcular caretodIstinguIsh aseriesfromase­ quence 20:Aseriesisanewsymbolforaseqllence deducible byadefimterulefromit. 5.ThesymbolwiththesIgnofsummatIOn "L'"canofcourseonlybeused whenthetermsofthesenesareformed byanexplicitly assIgned law,orwhena particular notation isavallablc forthem.IfforInstance thenumbers111111) 2'3'5'7'Il'I:\,17' orthenumhers 3'7'8'15'24'2(j':U' arctobetheterm,ofasenes,weshallhavetousetheexplicitsymbols 1 1 1 1 1 1 2+3+5-I-7/.11+13+... and1111111 :JI'7+"8+15+2l+2{j+:11+... andwntedownasmany tcrm~asnece,~ary, tIllwcmaya,sumc thatthereader hasrecogl1l~ed theI.l\vofformatIOn. Forthefirstofthesetwo,enes,thismay beexpected aftertheterm11\:theterm,arcthereelprocals ofthesuccessive prime numbers. Inthesecondl"dlmple ItWIllnotbeknownevenaftertheterm,\how toproceed: thedenommators oftheterm~aremeanttobetheIntegers oftheform pq-1 (p,q~2,:J,4,...) Inorderofmagnitude. Wenowadoptthefurther convention thatallexpressions usedto ~lcscribe thebehaviour, inrespectofconvergence, ofasequence arcto becarriedoverfromthesequence (s,,)totheinfiniteseries}; anitself. Thereby weobtaininparticubr thefollowing DefinitIOn. Aninfiniteseries.Eanissaidtobeconvergent, definitely69. divergent orindefinitely divergent, according asthesequenceofitspartial sumsshowsthebeha'L,iour indicated bythosenames. If,inthecaseofcon­ vergence, Sn-)-s,then'[oesaythatsisthevalueorthesumoftheconvergent infiniteseriesandwewriteforbrevity 'r ~a.=s, •lJ 00 , sothat2:aydenotesnotonlythesequence(s,,)ofthepartialsums,aslaiddown y~lJ inthepreceding definition, butalsothelimitlims'"whenthisexists 21.III thecaseofdefinitedivergence of(sn),'[oealsosaythattheseriesisdefinitely divergent andthatitdiverges to+00or-00according asSn-.)-+00 or-.)--00.Iffinally,inthecaseofindefinite divergence of(sn),Itandp. arethe10wfJrandupperlimitsofthesequence, then'[oealsosaythattheseries isindefinitely divergent andoscillates between the(lowerandupper)limits Itandp.. 20Theadditional epithetof"infil1lte" maybeomitted whenobvious. 21Exactly aswemaynow, IIIaccordance Withthefootnote 9to41,1,write (~n)=S. 102 Chapter 11.Sequences otrealnumb.:rs. Remarks andexamples. 1.Itisatonceobvious thattheserie~68,2a,bandhconverge andhave forsums+2,]and2.-re&pectively; 2 canddaredefinitely divergent towards 0:+00; 2 eisconvergent andhasforsumthenumber 5defined bythenest22 (5.,.'_1I52");2f,finally, oscillates between 0andI,and2 gbetween -00and +00. 2.Asregards thetermsumthereadermustbeexpressly cautioned about apossible mi"under&tandmg: Thennmber si.'inotasuminanysenseprevionsly inuse,blltonlythelImztofanznfzmt~ sequence ofsums;theequation en2.'a"=sorao+at+...+all+...=S n~O istherefore neither morenorlessthananother wayofwriting lims"=sors"-+-s. Itwouldtherefore seemmoreapproprIate tospeaknotofthesumbutofthe lzmltorvalueoftheseries, However theterm"sum"hasremained inlIse fromthetimewheninfinIte seriesfirstappeared inmathematical science and whennoonehadaclearnotion oftheunderlying limiting processes or, generally, ofthe"infinite" atall. 3.Thenumber szstherefore nosum,butisonlysonamed, forthesake ofbrevity. InpartIcular, calculations involvlDg series WIllinnowiseobey alltherulesforcalculating wIthsums.Thusforinstance inan(actual) sum wemayintroduce oromitbrackets inanymanner, sothatforinstance, 1- 1+1-1=(1-1)+(1-1)=1-(1-1)-1=O. Butonthecontrary i(-1)·==1- 1+1-]+-... '1=0 isnotthesamethingas (1-1)+(1-1)+(1-1)+..,~0+0+0+... oras 1-(1-1)-(1-1)-(1-1)-...=1-0- 0- 0- •••• Nevertheless, calculations involving serieswill'havemanyanalogies withthose involving (actual) sums.TheeXIstence ofsuchananalogy has,however, m everypartzcular casetobefzrstestabhshed. 4.Itisalso,perhaps, notsuperfluous toremark thatitisreallyquite 001 paradoxical thataninflOite serie&,say22.Ishouldpossess anything atall '1=0 a2Infacts2k1=(1_~)+(-!__~)+...+(_1__-.!-)=_1_ - 2 3 4 2k-l 2k1·2 +3\+"'+(2k~1)2k' sothatSl.<S3<S6<···; similarly fromSu =1-(~-~)-...-(;k-2k~1)wededucethatso>s.>s,>..'.Finally 'u-Slll!:_l=+2/+1'i.e.pOSItive andtending toO.By46,4and41,5, wehaves"-+-(sI!k-lIsI!It:).Cf.!nC,3and82,5wheretheseconsiderations aregeneralised. (n=0,1,2,.••)§11.Tnfiniteseries,infinite product~, andinfinite contInued fractions. 103 capableofbeingcalleditssum.Letusinterpret itinfourth-form fashion by shillings andpence:Igivesomeonefirst1s.,then'/9s.,then'/4s.,then'/Ss.,and soon.IfnowInevercometoanendwiththesegJfts,thequestion arises,whether thefortune oftherecipient mustthereby nC'cessanly incrcase beyond all bounds, ornot.Atfirstonehasthefeeling thattheformer mustoccur;for ifIcontmue constantly adding something, thesummust-itseems-ulti· mately exceed everyvalue. Inthecaseunderconsideration thiSISnotso, sinceforeveryn I I 1 1 Sn=1+2+4+...+2",--2 -2"remains<2. Thetotalgifttherefore neverreaches eventheamount of2s.Andifwenow,in spiteofthis,saythatE2~isequalto2,thenwearereallyonlyusinganabbreviated expression forthefactthatthesequence ofpartialsumstendstothelimit2.-Cf. thewell-known paradox ofAchIlles andthetortOise (Zenon's paradox). 5.Inthecaseofdefimte divergence wecanalso,inanextended sense,speak ofasumofthesenes,whichthenhasthe"value"+00or-00.Thu'lformstance thesenes 001 1 I 1 /E""']1--.+.-14+...n._111 2.3 isdefmitely divergent, andhasthe"sum"+00,because by46,3itspartialasums -++00.Wewriteforshort '"1;E-=+uo./n=1n,.- whichisonlyanother modeofwnting for Hm(1+1+...+.!.)=+co.2 tl 6.Inthecaseofanindefinitely divergent serieshowever, theword "sum"losesallsignIficance. Ifinthiscaselim5"=xandhm5"=P.(>x), thenwesaid,intheabove,thattheserieso5czllate5 between xand{"Butit mustbecarefully noted(cf.61,2),thatthisrefersonlytoadescription ofthe ultzmate behaviour oftheseries. Infactthepartialsums 5nneednotliebetween I(and1"Thus,forinstance, ifao=2,andforn>0, a=(_l)n[n+1+n+2J.. nn+l wecanatonceverifythat n+2 5"=aO+a,+...+an=(-1)"-­n+l andtherefore Hm5..=-1,lim5"=+1.Butallthetermsofthesequence (sn) • 2aIftherefore thepayments discussed in4.havethevalues1s.,1/..s., 1Iss.,1/4S.,...thefortune oftherecipient nowdoesincrease beyond all bounds. Itisnotatfirstatallobvious towhatitisduethatinthecase4,the sumdoesnotexceed amodest amount, whereas inthepresent caseitexceeds everybgund. Thedivergence ofthisserieswasdiscovered byJOhnBl!rngull! andpublished bylamesB,rnoulli in1689;butseemstohavebeenalready known toL~n 1673. 104 Chapter 11.Sequences ofrealnumbers. lieoutsidetheinterval -1...-+-1,alternately ontheleftandontheright,sothat aninfinitenumber oftermsofthesequence liesonbothsidesoftheinterval. 7.Asweemphasized abovethataseries1:anrepresents merelythesequence (sn)ofitspartialsums,-andtherefore ismerelyanother modeofsymbolising a sequence, sowemayeasilyconvince ourselves thatconversely everysequence (XII'X"•••)maybewritten asaseries.\Yeneedonlywrite ao~='X-o,tl1'-O:Xl-Xo,tlz~X2-Xl'•••,an==:Xn--Xn._h•••(n21). ,":f' fT) Forthentheseries}; artc.-"11-,-};(x/"-.'1'/,_1)hasforpartialsums Sll0-.'X"'SI=Xo ,,=,0 k-~l -+-(Xl-XII),-cXlandgenerally for11::~1 s"x"-+-(x,-x,,)--f-(x,-Xl)-+-...-+-(."n-1-Xn_2)-+-(Xn-Xn_1)=XTt' SOthattheabovewritten seriesdoesactually standforthesequence (xn).The newsymboloftheinfiniteseriesistherefore neithermorespecialnormoregeneral thanthatoftheinfinite sequence. Itssignificance residesprincipally inthefact thattheemphasis isonthe'/zfierence an,~Sn-sn_1ofeachtermofthesequence (sn)fromthepreceding, ratherthanonthesetermsthemselves. Theconvention laiddownin68,I,bywhichforimtance ~ ~ };anandall-+-a,-I-..•-+-am-+-};an 1"1"°_0 n--:--~m-l1 aretomeanthesamething,nowbecomes thetheorem (cL70and82,4)thatthe twoseriesinvolved converge anddiverge together, andthat,whenconvergent, thetwoexpressions havethesamevalue. 8.WithregardtotheHistoryofInfiniteSeries,anexcellent account isgiven inalittlebookbyR.Reif/(Tubingen 1889).Hereitmaysufficetomention the following facts:Thefirstexample ofaninfinite seriesisusuallyascribed toArchi­ medes(Opera, ed.].L.Heiberg, Vol.2,pp.310seqq.,Leipzig H1I3).He,however, merelyshowsthat1-+-~-+-...--I-:Inremains lessthan~,whatever valuenmay d- b I . 1 1have,andthatthe,fference etween tletwovalues IS:3-4'"andconsequently lessthanagivenpositive number, provided 11betakensufficiently large.Hetherefore proves-inourphraseology -thattheseriesl']"isconvergent, andshowsthat n-·O 4itssumequals3'Amoregeneral useofinfinite seriesdoesnot,however, begin tillthesecondhalfofthe17thCentury, whenN.Mercator andW.Brouncker, in 1668,whileengaged onthequadrature ofthehyperbola, discovered thelogarithmic series120,andwhenI.Newton, inHHi9,inhisworkDeanalysiperaequationes numeroterminorum infinitas placedtheiruseonafirmerbasis.Inthe18thCentury theconsideration ofprinciples was,itistrue,entirely neglected, butthepractice ofseries,ontheotherhand,wasdeveloped, aboveallbyEuler,inamagnificent manner. Inthe19thCentury, finally,thetheorywasestablished byA.L.Cauchy (Analyse algebrique, Paris1821)inanirreproachable manner, exceptforthewant ofclearness whichthenstillattached totheconceptofnumber assuch.(Forfurther historical remarks, seeIntroduction to§59.) n.Infinite products. Hereweareconcerned withproducts of theform '"U1·U2·Ua··.un•••orflun; 11=1 §11.Infiniteseries,infiniteproducts, andinfil11tecontinued fractions. 105 theymustbetaken,inaprecisely similarmanner totheinfiniteseriesjust considered, simplyasanewsymbolic formforthewell-defined sequence ofthepartialproducts PI=Ut;pz_CUt•uz; Pn=ul•Uz...Un; However weshalllater,withreference totheexceptional partplayedby thenumber 0inmultiplication, havetomakeafewspecialconventions inthisconnection. 1.Ifformstance wehave,foreveryn::;;;J,"1' product(n-I-I)'thentheinfimten(n~I2)' fj~(n:f-I)' 2'32,1',,2 (111-1)2 n._1n(n-I-2)or132·43"-1.li...n(n-I-2)••. repr('sent" thesequence ofnumbers 4 2 . :~ 2 . 4 2(n1-1) PI~~3;P.= -4-;p,=~,,-:•..;PI'=--n-:t-2-; 2.TheadchtlOns andrem.lrks Ju~tm,ldeinIret.unmutatis mutandis their significance here.Allfurther detaJl~willbecon"dered later(Chapter VII). m.Infinitecontinu!,dJracti9ns. Herethesequence (x,,)undere'.ammatlOn isformed bymC';msoftwoother sequence~ (a"a,...)and(b",b....),bywntmg; X3bo-I----(/'--- b I1_(I, (13b,-Ib J andsoon,""nointhegeneral LISC',bemgdeduced from""1'__1bysuhstltutmg for thelastuenOlninator bll_1of'\"n-1thevaluehn__1-I-~n,andproceedmg thusad "lflfimtum. Forthe"mflmte continued fr.lctIOn" sofornle,1 thenotatlO:l aIItl2I tini b"+Ih,+Ib,+...+Ibit1-••• isfairly u~ual.The mo~tnatural notatIOn forItwouldbe 00 buI-Kt- 11=-=1 Herealsoafewspecialcon\entlOns havetohemade,totakethefactintoaccount thatmdiVision thenumber 0agamplaysanexceptional part.Thesubjectofcun­ tinucdfractions wcshallnot,howcver, entcrmtomthl~trcatlsc 24. Ofthethreemodesofassigning asequence discussed above, thatbyinfinite seriesisbyfarthemostimportant forallapplications inhighermathematics. \Veshalltherefore havetodealmainlywith these.-Sinceseriesmerely represent sequences, theintraductory developments of§9provide uswiththepointsofVIewfromwhich agivenserieswillhavetobeinvestigated: Together withthe problem Awhichconcerns theconvergence ordivergence ofagiven series,wehaveagaintheharderproblem B,whichrdatestothesum ofaseriesalready seentobeconvergent. Andforexactly thesame 24Acomplete account oftheirtheoryandapplIcatIOns isgivcnbyO.Perron, DIeLchrevondenKettenbruchen, 2ndEditiOn, Lelpzlg HI29. 106 Chapter U.Sequences ofrealnumbers. reasons aswethereexplained, thesecond problem willgenerally present itselfintheform:Aseries ~anisknowntobeconvergent; doesitssumcoincide withthat0/anyotherseriesOrwiththelim~t 0/anyothersequence, ordoesitstandinanyassignable relation f() suchanothersumorlimit?2S Sincetheproblem Aistheeasierandsince-incontradistinction toproblem B -itadmitsofamethodical solution, wewillproceed 111thefirstplacetogiveourattention tothisindetail. Exercises onChapter 1126• 9.ProveTheorems Uito19ofChapter Ibythemethod indicat~d in thefootnote to14. 10.ProveinalldetaIlsthatthe ord~red arrangement, defmed by14 and1:S,ofthesystem ofalln('st~ofintervals, obeyseachofthetheorems of order1.(ForthIScf.14,4and1:S,2.) 11.Carryoutthedet'lIlsoftheproofreqUIred onp.:J2;i.e.provethat thefourmodesofcombinmg nestsofmtervals. defined by16to19,obey allthefundamental laws2. 12.Forfixede,wIthe<1. Xn=(n+I)"-ne_O. 13.Forarbitrary positive etandP. (logJ()gn)a .-+0. (lognll V3 14.\\'hich ofthetwonumbers(i-)and""({"2)2"isthelarger?--25Thuse.g.theseries 1+1+.!+-!+...+~+ ...willeasilybe2131 nl showntoconverge, Howdoweseethatitssumcoincides withthenumber, givenbythesequence(1+~)n?Similarly wemayverysoonconvince our. selvesoftheconvergence ofthetwoseries Buthowdowediscover thatifsands'aretheirsums,s=:s,gand45'=;or (i.e.equaltothelimitinathirdlimiting process, whichoccursinrelation to thecirclejcfpp.200and214)? 26Inseveral ofthefollowing exercises, afewofthesimplest results with T~gard tologarithms, andthenumbers •and:n:,areflssumed known, although theyareonlydeduced lateroninthe t~xt. Exercises onChanter H. 13.Provethefollowing hmitlOg relations;107 d)[log(1+:i)+log(1-+-;7~)+...+log(1+;f;.)]-+-~-; Ln.\:1+vn{+2++Vln-~+~J--1; [»2:i2+n.f2'+ +n-'-:-n']-.~-; [(n)"(n-1)"(1'"]e n+-,,-+...+n)--e=i; 1n______ ---- 4 -y(n+1)(n+2)..•(n+n)--.--. " e Notethatinexamples a)tod)atermbytermpassage tothelimitgives awrongresult,whf'rl'as ine)itgivesacorrect result 16.LE'tabe>0,x,:;>0lmdthesequence (Xl'XlJ,•••)defllled bythe convention thatforn>2 a) b)aX'II="'=---· 1+x"-l Shewthatincasea)thesequence tendsmonotonely tothepositive rootof x'-x--a=O; thatincaseb)ittendstothatofx2+x-a=O, butwith:2:" lyingalternately totheleftandtotherightofthebmit 17.Investigate theconvergence ordivergence ofthefolIowing sequences' a)xo,Xlarbitrary; foreveryn>2,x"=-}(X"-1+x,,_.): b)xO'Xli••"xp-1arbitrary; foreveryn>p x"=alX"-1+QlJxn-lJ+...+apX"_II (a"aBI•••,apgivenconstants, e.g.al1equalto{-): c)xO'x,posit,ve iforeveryIl~2,x"=J.r:"-, Xn_.i d)XO'Xlarbitrary jforeveryn2":2. 18.IfinEx.17,cweput,inparticular, Xo=1,Xl=2,thenthelimitof B_ thesequence is=y4 . 19.Leta"all.."apbearbitrary givenpositive quantities andletus write,for11=1,2,••• and 108ChapternSequences ofrealnumbers. Showthatx"always U1.Cl'eaSCS mon%nely andifonc,sayai,ofthegiven numbers isgreater thanalltheothers,thenx"_a1asIUllIt. (Hint:Firstshowthat s,:;S_:!<S3< ) SI=sJ=..... 20.Somewhat similarly tolastEx.,wnte " n__ ft,_ Va,+11a.+...+VaI' ,------p------ =s"and(5/)"=x.' 1';-- _ andshowthatX,.'decreases mOllo/vllely and-1a,ae.,.ap' 21.Dividethelllterval a...b(0<a<b)mtonequalparts;letXo=(I, Xl'xs•.•.,x"=bdenotethepointsofdivision. Showthatthegeometric mean 1 6 "'I,---1(bb)b--.(1fd)V'\o\,"!,··"n --> .,--expb----logxxeCl Clrl n-j-l b-<1 andtheharmonic mean -1--i------1 ->logb--=--loga' --I-+...I-. ~oXl ~n 22.Showthatinthecaseoftl]('g-eneral seqm'nre ofEx5 :r" Xl-{JXn-;;"--~-=-(J)• 23.Setx>0andletthesequence (x,,)bedefined by X.J=x~J, Xa=XX""l,..., Forwhatvalues ofXisthesequence convergent? (An~wer: Ifandonlyif 1 (-~Y<x~ei.) 24.Letlilllx"=X,hillx"=/',limx,,'=x',Ilmx,,'=/1.'.\Vhatmaybe saidoftheposItion ofthelImitsforthesequences (-x,.), (~-),(xn+xn'),(xn-xn'), (Xn·X,.'), "(x")?x '" Discuss allpossible cases. 23.Let(an)bebound.ed and(withthepossIble exception ofafewinitial turns)letusput Then(an)and(Pn)havethesameupperandlowerlimits. Thesameholds ifweput (11an) 1Pn log+n+nlogn=-n+nlogn' 26.DoesTheorem 43,3stillholdif'1=0or=+00? 27.Ifthesequences (xn)andCYn)givenin43,2and3aremonotom, tbensoarctheseqnences (xn')and(y,,')mentioned there. Exercises onChapter H. 109 2S.Ifthesequence(:n)ismonotone andbn>0,thenthesequence n baving ntbterm al+a2+···+an bl+-b2+~+ b,. isabomonotone. 29.Wehave lim_a2'=lim~,.-_a-"±_l, bn b"-bll+t provided thelimitontheTIghtexistsand(an)and(b,.)arenullsequences, with(b,.)mOllololll'. 30.Forpositive, monotone c,.'s, xo+x,+~+X~_; n+l implies CoXo+c.x,+ ±~.T!._.; Co+ct++cn provided (:;:)isbounded andCn-+00.(HereCn=Co+Cl+...+cn.) 31.Ifbn>0,andbo+bi+...+bn=BIl-+00,andxn_+00,then imphcs [>:ntl-~o~;,o~f\~:: ;t-b:~>:!']->g. 32.Forevcrysequence (xn).'weInvariably have -x+x+..·+x -hmXn<Hm 0 1 1 n<IimXn•--- n+ - (Cf.Theorem 161.) 33.Showthatifthecoefficients al"oftheTheorem ofToeplitz43,5 arePOSitwe,thenforeverysequence (xn)therelation HmXn<11111x,.'<l-mixn holds,wherexn'=all0 Xo+antXl+...+annX•. Partn. Foundations ofthetheory ofinfiniteseries. Chapter Ill. Seriesofpositive terms. §12.Thefirstprincipal criterion andthetwo comparison tests. Inthischapter weshallbeconcerned exclusively withseries,allof whosetermsarepositive oratleastnon-negative numbers. 1f.Eanis suchaseries,whichweshalldesignate forbrevityasaseriesofpositive terms,then,sinceall?;0,wehave Sn=sn_l -t~an~sn-l' sothatthesequence (Sll)ofpartialsumsisamonotone increasing sequence. Itsbehaviour istherefore particularly simple,sinceitisthendetermined bythefirstmaincriterion 46.Thisatonceprovides thefollowing simple Jdfundamental '10. Firstprincipal criterion. Aseriesu'ithpositivetermseithercon- vergesorelsediverges to+00.Anditisconvergent if,andonlyIf,itspartial sumsarebounded I. Beforeindicating thefirstapplications ofthisfundamental theorem, wemayfacilitate itsusebythefollowing additional propositions: Theorem 1.IfPisanypositiveinteger,thenthetwoseries '" '"EanandEan 710 l1=P converge anddivergetogether 2,andwhenbothseriesconverge, 00 ~ Ean=ao-j--at-1-•••+ap-1+Ean• n-0 n~p 1OnlybOllndedness ontheri!!ht(bollndedness above)comesinto que~tion, sinceanmcreasmg sequence isinvanably bounded ontheleft. 2Moreshortly: We"may"omitanarbitrary mitlalportion. -ForthiS reason,ItISoften unnece~sary tomdicate thelimitsofsummation (between which theindex 11ISmadetovdry). 110 §12.Thefirstprincipal criterion andthetwocomparison tests.III Proof.Ifsn(n=0,J,. ..)arcthepartialsumsofthefirstseries, ands,,'(n--0p,P+1,...)thoseofthesecond,then,forn2p, Sn=cao+-a1+...-+ap_1-1-s,.', whence, forn-+00,bothstatements follow,-evenwithout requiring thetermsantobenon-negative. Theorem 2.IfECnisaconvergent serieswithpositiveterms,thenso isEy"cn,ifthefactors Ynareanypositi've, butbounded, numbers 01. Proof.Ifthepartialsumsof~cnremain constantly<Kand thefactorsr"<r,thenthepartialsumsof~rncnobvIOusly remain always<rK,which,bythefundamental criterion, provesthetheorem. Theorem 3.It2,'d"isadivergent serieswithpositiveterms,then sois2,'(Jndn' itthefactors (J"areanynumbers withaposit~ve lowerbound(J. Proof.IfG>0bearbitrarily chosen, thenbyhypothesis the p:trtialsumsof2,'dn,fromasUItable mdexonwards, areall>G:(J. Fromthesameindexonwards, thepartialsumsof2,'(J"d"arethen >G.Thus.x ()"d"isdivergent. Boththeorems aresubstantially contained inthefollowing Theorem 4.11thefactorsansatisfytheinequalities o<a'<a"<cl', thenthetwoseneswithpositive terms2,'anand~IXnanconverge and divergetogether. Orotherwise expressed. Twoserieswithposllive terms 2,'a"and2,'a,,'converge anddl'verge together iftwopositivenumbers a'anda"canbe,assigned forwhich,constantly, (oratleastlromsome t~onwards)'~, a's:!!!'-s:cl'-all- 1Ilparticular therefore ifan''"anor,afortiori, ifan'~an(v.40,5~ Examples andRemarks. 1.If[(isaboundaboveforthepaltialsumsoftheserips:::anwith pO~ltive terms,thenthesumsoftinssenes IS;;:;!((v.-16,1). 2.Tilegeollutt'lc series. Givena>0,andtheso-called gcometru scnes iJan=1+a+a2+...+an+.." n=O wehave,ifa>1,thcnsn>nandso(5,.)isccrtainly notbounded itheseries 8'Veshallinfutureusually denotebyenthetcrmsofaseriesassumed convergent, andbydnthoseofaseriesassumed divergent . •Since,inthisformulottion ofthehypotheses, dIvision byallocellrs,the assumption isofcourseimplied thatan>°andnever=0.-Corresponding restrictions shouldbeobserved inthemorefrequent ca~esinthesequel.71. 112 Chapter Ill.SeriesofpositIVe terms. istherefore inthatcasedivergent. Butifa<1,then l_a"+1 5..=1+a+a2+...+an=----, (cf.p.22,(ootnlJtp 13)I-a andtherefore wehave,foreveryn, sothattheseriesisthenconvergent. SIncefurther Is-_1_1 =_1_.an+1 "I-a I-a formsanullsequence, by10,7and26,1,weatthesametimeobtain-this israrelythecase- asimpleexpression forthesumoftheseries: 00 tIan= . n=OI-a 3.The.«>1 1 1 1senes-2-------~+--+-+...hasthepartialsumsn=tn(n+I) 1·22·;3 3·4 5=(1__~)+(~_~)+...+(~__1_)=1 1 . " 2 2 3 nn+l n+l Theseareconstantly<1,theseriesistherefore converg-cnt. Asithappens, wecanseeatoncethats"-+1,sothats=1.J«>1 1 14.IIarmonic series.-2-=1+-+...+-+...isdIVergent, for, ~. ----------n=l'l 2 n aswesawin46,3,itspartialsums diverge 5to+00.Buttheseries cc;1 1 1 12;-=1+.--\--+-+... n=1n24 9 16 isconvergent. Foritsnthpartialsumis hence1 1 1 1 1 1s=1+-+-+ ..·+-<1+- +-+...+-~--..2·23 3 n·n 1·22·3 (n-l)n =1+(1-~)+(~-~)+...+(---!--~)=2 -.!..2 2 3 n-l nn' andtherefore snISconstantly<2,sothatthegivenseriesisconvergent. -The sumSisnotsoreadilyobtainable inthiscase;wehavehowever atanyrates<2, indeedcertainly s<~.Weshallfindlater(see136,156,189and210)thats='ii.- AseriesoftheformE~iscalledanharmonic series.~_ n «>1 1 1 5.TheseriesE"I==1+1+2--1+3"-1+...hasthepartial sums So=1, n~on. '1=2,andforn::s:2, 5Cf.footnote 2:l,p.1113. ~12.Thefirstprincipal criterion andthetwocomparison tests.113 1 1 1 sn=2+"2+2.3+...+~-3~' Replacing eachfactorinthedenominators bytheleast,namely 2,wededucethat ..--')1 1 1 sn~-~+2+2·:.!+...+2·2...2 1 1 1 1=2+2+2"+...+2ii=i=3 -2,,=1<3. Theseriesistherefore convergent, withsum:::;3.Weshallseclaterthatthissum------- -~ 1n coincides Wlt~_~~:. hmlteof~~_~_ers (1+n). 6.Asweremarked abovethatevcryseneswithpO,ltlve termsrepresents amonotone increasing sequence, sowesce,conversely, thateverymonotone in­ creasing sequence (xu,Xl>•••)maybeexpressed asaseneswithpositive terms, provided Xuispositive. Wencedonlywnte for,actually, Sn~Xo-I-(XI-xu)+...+(xn-xn_l)-xn andallthelIn'Sare;:::O. Fromourfundamental theorem weshallinduecoursededucecriteria whicharcmorespecial,butarealsoeasiertomanipulate. Thisweshall beenabledtodochieflybythcinstrumentality ofthetwofollowing "com­ parisontests"*: Com-p~n test oL~h_e_~t kin.d. 72...--- LetECnandEdnbetwoserieswithpositiveterms,alreadyknownto bethefirstconvergent, theseconddivergent.!fthetermsofagivenseries Ean>also?oithpositiveterms,satisfy,foreveryn>acertainm, a)thecondition thentheseries};a"isalsoconvergent. -If,however, foreveryn>acer­ tainm, b)?vehaveconstantly thentheseries};a"mustalsodiverge 6. Proof. By70,1,itsufficestoestablish theconvergence ordi- '"vergence of};an'Incasea)theconvergence ofthisseriesresults n=m+l 00 atonce,by70,2,fromthatof}; Cmbecause byhypothesis wemay, iI=m+l *German: Vergleichskriterien. (Tr.) RGarususedthiscriterion in1812(v.WerkeIII,p.140).Itwasnot,how­ ever,formulated explicitly, norwasthefollowing testofthe2ndkind,beforeCaltchy, Analyse algebrique (Pans1821). 114- Chapter Ill.SeriesofpO&ltive terms. foreveryn>m,writean=I'ncn'withI'n<1.Incaseb)thedi· '"vergence resultssimilarly 7fromthatof.Edn,because herewemay n=m+l writea"=b"d",with (~"~1. 73. Compm'ison testofthe2ndkind. Let:Ec"and2:dllagaindenoterespectively aconvergent anda divergent seriesofpositive terms.Ifthetermsofagivensenes2'aIof positive termssatisfy, foreveryn~acertainm, a)theconditions thentheseries ~'anisalsoconvergent. If,however, foreveryn~ acertainm,wehave b)constantly then2allmustalsodiverge. Proof. Incasea),wehaveforeveryn~HI 4n+1.-:;::an. Cn+1-Cn Thesequence oftheratioI'=a"is,fromacertainpointon· o tlen wards,monotone descendmg, andconsequently, &inceallItstermsare positive, itisnecessanly bounded Theorem 70,2nowestabhshes the convergence. Incaseb)wehave,analogously, ,~n+l~=n,sothatthe n+\ n ratiosb"=~:increase monotonely fromapointonwards. Butasthey areconstantly positive, they then haveapositive lowerbound. Theo­ rem70,3nowprovesthedivergence. Thesecompanson testsorcriteriacanofcourseonlybeuseful tousifwearealready acquainted WIthalargeDumber ofconvergent anddi\ergentserieswithpositive terms.Weshalltherefore haveto layinaslargeastockaspossible, sotospeak,ofseries whose con· vergence ordivergence isknown. Forthispurpose thefollowing examples mayformanucleus: 7Orelse-almostmoreconcisely-:Incasea)everyboundabove ofthepartialsumsof~c"isalsooneforthepartial sumsof~an;andin caseb),thepartialsum&of2.'a"mustullimately exceedeverybound,since thoseof~;dndoso. §12.Thefirstprincipal criterion andthetwocomparison tests.Hi) Example~. en1 divergent,:E .,convergent. Bythefir<tcomparison 74. 11:..111"'T1I.:E wa~seentobe 11--:1n test,theso-called harmOniC series 't:1E--n<:l.n=l istherefore certamly divergent for <X<:I,convergent for"~2.Itiq,however, onlyknownmthecaseIX.~evenmtegerhowItssummayberelatedtonumbers occurnng inotherconnections; for,mtance we~hallseelateronthatfor!X=4 .77'thesumIS90' 2.Bythepreceding, theconvergence ordlvenl;ence of:E~onlyremainsnot questIOnable incaqe1<ex<2.Wemayproveaqfollowqth.Itthe,enescunverges forevery IX'1:ToobtainaboundaboveforanypartIal,umsnofthesenes, choo,eksolargethat:lk.>n.Then ,_,..(11)._(1 1 1 +1)+ _(1_ 1 ) Sll--"'!_I-l~ 2z+:j" I4'+5'rIP7'...f-(:!k"')Z+"'~ (:!k_I)Z • Herewegroupinoneparenthe,is thosetcrmswhosemdIcesrunfromapCl\\er of2(mc1usive) tothenextpowerof2(exclUSIve). Replace, meachp.urofparen­ theses,everyseparate termbythefirst;thiSmvolves anmcrea.,e atvalue,mdwe havetherefore 2 4 2'-1 sn:S1+2~+4"+'"+(21Hf'" 1Ifwenowwnteforbrevity:Ti-i=~,-apO'ItIve number certamly<1,since IX>I,-thcnwehave __ • !_,_1-~l.. 1 s,,~-,I+~+~ +...-I-~ -i-~-<I--~; and~incethISholdsforeveryn,thepartialsumsofoursenesarebounded, and theseriesitselfISconvergent, q.e.d.(Cf.77.) Allharmonic seriesE\Ior <X~1arediver{!ent, andfor <X>l,cOl/vergent.n Inthese,withthegeometric senes,wchavealready quiteauseful,tackofcom­ p.lrisonseries. 3.Senesofthetype en1E-- n~l(an+b)'" whereaandbaregivenpositive numbers, alsodiverge for0<<:1,converge for !X>1.Forsince not(I)!X 1 (an+b)"X=:+: ->-ai'wchave and70.4nrovesthetruthofourstatement. 116 Chapter Ill.Seriesofpositive terms. Accordingly theseries inparticular, areconvergent foret>I,divergent forot;:S:1. -F) 4.IfEcnISaconvergent serieswithpositive terms,andwededucefrom n-,0 itanewseries1:cn'byomitting any(possibly aninfinitenumber) ofItSterms,or byinserting inanywaytermswiththevalue0,thus"dllutmg" theseries,thenthe resulting "sub-series" 1:cn'ISalsoconvergent. Foreverynumber whieh ISabound aboveforthepartialsumsof1:cnisthenalsoaboundaboveforthoseofthenew senes. Inaccordance withthis,thesenesE}..'whereprunsthrough allpnmem- tegralvalues,I.e.thesenes P I I 1 1 1 2""+~+LIi+~+Het+... iscertainly convergent forex>1.(Ontheotherhand,ofcourse, wecannot conclude without further examination thatItdiverges forex;;;::;I!) 5.Since:Eanisalready recognised asconvergent for0;;;::;a<I,we!Dfer inparticular theconvergence of IfZI'E.,••••Zn,•'.denoteany"digits", i.e.ifeachofthembeoneofthe numbers 0,I,2•••'.9,andifZoisanyinteger ~0,then,by70,2,theseries <XlZ2J10nn n=O isalsoconvergent. -Thusweseethataninfinite decimal fraction mayalso beregarded asaninfinite series. Intliissensewemaysaythateveryinfi1l1te decimal fraction isconvergent andtherefore represents adefinite realnumber.­ Inthisformofseneswealsohave,according toourcustomary orderofideas, animmediate conception ofthevalueofitssum. vi'13. Theroottestandtheratiotest. Weprepare thewayforamoresystematic useofthesetwo comparison tests,bythetwofollowing theorems. Ifwetakeascom­ parisonseries,tobeginwith,thegeometric series2:a",with0<a<1, thenweimmediately obtainthe 75. Theorem 1.It.givenaseries~a"atpositive terms.wehave, fromsomeplaceonwards intheseries,a"<anwith0<a<1,i.e• ..- Va,,~a<lt §13.The ~oottestandtheratiotest. 117 thentheseriestSconvergent. Ifhowever, frumsomeplaceonwards> "/-1((..21, thentheseriesisdivergent. (Cauchy's roottests.) n_Supplementary note.Fordivergence itclearlysuffices thatVan~1 shouldbeknowntoholdforinfimtely manydistinct ';aluesofn.Forwethen alsohave,forthosevaluesof'11,an~1;andaparticular partialsum SOlwill consequently exceedagiven(positive integral) number G,ifmischosenso largethattheinequality an>IoccursatleastGtimeswhile0::;;'11:£m.The sequence (sn)istherefore certainly notbounded. Thesecondcomparison testgivesimmediately: Theorem 2.If,fromsomeplaceonwardsintheseries,an>0,and a"+l~a<l a.. ' thentheseriesIanisconvergent. Ifhowever, fromsomeplace onwards, tln+1~1a-, " thentheseriesIanisdivergent. (Cauchy's ratioteste.) Remarks andExamples. n_ 1.Inboththesetheorems, itisessenhal forconvergence thatyanand ~"+trespectively shouldbeultimately lensthanafixedproperfraction a.11 a" doesnotatallsufficeforconvergence thatweshouldhave76. 1foreveryn.Anexample presents itselfatonceintheharmonic series.2-I 11 forwhichwecertainly alwayshave andalso1 I 1--:-=1---<1,n+l '11 '11+1 thoughtheseriesdiverges. Itisquiteessential thattherootandratioshould notapproach arbitrarily nearto1. 2.Ifoneofthesequences(-va.:-)or(~i~)isconvergent, saywithlimitex, thentheorems 1Ilnd2showthattheseries ~allisconvergent ifa<1, •Analyse alg~brique, p.132seqq. •Analyse alg~brique, p.134seqq. 118 Chapter lIr.SenesofposItive terms. n- 1--adivergent ifa>1.ForsupposeVall-IX<1,forinstance; thens=~>0 andmmaybedetermined sothat,foreveryn>m,wehave Andsincethisvalueais<1,theorem 1provestheconvergence. Ifonthe a-IcontraryIX>1,thens'=-r>0Jandm'maybesodetermined that,for everyn>m',wehave n_ l+aVa">a-s'=-2-=IJ. Andsincethisvalueais>1,theorem 1provesthedivergence. -Theproof inthecaseoftheratioisquiteanalogous. IfIX=I,thesetwotheorems provenothing. 3.Thereasoning justapplied in2.isobviously alsolegitimate when _..n/_ _a"+1 ..n/_ a"+1limva"orlim--is<1,intheonecase,andlimva"orHm---is>I,a" - --an intheother.Ifoneofthe<;cupperorlowerlimitsis=1,ortheupperlJmit >I,thelower<I,thenwecaninferalmostnothingastotheconvergence ordiver­ genceof1:an-Thesupplementary noteto75,I,however, showsthat,intheroot test,ItISsufficient fordivergence 10that-IImV~:>1. 4.Theremarks justmadein2.and3.aresoobvious that,insimilar casesinfuture,weshallnotspecially mention them. 5.Therootandratiotestsarebyfarthemostimportant testsusedin practice. Formostoftheserieswhichoccurinapplications, thequestIOn of convergence ordivergence canbesolvedbytheirmeans. Weappend afew examples, inwhichx,forthepresent, represents apOSItive number. a)Inaxn(aarbitrary). Herewehave n+l 1as--=1+--1andispermanently positIve (v.SS,8).Theseriesisn n therefore -andthiswithout reference tothevalueofa-convergent if x<1,divergent ifx>1.Forx=lourtwotestsareinconclusive; however wethengettheharmonic series,withwhichwearealready acquainted. b)2(n+1')x"=.J:(_I)"(-l'-1)x",,=0n ,,=0 n Herewehave(paninteger>1). a"+1(n+1'+l)(n+1')...(n+2)1'1n+l'+]--= --- ·x= :z:_:z:.a"(n+p)(n-l+p) ...(n+l)·1'1 n+l 10Thereby thecriterion obtain& adisjunctiv, form.Za"isconvergent_n_ ordivergent according aslimvanis<1or>1.(Further details in§§36 and42.) §13.Theroottestandtheratiotest. 119 Hencethisseriestooisconvergent forx<1,divergent forx>1,whatever be thevalueofp.Forx=1andp~0itobviously diverges, sincethenan+l~1for an everyn.Inthecaseofconvergence weshalllateronfindforitssumthevalue (_~1)Pr1. I -x c) Herewehaveforeveryx>0 theseriesistherefore convergent foreveryx>0.Forthesumweshall lateronfmdthevalueeX. d)\'xn• f>0ISconvergent orx= ,~nn Vx n'nn=xas ---.O.n e) f)-,1. 11' ."/-0'5----- ISconvergent ,asagamVan-+•~(Iogn)n 1 1)'---convergent, because an<<;; .:;..Jl+n~ n' n! 1.2...n<2f >22:-;;convergent, because an=--.---- _-2oreveryn=.in nn...n-n 1divergent, because a,,:/---1-in+ 1convergent, because a"<-----;.;:-. n2 Ig)E--(pfixed>0),ISdivergent, SInceby38,4fromsomenon­(logn)P wards(logn)P<n. h)'"1 ..<.J ISconvergent, aswemayatoncerecogmze bywntmgthe(lolot n)lo~n generictermintheform 1 11Inthisseries,summation mayonlybeginwithn=2,sincelog1=O. Suchandsimilarobvious restrictions weshallinfuturenotalwaysexpressly men­ tion;itsuffices, forthequestion ofconvergence o"'i-dlVergence, thattheindicated termsoftheseries,fromsomeplaceonwards, havedeterminate values.-Inall thatfollows, asalreadyagreedonp.83,thesign"log"WIllalwaysstandforthe naturallogarithm, i.e.thattothebasee(46a). 120 OntheotherhandChaptcr Ill.Seriesofpositive terms 1: 1 =.2e-(lOlllOll11)' (logn)loglog71- isdivergent, because by3S,4andEx.13,(loglogn)9<lognfromsome'I 1onwards, sothatthegeneric termoftheseriesis>-n. §14.Seriesofpositive, monotone decreasing terms. Beforepassing fromthesequiteelementary considerations, we willmention aparticularly simpleclassofseriesofpositive terms, namely thoseserieswhose termsan'atleastfromsomeplace onwards, formamonotone sequence. Tothisclassbelongnearlyall theseriesgivenasexamples aboveandalsothemajority ofthose whichoccurinapplications. Forsuchserieswehavethefollowing: QC 77. Cauchy's theorem ofconvergence12•11.2anisase,.ieswhose 71=1 te,.mslo,.mapositive monotone dec,.easing sequence (an)'thenitcon­ ve"gesanddive"ges with Preliminary remark. Whatisparticularly remarkable inthistheorem isthatitshowsthatasmallproportion ofallthetermsoftheseriessuffices todetermine theconvergence ordivergence ofthewholeseries.Forthis reasonitisalsocalledthecondensatton theorem. Itshowsthattheharmonic serics.2~,forinstance, iscertainly dlver­n gent,foritconverges anddiverges withtheseries 2kZ2k=I+I+I+ ... 1whichisunmistakably divergent. Andspeaking generally, theseries is 'la inferred toconverge anddiverge withtheseries butthisisageometric seriesandtherefore converges ordiverges according asIX>1orIX::;1. These ex~ples alsoshowusthattheconvergence ordivergence of .E2k agkisoftenmoreeasilyascertained thanthatoftheseries.Eallitself; itisjustinthisthatthevalueofthetheorem lies. Proof. Wedenotethepartialsumsofthegivenseriesbys... thoseofthenewseriesbytk•Thenwehave(cf.74,2) 11Analyse algebrique, p.135. §14.Seriesofpositive, monotone decreasing terms. 12J a)forn<2k srI:-::;::al+(ag+aa)+ + (agk+'"+a,Ml-l) <al+2ag+4a4+ + 2ka2"=tk• i.e. b)forn>2k srI>al+a2+(aa+a4)+...+(a2IH+t+...+agh) >~al+ag+2a4+...+2k-1agk=~tk' i.e. 2s,,~tk' Inequality a)showsthatthesequence (s,,)isbounded ifthesequence(tJ isbounded; inequality b).conversely, thatif(s,,)isbounded, sois(tk). Thetwosequences aretherefore eitherbothbounded orbothun­ bounded, andtherefore thetwoseriesunderconsideration eitherboth converge orbothdiverge, q.e.d. Beforegivenfurtherexamples illustrating thistheorem, we:nay extenditsomewhat 13;foritisimmediately evident thatthenumber 2 playsnoessential partinthetheorem. Infactwehave,more generally, the Theorem.IfIanisagainaserieswhosetermsformapositive78. monotone decreasing sequence (an)'andif(go'gl'...)isanymonotone increasing sequence ofintegers,thenthetwoseries '"Za",,=0and areeitherbothconvergent orbothdivergent, pro'IJided gk'fa'every k>o.fulfilstheconditions gk>gk-l>0andgk+l-gks:M·(gk-gk-t) inthesecondofwhichMstandsforapositiveconstantH• Proof. Exactly asbeforewehave a)forn<gk'-denoting byAthesumofthetermspossibly preceding agD(orotherwise 0),- srI<Sgk~A+(ariD+...+ag,-l)+'"+(arl"+...+arlHl-l) ~A+(gl-go)agD+...+(gH1-gk)arlk, i.e. 18Schlomilch. 0.:Zeitschr. f.Math.u.Phys.,Vol.18,p.425.1873. UThesecoudcondition signifies thatthegapsinthesequence (gk),re­ latively tothe<iequence ofallpositive integers, mustnotincrease attoo greatarate. 122 ChapterIll.Seriesofpositive terms. b)forn>gk $">SIl">(aMI+...+all,)+...+(afTk_I+!+...+all,) :2(gl-go)all,+ + (g"-g~-l)au,,' Msn:2::(g2-gl)afT,+ + (g"+1-g,,)all" MSn2t"-to' Andfromthetwoinequalities thestatements Inquestion followin thesamewayasbefore. iD. Remarks. 1.Itsuffices ofcoursethattheconditions ineithertheorem befulfilled fromandafteradefinite placeintheseries.Therefore wemay,intheextended theorem, suppose, asaparticular case, wheregisanyrealnumber> 1and[gIllthelargestinteger notgreater thangk.Wealsosatisfytherequirements ofthistheorem bytaking g,,=k9,=k8,=k4,•••• 00Withg"=k9weobtain,forinstance, thetheorem thattheseries ~an,-if n=O (a..)isapositive monotone decreasing sequence, -converges anddiverges with Wemayalsoreplace thislastseries,according to70,4,bytheseries ~kak,=at+2a.+3ag+.... 001 2.2}--- isdwergent, -although itstermsarematerially lessthan n=2nlogn thoseoftheharmonic series;foraccording toourtheorem, thissenescon­ vergesanddiverges with 002" 001 k~2k.lOg-(2k)=\~(rOg2) k andistherefore, by70,2,liketheharmonic series,divergent. Thedivergence ofthiSseriesandoftho~econsidered inthenextexamples wasfirstdiscovered byN.H.Abel15(v.CEuvres 11,p.200). 3..i;1: isalsostilldivergent, although itstermsareagainn=3nogn·oglog11 considerably lessthanthoseoftheAbel'sseriesjustconsidered. Forby Cauchy's theorem itconverges anddiverges with 00 2" 00 12} =2} • k=22k.log2".log(log2")-k=2klog2·10g(klog2) , 15NlelsHenrikAbel.bornAug.5th,1802,atFindoenearStavanger (Nor. way),diedApril6th,1829,attheFroland ironworks, nearArendal. §14.Seriesofpositive, monotone decreasing terms. 123 logox=x, (Pinteger ~1),andthis,sincelog2<1,haslargertermsthanAbel'sseries.2__1_discussedklogk above,andmusttherefore diverge. 4.Thuswemaycontinue aslongasweplease. Toabbreviate, letus denotebylogrxtherp1•repeated or;teratedlogarithm ofapo~itive number x, sothat 10glX=logx,loggx=log(logx),••• logrx=log(logr_1x). Wemayalsotake10g_1xtodenotethevalueer. Theseiterated logarithms onlyhaveameaning ifxissufficiently large; thuslogxonlyforx>0,loggxonlyforx>1,log.xonlyforx>e,and soon;andweshallonlyplacetheminthedenominators ofthetermsofour seriesiftheyarepositive, i.e.logxonlyforx>1,loggxonlyforx>e, logsxonlyforx>eC,andsoon.Iftherefore wewishtoconsider theseries .2 1 nnlogn·10g"gn...logpn thenthesummation mustonlybeginwithasuitably largeindex,-whose exactvalue,however, (by70,I),doesnotmatter. Sincethelogarithms increase monotonely withn,andthetermstherefore decrease monotonely, theseries, byCauchy's theorem, converges anddiverges with 1..2 k k kklog2 .log,2..,logp2 IIndthis,since2<e,mustcertainly diverge,if ..2 1 kklogk..•logp_lk diverges. Sincethedivergence ofthelatterserieswasproved forp=1(nnd p=2),itfollows byMathematical Induction (2,V)thatitdiverges for liveryp>1. 5.Theseriesaboveconsidered, however, become convergent ifweraise thelastfactorinthedenominator toapower> 1.That.2-!.-converges for na ">1,wealready know.Ifweassumeprovedforaparticular (integer) p>1, thattheseries16 ~ 1(*).oC.J (a>I) kk·logk•••10g'p_2k·(logp_l k)O isconvergent, itfollows justasbeforethattheseries ~ 1 .oC.J (">1)" n·Iogn...logp_ln·(logpn)° isalsoconvergent. Forthis,bytheextended Cauchy's theorem7S,converges anddiverges withtheseries-wechoose gk=3k - ..2ak+1_ak kaklog3k•••(logp3k)o' 18Forp=I,thisreduces totheseries.2!.. ka 124 Chapter Ill.Seriesofpositive terms As3>"thisserieshasitstermslessthanthoseoftheseries(.)(assumed convergent), ifthetermsofthelatteraremultiplied by2(whichby70,2 leavestheconvergence undisturbed). Theseriesbrought forward inthetwolastexamples willlateronrender usmostvaluable services ascomparison series. Wewillproveonemoreremarkable theorem onseriesofpositive monotone decreasing terms,although itanticipates toacertainextent thegeneral considerations onconvergence ofthefollowing chapter (v.82,Theorem 1)- 80.:.-Theorem.Ittheseries~anofpositive monotone decreasing term.~ istoconverge, thenwemusthavenotonlyan-.0,but17 nan-O. Proof. Byhypothesis, thesequence ofpartialsumsao+at+..+an=snisconvergent. Havingchmene>0,wecantherefore so choosemthatforeveryv>mandeveryA2:1wehave J.e. 8 /ly+]+aYH+...+ayH<2' Ifwenowchoosen>2m,then,takingv=[~n],thelargestinteger notgreater than ~n,wehave,,2:mandtherefore E ay+I+ay+2+...+a,..<2; afortiori, therefore, (n-v)an<; and " 8-ir·an<2'i.e.nan<6. Therefore nan-.0,q.e.d. Remark.Wemustexpressly emphasize thefactthatthecondition na.-+0isonlyanecessary, notasuff~cient onefortheconvergence ofour presenttypeofseries,i.e.ifnandoesnottendto0,thentheseriesinquestion iscertainly divergent18,whilena.-+0doesnotnecessarily implyanything astothepossible convergence oftheseries. Inpointoffact,theAbel'sseries Z-1-1-diverges, although ithasmonotone decreasing termsandnogn 1nan=-l--+O.ogn 17OllVier.L.:Journ.fd.reineu.angew.'Math.,Vol.2,p.34.1827. 18Accordingly, theharmonic seriesZ"!'-,forinstance, mustdivergen because ithasmonotone decreasing terms,butn.1__doesnottendtoO. IJ Exercises on<'hapter 111. 125 Exercises onChapter Ill. 34.Investigate thebehaviour (convergence ordivergence) ofaseriesz:an,forwhicha",fromsomeindexonwards, hasthefollowing values: ............1---1' 1"t­nn../na ?iT'_(nl)B (2n)I' (vn+i-v'n), aloglogn J(3 S )-yn+l--yn. alogn, vna •1 1 (loglogn)IOlln' (logn)"I--n- ,1 1+a'" 35.If:J:dndiverges, soalsodoes.2)-~,,~-.Whatisthebehaviour ofl+d" .L;~d,,__and.2~,,~? (d,,>0). 1+ndll 1+n2d" 36.Underthesameassumption that:J:dndiverges andd">0,whatis .",d,,?thebehaviour ofthesenes £.J~+ 11 37.Suppose Pn--+00.Whatisthebehaviour oftheseries '"1? £.JP"lOlllogtI'\-'1'"1 £.JP,,'" £.Jp"IOIln' 3S.Supposei"--+~,butwith e<~im(Pnh-P,,)<+C'6• Whatmustbethe\perandlowerlimitsofthesequence (e..)sothat \,,---- ~_.2Pn1en converge orsothatitdiverge? 3D.Foreveryn>1,nIl 1 1-<1+-+~~+-+...+-----<'J2 2 3 4 2"-1 40.Thesequence ofnumbers x..=[1+}-+...+-~-lognJ2 n ismonotone descending. 41.IfIallhaspositive termsandisconvergent, thenIVa..a"+1isalso convergent. Showbyanexample thattheconverse ofthistheorem isnot trueingeneral, andprovethatitdoesnevertheless holdwhen(a..)ismonotone. '\Ia42.If1:anconverges, andan~0,thenE~alsoconverlzes. andalsoind....dn theseriesE(t~~T6' forevery 11>O. (061) 126 Chaptcr IV.Scricsofarbitrary terms. 43.Everypositivc realnumber Xlis,inoneandonlyoncway,ex­ pressible inthcform whereanisanon-negativc integerwithanSn-1forn>1,subjecttothecon­ ditIOnofnotbemg ~71-1foreverynafteradefimte no.IfXIISratzonal, andonly then,thesenesterminates. 44.If0<x;SI,thenthereisoncandonlyonescqucncc ofpositive integcrs(kv),with 1<hI~kJ:sh3;S-.., forwhich 1 1 1x=--+--+ ...+-----+ ....kIkIk2 kIk~.••k" zisrational if,andonlyif,thekv'sareallequalaftersomeindexvl" Chapter IV. Seriesofarbitrary terms. §15.Thesecondprincipal criterion andthealgebra of convergent series. Aninfiniteseriesi3a",-whosetermsarenownolongerassumed ,,~O subjectt'd toanyrestriction, butmaybearbitrary realnumbers, was,weagreed, tobeconsidered asessentially anewsymbol for thesequence (s,,)ofitspartialsums s"=ao+at+...+an(n=O,I,2, ...) andweproposed totransfer immediately totheseriesitselfthede. signations introduced tocharacterise theconvergence ordivergence of(sn)'Thecaseofconvergence againoccupies ourmainattention. Thesecond maincriterion(47-51), expressing thenecessary and sufficient condition forconvergence, atonceprovides thefollowing 81. 0Fundamental theorem (Firstform).Thenecessary andsufficient condition tortheconvergence oftheseries2)a"isthat,havingchosen anyII>0,wecanassignanumber no=no(e.)suchthattorevery n>nuandeveryk~1,wehave ISn+k-S"<E, thatlStosay,inthepresentcase,that IOn+l+an+:l+···+an+k I<E. §11i.Thesecondprincipal criterion andthealgebra ofconvergent series.127 Starting withthesecondformofthemaincriterion, wealsoob· tainforthepresent fundamental theorem thefollowing °Second form.Theseries ~'anconverges if.andonlyif.givenSla. aperfectly arbitrary sequence (k,.)ofpositiveintegers. -thesequence ofnumbers Tn=«((n+l+((,1/+2+...+a..+kn) invariably provestobeanullsequence!. Andasbeforewecan extendthissomewhat tothe °Thirdform.Theseries.2allconverges xf,andonlyif.givenSIb. twope1'fectly a1'bittary sequences IV")and(k,Jofpositiveintegc1's. of whichthefxrst.atleast,tendsto+00.-theseqltenCe 01numbers Tn=(a...+I+a"n+2+ ...+a.,,+k,,) invariably provestobeanullsequence. Remarks. 1.Asenesrepresents essentially anewsymbolic expression forse· quences ofnumbers, andinrarticular, aswcremarked, notonlyeveryseries repre~ents asequence, buteverysequence l~alsoexpreSSible asaseries;all remarks andexamples givenonp.84havetheirparallels here. 2.Thecontents ofthefundamental theorem ma)"bpformulated a~follows: GivenE>0,every1Jorlwn oftheseries,however long,prOVided onlyitsinitial indexbesufficiently large,musthaveasumwhoseabsolute value i~<E. Or:GivenE>O.wemustbeabletoassignanindextUsothatforn>tUthe additIOn, to.In'ofanarbitrary number oftermsimmediately consecutive toancan onlyalterthiSpartialsumbylessthane:. H.Ourpresent theorem~ andremarks ofcoursealsoholdforsenesof po,itlve terms Thisthereaderbhouldverifyineachseparate case. Alinitcpartoftheseries,suchas a..+1+a"+2+...-I-a"+J,, wemayforbrevitycalla1)01fionoftheseries,denoting itbyT.ifit beginsImmedIately afterthe1,thterm.WhenreqUIred, wemayfurtherex­ plicitlyindicate thenumberoftermsintheportionbydenoting thisby T.,I.'IfwearcconSIdering anarbitrary sequence ofsuchportions whoseinitialindex-++00.weshallrefertoitforshortasa"se­ tjllelll.'eofpm·tions" ofthegivenseries.Thesecondandthirdform ofthefundamental theorem maythenalsobeexpressed thus: °4tllform.These1'ies.2anconverges if.andonlyif,evel'ySJc. "sequence 01po1'tions" ofthesetiesisanullsequence. 1Itissubstantially inthisformthatN.lJ.Abele,tabli,hes thecriterion inhisfundamental lDemoir ontheBmomlal serIesOoum f.dicreineu.angew. Math.,Vol.I,p.311.1826). 128 Chapter IV.Seriesofarbitrary terms. ,Remarksand exa mpies. 1.Ia..isthusdivergent if,andonlyifatleastonesequence ofportions canbeassIgned which ISnotanullsequence, Fortheharmonic series IJE-,forinstance, wehaven 1 1 1 1 1 T..=T.....=n+I+n+2+"'+2-;> n'2n="2' Thesequence (T..)istherefore certalOly notanullsequence, andtherefore ~I'd'L.i-ISIvergent.n 12.ForJE----;\wehaven I 1 Tv=Tv,l=(v+1)'+..,+(v+J.)'111<v(v+1)+(v+1)(v+2)+..,+(v+J.-I)(v+J.) =(~-v~-1)+C~-1-v~-2)+,..+c.+~-1-v~i)=:-;+-i' 1therefore Tv<-,sothatIv--+0,whenv--++00.Thesenesthereforev converges. 3.Forthesequence 00(_I)"-1 1 1 1 1JE----==1- --+---.+- -+... n=l n 2 3 4 5 Whether kisevenorodd,theexpression 10brackets iscertainly positive and <_1_1-,Forifwetaketogether, inpairs,eachpositive termandthefollow-n+ ingnegative term,thesumofthetwoi,ineachcasepositive. IfkISeven alltermsareexhausted inthiSmanner,ifkisunevenapositive termremains, sothatineithercasethecomplete expression isseentobeposItive. If,on theotherhand,wewriteitintheform alltheterm,arenowexhausted whenkisoddandanegative termremains 1overif11iseven,sothatinbothcasesonlysubtractions fromn+1occur. andthustheexpression is thio;involves1<--1' Aswenowhaven+ 1IT..I=IT..,kI<n+1 T"--+0, andourseriesconverges. -Weshallseelater(cf.120)thatitssumcoincides withthelimitofthesequence 46,2andhasthevaluelog2. §15.Thesecondprincipal criterion andthealgebra ofconvergent series.129 usually, asthesocalledremainder of byrn(sothatsn+rn=s=thesum Nowwemay,intheinequalitysumis denoted series).Tothesefourseparate formsofthesecondfundamental criterion we mayatonceattachthefollowing simplebutimportant considerations: Sinceinthesecond form,byputtingkn=1,weobtain an+1-+0,wehavealso(by27,4),an-+0,i.e.wehavethe oTheorem 1.Inaconvergent series,thetermsannecessarily formS2, anullsequence: an_O. Thatthiscondition isnotsufficient forconvergence, weknow already, fromtheexample oftheharmonic series. If,on.theotherhand,wealready knowthat.2anconverges, Cl) thensodoestheseriesan+1+an+,+an+3+...-.2aF,whose 1'=n+1 theseries.2an' ofthecomplete Ian+1+an+2+...+an-H,I<8, validforn>noandeveryk::::::1,allowktoincrease beyondallbounds andsoobtain,foreveryn>no'rn::;;:e.Thuswehavethe Cl) °Theorem 2~The1"em ainde1"s1"..=2JaFofaconvergent series F=n+l i.e.thenumbers (r_1=s),r0'1"1'1"2'•••,rn'..., alwaysformanullsequence. InSO,wesawfurtherthatifthetermsofaconvergent series 2:an(ofpositive terms)aremonotone decreasing, then,overand abovethetheorem justproved, thecondition nan-+0musthold. Thatthisneednolongerbethecaseinseriesofarbitrary termsis alreadyshewnbytheseriesgiveninSIc,3.Wecan,however, show thatwemusthave Q,+2a2+...+nan_0 n i.e.thatthetermsofthesequence (nan)aresmallontheaverage. Infactwehave2themoregeneral co °Theorem 3.If.2a..isaconvergent seriesofarbitrary terms n=O andif(Po'P1'••.)denotes anarbitrary monotone increasing se­ quenceofpositive numbers tending to+00,thentheratio 20ao+PIap~...+Pnan_o. 2L.Krollecker, Comptes rendusdel'Ac.deParis,VD!.103,p.980.1886. -Moreover, thiscondition isnotonlynecessary, butalso,inaquitedetermmate sense,sufficient, fortheconvergence oftheseries1:an(cf.Ex.58a). 130 Chapter IV.Seriesofarbitrary terms. Proof. By44,2,s"-simplies "lPiSo+(pg-Pi)Si+...+(1'"=_~"-1)S"_l_S. P" Since :f1~~~-+0ands71-+s,wemusttherefore have (Pi-Po)So+(Po-Pi)51+...+(1'..-P..-1)Sn-10 SOl- p" -+• Butthisisprecisely therelatIOn wehadtoprove,asmaybeseen atoncebyreducing tothecommon denominator Pnandgrouping in succession thetermswhichcontainPo.PI'...,Pnrespectively 3. Asregards anycondition forconvergence whatsoever, wehave torepeatexpressly thatthestipulations madethereIDalwaysconcern oronlyneedconcern -thosetermsoftheserieswhichfollow onsomedeterminate one,whose indexmaymoreover bereplaced byanylargerindex. Indeciding whether asenes ISorisnotcon­ vergent. thebeginning oftheseries,-asitisusually putforbrev­ ity,-doesnotcomeintoaccount. Thisweexpress moreexactly inthefollowing co °Theorem 4.Ifwededuce. fromagivenseries.2an'anew n=Oco series.2an'byomitting afinitenumber ofterms,prefixing afinite n=O number ofterms,oraltering afinitenumber ofterms(ordoing allthreethingsatonce)andnowdesignating afreshthetermsofthe seriessoproduced byao',a/•.../'theneitherbothseriesconverge orbothdiverge. Proof. Thehypotheses implythatadefinite integerq>Oexists< suchthatfromsomeplaceonwards, sayforeveryn>m,wehave , an=an+q• Everyportion oftheoneseriesistherefore alsoaportion of theother,provided onlyitsinitialindexbe>m+IqI.Thefun­ damental theoremSIaimmediately proves thecorrectness ofour statement. 8Instead ofthepositiveP..wemay(cf.44,3and5)takeanyse­ quence(P..),forwhich,ontheonehand,IP..I-+00and,ontheother,a constant Kisassignable forwhich IPoI+IPI-PoI+...+IP..-P..-1I<KIP..I foreveryn. 41.e.inshort:".•.bymaking afinitenumber ofalterations (27,4)in thesequence (a,,)ofthetermsoftheseries.••" ~15.Thesecondprincipal criterion andthealgebra ofconver"ent senes.131 Remark. Itshouldbeexpressly notedthatforseriesofarbitrary terms,compari. 50ntestsofeverykindbecome entirely powerless. Inparticular, oftwoseries 2anand::::an'whosetermsareasymptotically equal(an""a,,'),theonemay . (-l)nqUItewellconverge andtheotherdiverge. TakcforInstance an=----­n and, 1an=an+-1--'nogn Finally weprovethefollowing criterion ofconvergence, which appears almostunique inconsequence ofitsparticularly elementary character, a'ndrelatestotheso-called nlternaUny series, i.e.toseries whosetermshavealternately positive andnegative signs: Theorem 5.[Leibniz's rule5.]Analternating series,forwhich theabsolute values ofthetermsformamonotone nullsequence, isinvariably convergent. Theproofproceeds onquitesimilar linestothatofSI(!,3. ForIf.2-'a,.isthegivenalternatmg series,thenallhaseitherthe SIgn(-It,foreveryn,ortheslf,n(_1)71+1, foreveryn.Ifwe write,therefore,IanI=IX,.,wehave AstheIX'Saremonotone decrea;,ing, wemayconVlllce ourselves precisely asintheexample referred to,thatthevalueofthesquare bracket isalways positive, butlessthanItsfirsttermIX..+1'Thus ITnI=IT".7'1<"71+1' which,sinceanformsanullsequence byhypothesis, involvesTn-.0 andtherefore convergence of.2an'bySIc. Thealgebra ofconvergent series. Already in69,2,3,ithasbeenemphasized thattheterm"sum", todesignate thelimitofthesequence ofpartialsumsofaseries, ismisleadmg insofarasitarouses abeliefthataninfinite serie& maybeoperated onbythesamerulesasan(actual)sumofadefinite number ofterms,e.g.oftheform(a+b+c+d),say.Thisisnot thecase,however, andthepresumption istherefore fundamentally erroneous, although someoftherulesinquestion doactually remain validforinfinite series.Theprincipal lawsinthealgebra of(actual) sumsare(according to2,IandIll)theassociative, distributive and commutative laws.Thefollowing theorems areintended toshowhow fartheselawsremain trueforinfinite series. &Letters toJ.Hermatln of26.VI.1705andtoJohnBernoulli or10.1.1714. 132 Chapter IV.Seriesofarbitrary terms. S3. uTheorem 1.Theassociative lawholdsforconvergent infiniteserie$ unrestrictedly inthefollowing senseonly: ao+a1+a~+...=s implies (ao+a1+...+a,.,)+(av,+!+a",+2-1-...+a.,,)+...=s, ifVI'V2,..•denoteanyi1tcreasing sequence ofdifferent infegers and thesumofthetermsenclosed ineachbracketisconsidered asone termof'anewseries where,thereforeJfork=0,1,2,..., Ak=avk+1+a"k+2+...+a"k+t (vo= -1).Theconverse ishowever notalwaystrue. Proof. Thesuccession ofpartialsumsSI.ofZA/,isob· viouslythesub-sequence sJ'"s".J•••JS"kJ•••ofthesequence ofpartial SUInssriof2'a...By41J4JS..therefore tendstothesamelimitassri' Remarltsandexamples. 00(_I)n-t 1 1 1 1.Theconvergence of.2--~-=1---+-- --"'"therefore im- ,,=1 11 2 ~{4 pliesthatof andalso,simJlarly, of l-(-}- ~)-(-~-- ~)-"'=1-2\-4~5-/7-·". andallthreeserieshavethesamesum.IfwedenotethisbySIthesecond 1 1 7seriesshowsthatinanycase,,>"[:2+3.4=12'andthethird,that 110,<1 -2:-3=12'Thus 2.Thatwemayintroduce brackets, butmaynotwithout consideration omit brackets occurring Inaseries,thefollowing simpleexample shows: Theseries o+0+0+...IScertainly convergent andhasthesumO.Ifwesubstitute every­ where(1-1)for0,weobtainthecorrectequality (1-1)+(1-1)+...==E(1-1)=O. Butbyomitting thebrackets weobtainthedivergent series 1-1+1-1+- .... §15.Thesecondprincipal criterion andthealgebraofconvergent series.133 whichtherefore maynotbeput..=0".Forweshouldthenbyagaingrouping theterms,thoughinaslightlydifferent way,obtain 1 -(1-1)-(1-1)-•••'=1-O.- 0 - O..•, whichagainconverges andhasthesum1.Weshouldtherefore finallydeducethat 0=I!!8. Weproceed atoncetocomplete Theorem 1bythefollowing 00 oTheorem 2.Ifthetermsofaconvergent infiniteseriesEAkare k=U themselves act!lalsums(say,asabove,Ak=aVk+1+...+aVk+l;k=0, 1,...;Vu= -1),thenwe"may"omitthebrackets enclosing theseif, co andonly ~f,thenewseries1:anthusobtained alsoconverges. ,,~O Infactinthatcase,bythepreceding theorem,Ean=1:Abwhilein thecaseofdivergence of1:an>thisequality wouldbecomemeaningless. Ausuallysufficient indication astowhetherthenewseriesconverges isprovided bythefollowing oSupplementary theorem. Thenewseries1:andeducedfrom1:Ak inaccordance withthepreceding theoremiscertainly convergentifthequantities Ak'=IaVk+lI+Ia"k+2I+...+IaVk-1-lI formanullsequence 7. Proof.IfEbegiven>0,choose mlsolargethat,foreveryk>mu wehave ISk-l-si<~ andchoosem2solargethat,foreveryk>m2,wehaveAk'<;.Ifm islargerthanboththesenumbers m1andm2,thenwehave.forevery n>Vm• ISn-si<E. 8Informer times -~beforethestrictfoundation ofthealgebra ofinfinite series(v.Introduction) -mathematicians foundthemselves fairlyatalosswhen confronted WIthparadoxes suchasthis.Andeventhoughthebettermathematicians instmctlvely avoided arguments suchastheabove,thelesserbrainshadallthe moreopportunity ofindulging intheboldest speculations. -Thuse.g.Guido Grandi(according toR.Reiff,v.69,8)believed thatintheaboveerroneous train ofargument whichturns0into1,hehadobtained amathematical proofofthe possibility ofthecreation oftheworldfromnothingI 7AsAk-+0,thisisofitselfthecaseifthetermswhichconstitute Akhave oneandthesamesign-inparticular, therefore, ifbyomission ofthebrackets weobtainaseriesofpositive terms.Furthermore, thisisalwaysthecaseiftheterms anformanullsequence andifthenumber Vk+l-Vkoftermsgrouped together in Akformsabounded sequence fork=0,I,2,••.(Anexample isafforded bythe seriesJ:(an-I-bn)inthenextTheorem 3.) 134 Chapter IV.Seriesofarbitrary terms. Fortoeachsuchncorresponds aperfectly definitenumberk,for which v"<n<'1'''+1 andthisnnmber kmustbe2m.Inthatcase,however, Sn=5k-1+a"k+1+...+an' Andsmce wethenhave,effectually, 00Is..-sI<eJ1.e..2a..=sJ n=O Example.q.e.d. isconvergent; forAkispositive, and,foreveryk>1,ill Sincesimilarly, foreveryk>I, A' 2 1 1 k<4k-4+2k<k-l' CA,,')isanullsequence. Therefore theseries 1 11111+---+-+----++- ...3 2 57 4 isalsoconvergent. -Itssum-callitS -iscertainly>A1+Ag>f~'as theseriesinitsfirstformhadonlypositive terms. oTheorem 3.Convergent seriesmaybeaddedtermbyterm.More precisely, ao.2a..=S n=Uao and.2b..=e n=O implyboth lCa..+b..)=s+t n=O andalso-without brackets! - §15.Thespcondprincipal criterion andthealgebra ofconvergent series.135 Proof.IfsnandtnarethepartialsumsoJthefirsttwo series,then(sn+tn)arethoseofthethird.By41,9,ittherefore follows atoncethatCSn+tn)-s+t.Thatthebrackets maybe omitted, intheseriesthereby proved convergent, follows fromthe supplementary theorem ofTheorem 2,since(Iani)and(Ibn1)and therefore also(Iani+IbnI)arenullsequences. °Theorem 4.Convergent seriesmayinthesamesensebesub­ tractedtermbyterm.Theproofisidentical. 0Theorem 5.Convelgent seriesmaybemultiplied byaconstant, thatistosay,fromXan=sitfollows,ifcisanarbitrary number, that I(can)=cs. Proof. Thepartialsumsofthenewseriesarecsn'ifthose oftheoldaresn'Theorem 41,10atonceprovesthestatement. ­ Thistheorem, tosomeextent,provides theextensIOn tomfiniteseries ofthedistributive law. Remarks andExamples. 1.Thesesimpletheorems areallthemoreimportant, astheynotonlyallow ustodeducetheconvergence ofthenewseriesfromtheconvergence ofknownseries, butalsosetuparelation between itssumandthatoftheknownseries.Theyform therefore thefoundation foractualcalculation IntermsofInfimteseries. 00(_I)"-1 2.Theseries.2---- wasconvergent. Letsdl'note itssum.Hy n=1n theorem 1,theseries84. oo(11) k~2/1-1-21 and arcthenalsoconvergent withthesums.Multiply thefirstbytoinaccor­ dancewithTheorem 5,-thisgiving --andaddthistermhytermtothesecond; weobtain ~(111)3L;4k-3+4k-1-2k="2"s ,k=1 ormoreprecisely: weobtaintheconvergence oftheseriesonthelefthandside andthevalueatitssum,-thelatterexpressed intermsofthesumofthe seriesfromwhichwestarted. Theconvergence wasalsoproved directly in connection withtheorem 2;thepresent considerations haveledhowever appre­ ciablyfurther, sincetheyaffordadefinite statement astothesumoftheseries. Beforeweexamine thevalidityofthecommutative anddistributive lawsandinvestigate, inrelation tothelatter,thepossibility offorming theproduct oftwoseries,westillrequireanimportant preliminary. 136 Chapter IV.Seriesotarbitrary terms. certainly convergent ittheseries(ot Andit~an=s,~IanI=Sthen§16.Absolute convergence. Derangement ofseries. Theseries1 -~+~-t+...proved(81c,3)tobeconvergent. Butifwereplace eachtermbyitsabsolute value,theseriesbecomes thedivergent harmonic series1+~+~+.... Inallthatfollows, it willusually makeaverymaterial difference whether aconvergent series ~anremains convergent orbecomes divergent, whenallits termsarcreplaced bytheirabsolute values. Herewehave,tobegin with,the 8ri. 0Theorem. Aseries~anis positiveterms) ~IanIconvergess• IsI:S:::S. Proof. Since Ian+1+anH+...+an+kIs:Ian+1I+...+Ian+leI thelefthandsideisherecertainly<Eiftherighthandsideis, whence bythefundamental theorem81ourfirststatement atonce follows. Sincefurther IsnI<IaoI+IatI+...+Ian'<S, wehavealso,by41,2,IsI<S. Bythistheorem, allconvergent seriesaredividedintotwoclasses and~anbelongs totheoneortheotheraccording as~'IanIisor isnotalsoconvergent. Wedefine 86. 0Definition. Ifaconvergent series1:anissuchthat1:IanIalso converges, thenthefirstserieswillbecalledabsolutely convergent, andother­ wisenon-absolutely convergent 9. Theseries 1:~~)n. ft-In2 'Examples. <Xl(_1)tn(n-l) ":1 n'"•(IX>1);<Xl Ea",O>a>-1; ,,~o 00xft. 00xn.1:-,x<O; 1:--I'x<O; n-Onn n~1n. areabsolutely convergent. -Everyconvergent seriesofpositive termsisofcourse absolutely convergent. Theverygreatsignificance oftheconceptofabsolute convergence willfirstappearinthis:theconvergence ofabsolutely convergent series ismuchmoreeasytorecognise thanthatofnon-absolutely convergent series,-usually, infact,bycomparison withseriesofpositive terms, 8Cauchy. Analyse algebrique, p.142.(Theproofisinadequate.) -On theotherhand,theexample justgivenshowed thattheconvergence ofIan neednotinvolvethatofIIa..,. DAseriesisthus"non-absolutely convergent" ifitconverges, but notabsolutely. Thede~ignation "non-absolutely convergent" appliestherefore toconll81"gml seriesonly. §16.Absolute convergence. Derangement ofseries. 137 sothatthesimpleandfar-reaching theorems ofthepreceding chapter become available forthepurpose. Butthissignificance willimme­ diatelybecome furthervisibleinthatwemayoperate onabsolutely convergent series,onthewhole,precisely asweoperateon(actual)sums ofadefinite number ofterms,whereas inthecaseofnon-absolutely convergent seriesthisisingeneralnolongerthecase.-Thefollowing theorems willshowthisindetail. °Theorem 1.ll.Icnisaconvergent series01positive termsand87. ifthetermsol-agivenseries~an'loreveryn>m,satisfythecondition IanI<c..ortheconditionIan+1I~C..+1 •an ell then.Ia..is(absolutely) convergent.1O Proof. Bytheptand2ndcomparison tests,72and73,respec­ tively,};IanIisineithercaseconvergent 11,andsotherefore, bySli, is};an' Inconsequence ofthissimpletheorem thecomplete storeofcon­ vergence testsrelating toseriesofpositive termsbecomes available forseriesofarbitrary terms.Weinferatoncefromitthefollowmg °Theorem 2.ll.Ianisanabsolutely convergent seriesandil thelactorsanformabounded sequence, thentheseries .Ianan isalso(absolutely) convergent. Proof. Since(IanI)isabounded sequence simultaneously with (an)'itfollowsfrom70,2thatZIa..I·\a..I=.1:IananIisconvergent simultaneously with.IIa..I. Examples. 1.IfIc..isanyconvergent seriesofpositive termsandiftheer"'sare bounded, thenSCl..Cnisalsoconvergent, forthenIC..isalsoabsolutely con­ vergent. Wemaythus,forinstance, insteadofjoiningtheterms COlCl'CB"" withtheinvariable sign+,replace thisbyquitearbitrary +.and-signs,­ ineverycasewegetaconvergent series;forthefactors±1certainly form abounded sequence. Thusforinstance theseries I(-I)"c ..,I(-I)[yti]cn,I(_I)[!Ollftl c..,... areallconvergent, where[z],asusual,standsforthelargest integer not greater than1:. 10Inthesecondcondition, itistacitlyassumed that,forevery">m, a..9=0andCn+O. 11Thecorresponding criteria ofdivergence, and areofcourseabolished, sincethedivergence ofIIanI,notnecessarily ofIll.., i~allthatfollows. Cf.Footnote 8. 138 Chapter 1V.Seriesofarbitrary terms. 2.IfIanisabsolutely convergent, thentheseriesobtained fromitby anarbitrary alteration inthesignsofitsterms,isinvariably anabsolutely convergent series. Weshallnow-returning thereby tothequestions putasideat theendoflastsection(§15),-showthatforabsolutely convergent series thefundamental lawsofthealgebra of(actual) sumsareinallessen­ tialsmaintained, butthatfornon-absolutely convergent seriesthisis nolongerthecase. Thusthecommutative law<la+b=b+a"doesnotingeneral holdforinfinite series.Themeaning ofthisstatement isasfollows: If('Po''PI''P2,•••)isanyrearrangement (27,3)ofthesequence (0,1, 2,.••)thentheseries (i.e.witha'=aforn=0,1,2,...)n On willbesaid,forbrevity, toresultfromthegivenseries1;anby ..=0 rearrangement orderatlyement ofthelatter.Thevalueof(actual) sumsotadefinite number oftermsremams unaltered, however the termsmayberearranged (permuted). Forinfinite seriesthisisno longer the case12•ThiSisshownalready bythetwoseriesconsidered asexamples inSIc,3andS3Theorems 1and2,namely 1 -~+l-i+-...and1+~-~+~+~-~++-... whichareevidently rearrangements ofoneanother, buthavedifferent sums.Thesumofthefirstwasinfacts<i~,whilethatofthesecond wass'>g;andindeedtheconsiderations ofS4,2showed more precisely thats'=i8. Thiscircumstance ofcourseenforces thegreatest careinworking withinfiniteseries,sincewemust-toputitshortly-takeaccount ofthe01deroftheterms 13.Itistherefore allthemorevaluable to knowinwhichcaseswemaynotneedtobesocareful, andforthis wehavethe SS. °Theorem 1.Forabsolutely convergent series,thecommutative law holdsunrestrictedly 14. Proof. LetEanbeanyabsolutely convergent series(Le.EIanI 18convergent aswell),andletEan'~Eay"beaderangement ofEan. 1>1Tbbwasfirstremarked byCauchy (Resumes analytiques, Turin1833). 18AsIanmerely represents thesequencp (Sll),andarearrangement of Ianproduces aseriesIall'withentirely d~fferene partialsumss,,',-these notmerely forming arearrangement of(sn),butrepresenting entirely diffe­ rentnumbers!! -itseemsapriorimostimprobable thatsuchaderangement willbewithout effectonthebehaviour oftheseries. ..Lejeune-Dmchlet, G.:Abh.AkadBerlin1837,p.48(Werke I,p.319). Herewealsofindtheexample giveninthetext,ofthealteration inthesum oftheseriesbyderangement. Butthisimplies that i.e.I:an'isconvergent§16.Absolute convergence. Derangement ofseries. 139 TheneveryboundforthepartialsumsofI:IanIisclearlyalsoabound forthepartialsumsofI:Ian'I.SOI:an'isabsolutely convergent with I:an'LetSndenotethepartialsumsofI:amands11'thoseofI:an'.Then ifEisarbitrarily given> 0,wemayfirstchoosem,inaccordance with 81,solarge,thatforeveryk>1 Iam+lI+Iam+21+...+Iam+kI<E andnowchoosenosolargethatthenumbers vo,VI!V2'•••,1)"0comprise 15 atleastallthenumbers 0,1,2,...,m.Thenthetermsao,ai'a2,..., amevidently cancelinthedifference sn'-s",foreveryn>no,andonly termsofindex> mremain,--thatis,only(afinitenumberofthe)terms amH'am12'•••Since,however, thesumoftheabsolute valuesofa:1Y numberofthesetermsisalways<E,wehave,foreveryn>no, Is,.'-s"I<E, and therefore (s,,'-sn)isa'nullsequence. s,,'=s"+(s,,'-s,,)hasthesamelimitasSm andhasthesamesumasI:amq.·c.d. Thisproperty ofabsolutely convergent seriesissoessential thatit deserves aspecialdesignation: oDefinition. Aconvergent infiniteserieswhichobeysthecommutation 89. lawwithoutanyrestriction, -i.e.remains convergent, withunaltered sum, undereveryrearrangement, -shallbecalledunconditionally conver­ gent. Aconvergent series,ontheotherhand,whosebehaviour astocon­ vergence canbealteredbyrearrangement, forwhichtherefore theorderof thetermsmustbetakenintoaccount, shallbecalledconditionally con­ vergent. Thetheorem provedjustabovecannowbeexpressed asfollows: "Everyabsolutely convergent seriesisunconditionally convergent." ­ Theconverseofthistheorem alsoholds,namely oTheorem 2. tionally convergent 16.Everynon-absolutely convergent seriesisonlycondi­ Inotherwords,thevalidityoftheequality inthecaseofanon-absolutely convergent seriesI:andepends essentially ontheorderofthetermsoftheseriesontheleft,andmaytherefore, by asuitablerearrangement, bedisturbed. 16Thatsuchanumber noexistsfollowsfromtheverydefinition ofderange­ ment. 10Cf.Fundamental theorem of§44. 140 ChapterIV.Seriesofarbitrary terms. Proof. Itobviously sufficestoprovethat,byasuitable rearrange­ ment,wecandeducefromEanadivergent seriesEan'.Thiswemay doasfollows: ThetermsoftheseriesEanwhichare>0,wedenote, intheorderinwhichtheyoccurinEambyPI'Pz,Pa,...;thosewhich are<°wedenotesimilarly by-ql'-qz,-qa,...ThenEPnand Eqnareseriesofpositiveterms.Ofthese,oneatleastmustdiverge. For ifbothwereconvergent, withsumsPandQsay,thenweshouldobviously have,foreachn, IaoI+Iad+...+IanI<P+Q, henceEanwould,by70,beabsolutely convergent, incontradiction with ourassumption 17.IfforinstanceEPndiverges, thenweconsider aseries oftheform inwhich,therefore, wehavealternately agroupofpositive termsfol­ lowedbyasinglenegative term.Thisseriesisclearlyarearrangement ofthegivenseriesEanandwill,assuch,bedenoted byEan'.Nowsince theseriesEPnwasassumed todiverge, anditspartialsumsaretherefore unbounded, wecan,intheabove,firstchoosemlsolargethatPI+pz+...+Pm,>1+ql'thenmz>mlsolargethat PI+pz-I-...+Pm,+...+Pm,>2+ql+qz and,generally, mv>mv_lsolargethat (v=3,4,...).ButEan'isthenclearlydivergent; foreachofthosepartial sumsofthisserieswhoselasttermisanegative term-qvofEamisby theabove> v(v=1,2,...).Andsincevmaystandforeverypositive integer, thepartialsumsofEan'arecertainly notbounded, andEan' itselfisdivergent, q.e.d.18• HEqnisdivergent, weneedonlyinterchange EPnandEqnsuitably intheabovetoreachthesameconclusion. 17Itisnotdifficult toseethatactually boththeseries1:Pnand1:qnmust diverge(cf.§44);butthisisforthemoment superfluous. 181:an'clearlydiverges to+00. §16.Absolute convergence. Derangement ofseries. 141 Example. ~~=-D~=__1+1 _!+1_1_Ln::1n 2 3 4 5r...wasseeptobenon-absolutely wehave,forv"':I,2,.••, 1 1 1 12+4+6+...+28;;>2v. Iftherefore weapplytotheseriesE(-l)ntheprocedure described above,wen needonlyputmv~=28v,todeducefromitbyrearrangement thedivergent senes III 1 1 11 2+4+6+...+28-1+28+2+...+216-3+... ForthepartialsumsofthiSseriesterminating withthevthnegative termisgreater than2 vminusvproperfractIons, -i.e.certmnly>v. Theorem 88,1onthederangement ofabsolutely convergent series maystillbeconsiderably extended. Forthepurpose, wefirstprovethe following simple oTheorem 3.If};anisabsolutely convergent, thenevery"sub-series" Ea}.n-forwhichtheindicesA..denote,therefore, anymonotone increasing sequenceofdifferent positiveintegers,-isagainconvergent andinfactagain absolutely convergent. Proaf.By74,4,Ela}.nIconverges withE(an).By85,thestate­ mentatoncefollows. Wemaynowextendtherearrangement theorem 88,1inthefol­ lowingmanner. Webeginbypickingoutafirstsub-series Ea}.nofthe givenabsolutely convergent series};amandarranging thisfirstsub-series inanyorder,denoteitby letz(O)bethesumofthisseries,certainly existing, bythepreceding theorem, andindependent ofthechosenarrangement by88,119•Wemayalso allowthisandthefollowing sub-series toconsistofonlyafinitenumber ofterms,-i.e.nottobeaninfiniteseriesatall.Fromtheremaining 1&Theletter IlJisintended asareference totherowsofthefollowing doubly infinitr.array. 142 Chapter IV.Seriesofarbitrary terms. terms-asfarasispossible -weagainpickouta(finiteorinfinite) sub-series, anddenoteit,arranged inanyorder,by ao(l)+al(l)+a2(1)+...+an(l)+..., itssumbyz(1);fromtheremaining termsweagainpickoutasub-series, andsoon.Inthismanner, weobtain,ingeneral, aninfiniteseriesoffinite or(absolutely) convergent infiniteseries; ao(O)+al(O)+a2(0)+...+-an(O)+=z(O) ao(l)+a1(1)+a2(1)+.,,+ an(l)+ =z(l) ao(2)+a1(2)+a2(2)+-..'+an(2)+'"=Z(2) Iftheprocesswassuchastogiveeachnon-zero term 20oftheseries.Ean aplaceinone(andonlyone)ofthesesub-series, thentheseries z(O)+z(1)+z('!)+'", or,thatistosay,theseries mayinafurtherextended sensebecalledarearrangement ofthegiven series 21,Forthisagainwehave,corresponding totheorem 88,1; oTheorem 4.Anabsolutely convergent series"may"alsointheex­ tendedsenseberearranged. Moreprecisely; Theseries z(O)+Z(l)+Z(2)+'.. isagain(absolutely) convergent, anditssumisequaltothatof.Ean' Proof.Ife:>0begiven,firstdetermine msothat,foreveryk::?1, theremainderIam+lI+Iam+2I+ '.,<e:,andthenchoose nosothat inthefirstno+1sub-series 1:an("),J)=0,1,.., ,no,thetermsao,aI'a2, •..,amofthegivenseriescertainly appear.Ifn>noand>m,then theseries (z(O)+Z(1)+...+Z,<n»-Sn 20Theintroduction oromission ofzerotermsinXanorinthepartialsums isobviously WIthout influence onthepresent considerations. 21Putintothefirstsub-series, besides aoandaI,allthosetermsan'forin­ stance,inanyorder,whoseindicesnaredivisible by2jintothenextallthoseof theremaining termswhoseindicesaredivisible by3jintothenextagainallre­ maming termswhoseindicesaredivisible by5iandsoon,usingtheprimenumbers 7,1I,13...asdivisors. §16.Absolute convergence. Derangement ofseries. 143 contains onlyterms±anwhoseindicesare>m.Hence,bythechoice ofm,theabsolute valueofthisdifference is<E,andtendstherefore, withincreasing n,tozero,sothat lim(z(O)+z(l)+...+zen»~=limSn=S=Ean' n~~ n~~ Moreover, theconvergence ofEZ(k)whichisthusestablished isalso absolute, sinceforeachnwehaveobviously Iz(O)I+Iz(l)I+...-/-Izen)I::;:S=EIavI. Theconverse ofthistheorem is,ofcourse,evenlessvalidthan thatoftheorem 83,I,without furtherconsideration. Given,fork~-0, 1,2,.•.,theconvergent series <r> Z(k)=Ean(h), n0_0 iftheaggregate ofterms an(k)bearranged inanywayasasequence (cf. 53,4),thenEanneednotatallconverge, -evenshould1:Z(h)becon- k-O vergent. Toshowthisispossible wehaveonlytotake, for eachofthe seriesz(h),theseries1-1-/-0+0-/-0-/-....AndeventfEancon­ verges,thesumneednotbeequaltothatofEz(k). Ageneraldiscussion ofthequestion underwhatcircumstances this converse ofourtheorem doeshold,belongstothetheoryofdoubleseries. However, wemayevenhereprovethefollowing case,whichisapar­ ticularly important oneforapplications: oMainrearrangement theorem 22.Wesupposegivenaninfinite90. numberofconvergent series (A)[Z(o)=ao(O)+a1(0)+...+an(O)-/-••• Z(l)=~ao(1)-/-a1(1)-/-•••-/-an(1)-/-••• 1~(h)'_'a(~)-/-'a'(h)'-1-• •-/-'a'(h)'-,-' • ---0 1 .•. nI···,....... . . . .... andassumethattheseseriesarenotonlyabsolutely convergent, butsatisfy thestrictercondition that,ifwewrite theseries~Elan(h)I='(h) n=O <XlE'(h)=(] h=O(k=0,1,2,..•,fixed), 12AlsocalledCauchy's DoubleSeriesTheorem. 144 Chapter IV.Seriesofarbitrary terms. isconvergent. Thenthetermsstanding vertically onebelowtheotheralso form(absolutely) convergent series;andIfwewrite23 co1:an(k)=s(n) 10=0(n=0,1,2,.••,fixed), then1:s{n)isagainabsolutely convergent andwehave .... 1:s(n)=1:z(1'); ,.=0' k'~O inotherwords,thetwoseriesformedbythesumsoftherowsandbythesums ofthecolumns, respectively, arebothabsolutely convergent andhavethesame sum. Theproofisextremely simple: Suppose allthetermsin(A)arranged anyhow (inaccordance with53,4)inasimplesequence, anddenoted, astermsofthissequence, byao,aI'a2,• • • •Then1:anisabsolutely con­ vergent. Foreverypartialsumof1:IanI,forinstance IaoI+IalI+...+IamIt muststillbe::::;;u,sincebychoosing ksolargethatthetermsao,aI'a2, •..,amalloccurinthekfirstrowsof(A),wecertainly have i.c.::::;;u.Adifferent arrangement ofthetermsan(k)in(A)asasimple sequence ao',aI"a2',•••wouldproduce aseries1:an'whichwouldbe amererearrangement of1:an>andtherefore againabsolutely convergent, withthesamesum.Letthisinvariable sumbedenoted bys. Nowboth1:z{k)andalso1:s{n)arerearrangements of1:an=s, intheextended senseoftheorem 4,justproved. Therefore thesetwo seriesarebothabsolutely convergent andhavethesamesums,q.e.d. Thisrearrangement theorem maybeexpressed insomewhat more generalformasfollows: oSupplementary theorem. IfMisacountable setofnumbers andthereexistsaconstant Ksuchthatthesumoftheabsolute values ofanyfinitenumberoftheelements ofMremains invariably<K, 23Heretheletter Iisintended asareference tothecolumnsof(A). §16.Absolute convergence. Derangement ofseries. 145 (n=0,1,2,.00)thenwecanasserttheabsolute convergence -withtheinvariable sums-ofeveryseries:EAkwhosetermsAkrepresent sumsofa finiteorinfinite number ofelements ofM(provided eachelement atMoccursinoneandonlyoneofthetermsA,,).Andthisremains trueifweallowarepetitionattheelements ofM,provided eachele­ mentoccursexactlythesamenumber oftimesinalltheAk'staken together, asinMitself24. Examples oftheseimportant theorems willoccuratseveral crucial pointsinwhatfollows. Herewemaygiveoneortwoobvious applications: 1.Let~a"=sbeanabsolutely convergent seriesandput ao+2a1+4a2+...+2"a" I 2"+1 =a" Thenwealsohave ~a,,'=s. Theproofresultsimmediately, bytheprevious rearrangement theorem, fromthcconsIderation ofthearray Ja=ao+ao+!,~+~,!+ ... o24816 a1=0+ 2a1+2a1+2a1+... I4 8 16 ag=0 + 0+4a2+4a2+...816 I • • • • • 1 1 1 2.Similarly, fromp(p-+1)+(p+I)(P+2)+...=p(v.68,2h),and thearray aoaoaoao=~+ 2.3+3.4+'" o+2~+2~+ ..•2·33·4 o wededucetheequality, validforanyabsolutely convergent series ~a,,: ~ aoao+2a1ao+2a1+3a2 ,,--;;:oa"=~+-2.3· +--34 +... s.Thepreceding rearrangement theorem evidently holdswhenever every a"lk)is>0andatleastoneofthetwoseries ~Z(h)and~5("1converges; itholdsfurtherwhenever itispossible toconstruct asecondarray(A')similar to(A),whosetermsarepositive and>theabsolute valuesofthecorresponding termsin(A),andsut'hthat,in(A'),eitherthesumsoftherowsorthesums ofthecolumns formconvergent series. 24Aninfinitenumber ofrepetitions ofatermdifferent fromzeroisel­ eludedfromtheoutset,sinceotherwise theconstant Kofthetheorem would certainly notexist.Andthenumber 0canproduce nodisturbance. 146 Chapter IV.Seriesofarbitrary terms. §17.Multiplication ofinfinite series. Wefinallyenquire towhatextentthedistributive law"a(b+c) =ab+ac"holdsforinfinite series.Thataconvergent infiniteseries :Eanmaybemultiplied termbytermbyaconstant, wehavealreaGY seenin83,5. Inthesimplest form thedistributive lawistherefore validforallconvergent series. Inthe caseofactualsums,itatoncefollows further, fromthedIstributive law,that(a+b)(c+d)=ac+ad +bc+bd, andmoregenerally, that (ao+al+...+al)(bo+b1+...+bm)=aobo+aob1+."+albm' orinshort,that wherethenotation ontherightisintended toconveythattheindices A.andflassume, independently ofoneanother, alltheintegral values from0toland0tomrespectively, andthatall(l+1)(m+1)such products a~bllaretobeadded,inanyorderweplease. Doesthisresultcontinue toholdforinfinite series?IfLan=s andLbn=taretwogivenconvergent infiniteseriesofsumsandt, isitpossible tomultiply outintheproduct inanysimilarway,andinwhatsenseisthispossible? Moreprecisely: Lettheproducts (.i=0,1,2,...\ 1-'-=0,1,2, .../ bedenoted, inanyorderwechoose 25,byPo,PI'P2'.••;istheseries EPnconvergent, andifconvergent, doesithavethesums.t?-Here againabsolutely convergent seriesbehavelikeactualsums.Infactwe havethe 91. 0Theorem 26.IftheseriesEan=sandEbn=tareabsolutely convergent, thentheseriesEPnalsoconverges absolutely andhasthe sums·t. 2SWesuppose, forthIS,thattheproducts a"bl'arewrittendownexactlyin thesamewayasan(k)ora,,(/')for53,4and90,toformadoublyinfinitearray(A). Wecanthensuppose inparticular thearrangement bydiagonals orthearrangement bysquarescarriedoutfortheseproducts. 2.Cauchy: Analyse algebrique, p.147. §17.Multiplication ofinfinite series. 147 Proof. 1.Letnbeadefinite integer>0andletmbethe largestoftheindiceslandfLoftheproducts a;.bitwhichhavebeen denoted byPo'PI'...,P..,Evidently i.e.<(J.7:,if(Jand7:denotethesumsoftheseries2"a..Iand..EIbit\. Thepartialsumsof..EIP..Iaretherefore bounded and..EP..isab­ solutely convergent. 2.Theabsolute convergence of..EPnhavingbeenproved, weneed onlydetermine itssum-callitS-foraspecial arrangement oftheproducts a..bp,forinstance thearrangement "bysquares". For thiswehave,however, obviously, aobo=Po' andingeneral (ao+...+aJ(bo+...+b,,)=Po+...+P,n+l)"-l' anequality which,by41,10and4,becomes, whenn-.00, s·t=S whichwastherelation tobeproved. Remarks andExamples. 1.Asremarked, forthevalidity oftherelation2p"=s.'underthehypo­ thesesmade,itisperfectly indifferent inwhatmanner theproducts a..bpare enumerated, thatistosayarranged inorderasasimplesequence (P,,).The arrangement bydiagonals isparticularly important inapplications, andleads, jftheproducts ineachdiagonal aregrouped together (S3,1),tothefollowing relation: ia,,'ib"=aobo+(aobl+atbo)+(aob~+alb1+aJbo)+... ,,=0n=O <Xl=,]Jc"n=O writing forbrevity aobIt+a1bIt-1+asb"_s+...+a"bo=c".Thevalidity of thisrelation istherefore secured whenbothseriesontheleftconverge ab­ solutely. Wearealsoledtothisformorarrangement ofthe"product series", sometimes calledCauchy's product ofthetwogivenseries27Ibytheconside­ rationofproducts ofrational integral functions andthoseofpowerseries,which latterwillbediscussed inthefollowing chapter:Ifinfact,weformthepro­ ductoftworational integral functions (polynomials) ao+a1x+asXS+...+a,xlandbo+b1X+bvXS+...+brnxm 27Cauchy loc.elt.examines theproduct seriesinthisspecial formonly. 148 Chapter IV.Seriesofarbitrary terms. andarrange theresultagaininorderofincreasing powers ofx,thenthefirst termsare aobo+(aobl+a,bo)x+(aobg+albl+agbo)x2+..., sothatwehavethenumbers co,Cl'Cg,••"aboveintroduced, appearing as coefficients. Itisprecisely duetothisconnection thatCauchy's productoftwo seriesoccursparticularly often. 2.SinceIx"isconvergent forIxI<1Jwehaveforsuchanx (_l_t=.2 x'".2x"=i(n+l)xft •1-00 n=O n=O ft_O x"3.Theseries.2ni'cf.76,5cand85,isabsolutely convergent for everyrealnumberx.Iftherefore Xlandxgareanytworealnumbers, we mayformtheproduct of ~ ~X,".ttCJQI1==,£..J­n! according toCauchy's rule.Weget n nXvXn-v1" n' (+)"c=.2:a"bn-I'=.L; ~]2__= -,2;'---"-x"x'aJ"--'"=~~ . ..v=o v=ovl(n-v)! nlv=ovl(n-v)11nl Therefore wehave-forarbitrary Xlandxg-putting Xl+XI=x,: forn~1, and(_1)"-1a"=bn=----"-fn sothatIa..andIbnareconvergent inaccordance withLeibnitz's rule82,5. ThenCo=Cl=0Jandforn~2, cn=C-1)"[V1~+V2v'~-2+"'+ vn-\v'll denominators bythelargest,Vn-1, Replacing eachrootinthe itfollows that,forn~2, In-lIcn~vn:::r~=1n-l'n-l andtherefore theproduct seriesIcn=,2'(aobn+aJb"-l+...+anbo)Byourtheorem, wehavenowestablished thatthedistributive lawmayatanyratebeextended without change toinfinite series, -andthis,moreover, withanarbitrary arrangement oftheproducts a.tbp-,ifboththetwogivenseriesareabsolutely convergent. Itis conceivable thatthisrestricting assumption isunnecessarily strict.On theotherhand,thefollowing example,givenalready byCauchyZ8 forthepurpose, showsthatsomerestriction isnecessary, orthetheorem nolongerholds:Let ao=bo=0 11Analyse algebrique, p.149. Exercises onChapter 1V. 149 iscertainly divergent inaccordance withS2,1.Thisistherefore a fortiori thecasewhenweomitthebrackets. Nevertheless, thequestion remains open,whether wemaynot beable,underlessstringent conditions thanthatofabsolute conver· genceofboththeseries2:anand2,'b",toprovetheconvergence of theproduct series2,'Pn-atleastforsomespecialarrangement of theterms aJ..bp,forinstance asintheseries2:e,.above, Tothis question weshallreturnin§45. Exercises onChapter IV. 4:S.Examine theconvergence ordivergence oftheseries:z(-I)"a", [orwhichan>fromsomenonwards, hasoneofthefollowing values: 1 1 a+n'an+b'1 ..j-n'1 1 logn'loglogn' 46.Whatalterations havetobemadeintheanswers toEx.34,when thebehaviour of:z(-I)"anisrequired? 47.Let Thentheseries,,,={+1 for22k<n<22k+1, -1for22k+1<n<2~k+2.(k=0,1,2,...) J;_8_,,_ k=2nlogn -:onverges. Whatisthebehaviour of.2:Ell?n '" 2n+l .48..L;(-1)"-1.( )ISconvergent andhasthesum1. n=l nn+l 49.Letthepartial sumsoftheseries1 -~+~--~-+-...be111denoted bys",andItssumbys,andput--+---+...+-=x.~hown+ln+2 2n" that,foreveryn, Xn=52" . a;(-1)"-1sothathmx"=~ =S(=log2). n=l n :SO.Lets(=log2)denoteasabovethesumoftheseries1-{-+~- +... Provethefollowing relations: a) b) c) d) e)111111111 11-----+ ----+------+-- ...=---log\!'3 5 7 5 9 11713 15 3 2 '11111111 11----+- -----+--- ---+--...=-log2'2 4 3 6 8 5 10122'111111 21- - - -+-+-- ----++- -...= -log2'2 457l:l10a' 1 1 1 1 1 1+3+5-2-4+++--'" =2log6; 111111+-+-------+++---...=log235246 . 150 Chapter IV.Seriesofarbitrary terms. 51.Withreference tothelasttwoquestIOns, showgenerally thatthe seriesremains convergent whenwealternately writethroughout ppOSItive terms 1Pandqnegative ones,andthatthesumisthen=log2+210gq' 52.Theharmonic series1+~+~+~+...remains divergent, whenthe signsaresochanged thatwehavethroughout alternately Ppositive terms andqnegative ones,withP=l=q.IfP=qtheresulting seriesisconvergent . ..,(_l)n-153.Consider therearrangements oftheseries.2 exactly corre- ,,=1Vn (_l)n-1sponding tothoseoftheseries.2: inEx.50and51.Whenisthen resulting seriesconvergent andwhenisitnot?Whenisthesumexpressible intermsofthesumofthegivenseries? 54.Consider, withtheseries.2:_1_,thesamealterations insignsasinvn Ex.52,fortheseries.2:-!...Whenisaconvergent seriesobtained?n 55.Forwhichvaluesofet:dothefollowing twoseriesconverge: I I 1 I 11--+---- +---+_ ...•2aS 4a5Ga 1 11111+- ---+-+----++-...?Sa2a5a7a4a 56.Thesumoftheseries1---!..+-.!..--.!..+-...2uSa4u forevery et:>O. G7.Given ..,1(n;9).2:9=5=­ ,,=1n 6 showthat1liesbetween2andI, 1lIS1+32+5"+72-+...=4:5• 1 1 1 1 21+59+72-+li2+132-+...=3"s , 111111 4, 1-22-49-+59+72-~9-102++- -...=9s. (Withthelatterequality cf.Ex.SOc.) 58.Tnevery(conditionally) convergent seriesthetermscanbegrouped together Insuchamanner thatthenewseriesconverges absolutely. 58a.Thefollowing complement toKronecker's theorem82,3holdsgood: IfaseriesIa..issoconstituted thatforeverypositive monotone sequence (p..) tending to+00Ithequotients Poao+P1a1+...+P..a" P.. tendto0,thenIanisconvergent. -Inthissense,lherefore, Kronecke,,'s condition isnecessary andsutlJcient fortheconvergence. §18.Theradiusofconvergence. 151 C'SD.IffromagivenseriesIa",withthepartialsumss".wededuce, byassociation ofterms, anewseriesIAkwiththepartialsumsSk,then theinequalities invariably holdgood,whether Ianconverges ornot. 60.If:::a",withthepartialsumssn,diverges indefinitely, ands'isa valueofaccumulation (:i2)ofthesequence (s,,),thenwecanalwaysdeduce fromIala,byassociation ofterms,aseriesIAkconverging tos'assum. 61.If~an,withthepartialsumssn'dIverges indefinitely, anda,,-+O, theneverypointofthestretch between theupperandlowerlimitsofs"isa pomtofaccumulation ofthisseqnence. 62.Ifeverysub-series ofIa"(;onvcrges, thentheseriesitselfisabsolutely convergent 63.Cauchy's product ofthetwodefinitely divergent series 1-~_(~)2_ (~)"_...2 2 2 and is 3(3)2(3)"1+4+4;+-1+...; thatofthetwoseries3+1:3"and- 2+1:2"is- 6+0+0+0+....n-l n~l Inbothcasesitisabsolutely convergent. Howcanthisparadox beexplamed? Chapter V. Powerseries. §18.Theradiusofconvergence. Thetermsoftheserieswhichwehaveexamined sofarwere, forthemostpart,determinate numbers. Insuchcasestheseries maybemoreparticularly characterised ashavingconst,mt terms.This however wasnoteverywhere thecase.Inthegeometric series ~a", forinstance, thetermsonlybecome determinate whenthevalueofa isassigned. Ourinvestigation ofthebehaviour ofthisseriesdidnot, consequently, terminate withamerestatement ofconvergence or divergence, -theresultwas:2'a"converges illaI<1,butdiverges illaI21.Thesolution ofthequestion ofconvergence ordivergence thusdepends, asdothetermsoftheseriesthemselves, onthevalue ofaquantity leftundetermined - avariab!e. Serieswhichhavetheir terms,-andaccordingly theirconvergence ordivergence, -depending onavariable quantity {suchaquantity willusuallybedenoted byx 152 Chapter V.Powerseries. andweshallspeakofseriesofvariable termsl)willbeinvestigated laterinmoredetail.Forthemoment wepropose onlytoconsider seriesoftheabovetypewhosegeneric term,instead ofbeinga number a..,hastheform i.e.weshallconsider seriesoftheform2 <Xl ao+a1x+a:jx2+...+a"x"+...--.1'anx". n-=O Suchseriesarecalledpowerseries(inx),andthenumbers a"are theircoefficients. Forsuchpowerseries,wearethusnotconcerned simplywiththealternatives "convergent" or"divergent", butwiththe moreprecise question: Forwhatvaluesofxistheseriesconvergent, andforwhatvaluesdivergent? 92. Simpleexamples havealready comebeforeus: 1.Thegeometric series;rx"isconvergent forIxI<1,divergent for IxI~1.ForIxI<1,indeed, wehaveabsolute convergence. 2..L;':~is(absolutely) convergent foreveryrealx;lIkewise theserie!> QC 2k .L;'(-I)k(_~k)1 and k=O 8..L;'X:'becauseIxn",:;;:::;1xI",isabsolutely convergent forIxI<1. ForIxI>1,theseriesisdivergent, because inthatcase(byas,1and40),Ixn"l_+00•Forx=1itreduces tothedIVergent harmonic series,andfor x=-1,toaseriesconvergent byS2,Theorem 5. <XlX. 4..L;'~2nis(absolutely) convergent forIxI;;;;2,butdivergent for n=1 Izl>2. <Xl5.E~ftxftisconvergent forx=0;butforeveryvalueofz=r°itIS n=l divergent, forifx=f0,Inx1-+00andafortIOriInftxft1-+00,sothat (byS2,Theorem 1)therecanbenoquestion oftheseriesconverging. Forx=0,obviously everypowerseries2:a"x"isconvergent, whatever bethevaluesofthecoefficients a",Thegeneral caseis evidently thatinwhichthepowerseriesconverges forsomevalues ofx,anddiverges forothers, while,inspecial instances, thetwo extreme casesmayoccur,inwhichtheseriesconverges foreveryx (Example 2),orfornone9=0(Example 5). ITheharmonic series.L;'.!.-isalsoofthistype:itconverges forx>1,nit' diverges forx::;l. •Weherewrite,forconvenience, XO=1,evenwhenx=O. §18.Theradiusofconvergence. 153 Tnthefirstofthesespecial caseswesaythatthepowerseries iseverywhere convergent, inthesecond-leavingoutofaccount the self-evident pointofconvergence x=0 -wesaythatitisnowhere convergent. Ingeneral, thetotalityofpointsxforwhichthegiven series2,'anx..converges iscalleditsregionofconvergence. In2.thisconsists therefore ofthewholeaxisofx,in5.ofthe singlepoint0;intheotherexamples, itconsists ofastretchbisected attheorigin,-sometimes with,sometimes without oneorbothof itsendpoints. Inthiswemayseealready thebehaviour oftheseriesinthe mostgeneral case,forwehavethe °Fundamental theorem. If2'anx"isanypowerserieswhich93. doesnotmerelyconverge everywhere ornowhere, thenadefinitepositive numberrexistssuchthat2,'anxnconverge's foreveryIx1<r(indeed absolutely), butdiverges foreveryIxI>r.Thenumberriscalledthe radiusofconvergence, orforshorttheradius, andthestretch -r ...+rtheinterval ofconvergence, ofthegivenpou'erseries3. -Fig.2schematizes thetypicalsituation established bythistheorem. r----conv. •; f.lIy.-ro Fig2. Theproofisbasedonthefollowing twotheorems. °Theorem 1.IfagivenpowerseriesLa..x..converges forx=Xo (xo+0),oreveniftltesequence (anXon)ofitstermsisonlybounded there,then2:anx"isabsolutely convergent foyeveryx=Xlnearey totheoriginthanxo'i.e.WithIXli<IXo/. Proof.IfIanxon1<K,say,then where{}=theproperfraction5..By.87,1theresultstatedfollows Xo immediately . .°Theorem 2.Ifthegivenpowerseries2,'anx..diverges forx=Xo thenitdiverges afortiorifoyeveryx=Xlfurtherfromtheorigin thanxo'i.e.withIXlI>IXo,. ,Inthetwoextreme caseswemayalsosaythattheradiusofconver­ genceoftheseriesisr=0orr..+00.respectively. 154 Chapter V.Powerseries. Proof.Iftheserieswereconvergent forXl'thenbytheorem 1 itwouldhavetoconverge forthepointxO'nearer0thanXl'­ whichcontradicts thehypothesis. Proofofthefundamental theorem. Byhypothesis, there existsatleastonepointofdivergence, andonepointofconvergence =t=O.Wecantherefore chooseapositive number Xonearer0than thepointofconvergence andapositive number Yofurtherfrom0 thanthepointofdivergence. Bytheorems 1and2,theseries2:anXli isconvergent forX=xo'divergent forX=Yo'andtherefore we certainly haveXo<Yo'Totheinterval10=Xo...Yo'weapplythe method ofsuccessive bisection: wedenoteby11theleftortheright halfof11according as2,'anx"diverges orconverges atthemiddle pointof10,Bythesamerule,wedesignate aparticular halfof11 by12,andsoon.Theintervals ofthisnest(In)allhavetheproperty that2:anx..converges attheirleftendpoint(sayXII)butdiverges at theirrightendpoint(sayyn).Thenumber r(necessarily positive), whichthisnestdetermines, isthenumber required forthetheorem. Infact,ifx=XisanyrealnumberforwhichIXI<r(equality excluded), thenwehave'x'I<xk'forasufficiently largek,i.e.such thatthelengthoflkislessthanr-IxI.Bytheorem 1,x'isa pointofconvergence atthesametimeasxkis;andindeedatXwe haveabsolute convergence. If,onthecontrary, x"isanumber for whichIx'1>1',thenIx'I>Y'n'provided mISlargeenoughforthe lengthofImtobelessthanIx"I-r.Bytheorem 2,x'isthena pointofdivergence atthesametimeasYmis.Thisprovesallthat wasdesired. Thisproof,whichappeals tothemindbyitsextreme simplicity, isyetnotentirely satisfying, inthatitmerelyestablishes theexistence oftheradiusofconvergence without supplying anyinformation asto itsmagnitude. Wewilltherefore provethefundamental theorem by analternative method, thistimeobtaining themagnitude ofthe radiusitself.Forthispurpose, weproceed -quiteindependently ofourprevious theorem, -toprovethemOleprecise 14. °Theorem4:Ifthepowerseries2:anXliisgivenandp.denotes theupperlimit0;the(positive) sequence ofnumbers S__ n__ Vlasl,·..,Vla..!,..., i.e._n__ p.=Hmv'lanI' ~Cam")': Analyse algebrique p.151.-Thisbeautiful theorem remained forthetimeentirely unnoticed, till].Hadamarda.demath.puresetappl.,(4) Vol.8,p.107.1892)rediscovered itandmadeuseofitinimportant appli cations. §18.Theradiusofconvergence. 155 1IX1</l' 1Ixl>-·p.forevery interpretation, 1 1'1'=-=----IL n--- UmVIanithen a)ifft=0,thepowerseriesiseverywhere convergent,' b)iff£=+00.thepower series isnowhere convergent; c)if0<ft<+00,thepowerseries converges absolutely forevery butdiverges Thus-withthesuitable istheradiusofconvergence ofthegivenpowerseriesr'>. Proof.Ifincasea)Xoisanarbitrary realnumber+0.then 12lXal>0andtherefore by~9, Vra:-r<2r~~orIanXonI<21n foreveryn>m.By87,1,thisshowsthat~anxo"converges ab· solutely, -whichprovesa). Ifconversely ~a"x"converges forx=x1=l=0,thenthesequence (a"x/') and,afortiori, thesequence (VTa"x1n,).arebounded. If ".---- n- J(V\a"x1"I<[(1'say,foreveryn,thenviani<1:1::I=K,foreveryn, i.e.(VIallI)isabounded sequence. Incaseb).inwhichthesequence is~lssumed unbounded above,theseriestherefore cannotconverge for anyx=+=O. Finally, incasec),ifx'isanynumber forwhichIx'I<..!.... f.' thenchooseapositiveeforwhichIx'I<e<..!...,andso..!...>ft.By p. e thedefinition offt'wemusthave,foreveryn>someno' VIa"I<..!...andconsequently IfIanx'''[<r<1-e e By7iS,1,~an'Jinistherefore (absolutely) convergent. 6)Forconvenience ofexposition, wehereexceptionally write~=+00• _1_=O._Furthermore itshouldbenoticedthat 1isnotforinstance+00 _,,-­ limVianI - 1thesameaslim--- I -asthestudent shouldverifybymeansofobvious n.__ VianI examples. (Cf.Ex.24.) 156 Chapter V.Powersenes. Ontheotherhand,ifIx"I>~,sothatI~,I<p,thenwemust have,foraninfinitenumber ofn's(againandagain;v.59) n_/1IVla"l>x'or By82,Theorem 1,therefcre theseriescertainly cannotcon­ verge 6. Thusthetheorem isprovedinallitsparts. Remarks andExamples. 1.Sincethethreepartsa),b),c)ofthepreceding theorem aremutually exclusive, itfollows thattheconditions arenotmerely sufficient, butalso necessary forthecorresponding behaviour of~allxn•,,--2.Inparticular, wehaveVIanI-.0foranypowerserieseverywhere convergent. Forbytheremark above, p.=0,andsinceweareconcerned withasequence ofpositive numbers, thesecerta1ll1y havetheirlowerlimit xZ;p..Sinceontheotherhandxmustbe;;;;p..wemusthavex=jL=O. By63thesequence(VTa:T) istherefore convergent Withlimit0 Thusforinstance _"rf--.0,orVnT-.00,V'1'1' x"because '"converges everywhere. (Cf.43,Example 4.)~nl 3.Theorems 93and9-1gIVeusnoinformation astothebehaviour of theseriesforx=+randforx= -";thisdiffersfromcasetocase:2)x", x" xfl. '\~--,'"---;-allhavetheradius1.Thefirstconverges neitherat1norat~n~n~ - 1,thesecondonlyatoneofthetwo,thethirdatboth. 4Further examples ofpowerseriesWilloccurcontinually inthecourseof thenextparagraphs, sothatweneednotindicate anyparticular examples here. Wesawthattheconvergence ofapowerseriesintheinterior of theinterval ofconvergence is,indeed, absolute convergence. We proceed toshowfurtherthattheconvergence issopronounced asto beundi&turbed bytheintroduction ofdecidedly largefactors.Wehave infactthe 95. °Theorem. It.i:a"x"hastheradiusotconvergence 1',thenthe n=O <s> powerseries.L;nanX"-l,orwhat n=O hasprecisely thesameradius.<s> isthesamething,,2(n+l)a ..+1x",,,=0 •Casec)maybedealtwithsomewhat moreconcisely: If limVIa"1=1-',thenlim-VIa"x"-' =fiffiyra-..l·j x1=p."xI (forwhatreason?).By76,3theseriesistherefore absolutely convergent for I-'•IxI<I,andcertamly divergent forI-'•IxI;.>1,q.e.d. §18.Theradiusofconvergence. 157 Proof. Thistheorem maybeimmediately inferred fromTheo· rem94.Forifwewritenan=an',then ft___ fJ.--tI_ 11Ian'1=Va,,·Vn. Since(by3S,5),Vn-+-1,itfollows atoncefromTheorem62that thesequences(V-ra:1) and(Vra::1) havethesameupperlimits.For ifwepickoutthesamesub-sequences fromboth,ascorrespondmg termsonlydifferbythefactorV'n,which-.+1,thesesub-sequences eitherbothdiverge orbothconverge tothesamelimit 7. Examples. 1.Byrepeated applIcation ofthetheorem, wededuce thattheseries 2:na"x"-I, 2:n(n-1)a"x"-2,•••,2,'n(n-l) •..(n-k+1)a nx"-1: or,what i~exactly thesamething,theseries 2:(n+1)a"+lx",:E(n+1)(n+2)a.+2x·,..•, 2)(n+1)(n+2)•••(n+k)a"+kx"=k!2)(ntk)all+kx" allhavethesameradiusas::::anx",whatever positive integer bechosen fork. 2.Thesameofcourseistrueoftheseries 2Jn':;:1x"+1,2J(n+ff(n+2)x"+2,"',2)(n+1)(n+~...(n+k)x"+". Thusfarwehaveonlyconsidered powerseriesoftheform :Eanxn.Theseconsiderations arescarcely altered, ifwetakethemore general type Putting x--xo=x',weseethattheseseriesconverge absolutely fOI Ix'I=Ix-xoI<~, butdiverge forIx-XoI>~,ifl'agallldenotes thenumber deter­ minedbyTheorem 94-.TheregIOn ofconvergence ofthisseries except intheextreme cases, inwhichitconverges onlyfor x=xo'orforeveryx, istherefore astretch bisected bythe pointxo'sometimes with,sometimes without oneorbothofitsend· points. Except forthisdisplacement oftheinterval ofconvergence, allourconsiderations remain valid.Thepoint Xowillforbrevitybe calledthecentre01theseries.IfXo=0,wehavetheprevious form oftheseriesagain. ~Alternative proof. By76,5aor91,2,theseries2:nlJ"-l is convergent foreveryI{}I<1.IfIXoI<r,andeissochosen that IXoI<e<r,thenIa..e"converges, (alle")isthereforebounded, say !alle"I<K. WeinferthatIna"xo..-ll<:.nl:ol"-l, which, since I;0I<1Iproves theconvergence. 6- (061) 158 Chapter V.Powerseries. Intheinterval ofconvergence, thepowerseries ~an(x-xo)n hasadefinitesums,foreachx,andusuallyofcourseadifferent sum foradifferent x.Inordertoexpress thisdependence onx,we write <Xl ~an(x-xo)n=S(x) no~O andsaythatthepowerseriesdefines, initsinterval ofconvergence, a function ofx. Thefoundations ofthetheoryofrealfunctions, thatistosaythe foundations ofthedifferential andintegralcalculus, weassume, asremarked intheIntroduction, tobealreadyknowntothereaderinallthatisessential. Itisonlytoavoidanypossible uncertainty astotheextentofthefacts required fromthesedomains, thatweshallrapidlyindicate, inthefol­ lowingsection, allthedefinitions andtheorems whichweshallneed, without goingintomoreexactelucidations orproofs. §19.Functions ofarealvariable. Definition 1(Function). Iftoeachvaluexofanintervalofthe x-axis,byanyprescribed rule,adefinite valueyismadetocorrespond, thenwesaythatyisafunction ofxdefinedinthatintervalandwrite, forshort, y==f(x), where''/''symbolises theprescribed ruleinvirtueofwhicheachxhas corresponding toittherelevant valueofy. Theinterval, whichmaybeclosedoropenononeorbothsides, bounded orunbounded, iscalledtheintervalofdefinition off(x). Definition 2(Boundedness). IfthereexistsaconstantKtsuch thatforeveryxoftheintervalofdefinition wehave thenthefunctionf(x)issaidtobebounded ontheleft(orbelow)inthe interval, andKtisaboundbelow(orlefthandbound)off(x).Ifthereexists aconstant K2suchthatforeveryxoftheintervalofdefinitionf(x)::;:K2, thenf(x)issaidtobebounded ontheright(orabove)andK2isabound above(orrighthandbound)off(x).Afunction bounded onbothsidesis saidsimplytobebounded. Therethenexistsaconstant Ksuchthatfor everyxoftheintervalofdefinition, wehave If(x)I<K. §19.Functions ofarealvariable. 159 Definition 3(Upper andlowerbound, oscillation). Thereis alwaysaleastoneamongalltheboundsaboveofabounded function, and alwaysagreatest amongallitsboundsbelow 8.Theformerwecallthe upperbound,thelatterthelowerbound,andtheirdifference theoscillation ofthefunctionf(x)initsinterval ofdefinition. Corresponding desig­ nationsaredefinedforasub-interval a'.•.h'oftheintervalofdefinition. Definition 4(Limitofafunction). If~isapointoftheinterval ofdefinition ofafunction f(x),oroneoftheendpoints ofthatinterval, thenthenotation limf(x)=C x-;.t or f(x)-;.cforx-+~ meansthat a)foreverysequence ofnumbers Xnoftheintervalofdefinition which converges tog,butwithallitstermsdifferent from"thesequence ofthe corresponding values Yn=f(xn) (n=1,2,3,...) ofthefunction converges toc;or h)anarbitrary positive number e:beingchosen, another positive number I)-=I)(e:)canalwaysbeassigned, suchthatforallvaluesofxin theintervalofdefinition with Ix-~I<.3butx=!=~. wehave 11 If(x)-cl<e:. Thetwoformsofdefinition a)andb)meanprecisely thesamething. Definition 5(Righthandandlefthandlimits). If,inthecase ofdefinition 4,itisstipulated besidesthatallpoints Xnorxtakeninto account lietotherightof~(whichmustnotofcoursebetherighthand endpoint oftheinterval ofdefinition off(x»,thenwespeakofaright handlimit(orlimitontheright)andwrite limf(x) =c; x-;.f+O similarly wewrite limf(x) =c, x-+E-() andspeakofalefthandlimit(orlimitontheleft),ifgisnotthelefthand endpoint oftheintervalofdefinition off(x),andifpoints Xnorxtothe leftoftarealonetakenintoaccount. 8Cf.8,2,andalso62. 9Theoldernotation limf(x) forlimf(x) shouldbeabsolutely discarded since x=E x-+f thewholepointisthatxistoremain'*~. 160 Chapter V.Powersenes. orDefinition 5a(Further typesoflimits). Besides thethreetypes oflimitalready defined, thefollowing mayalsooccur10: limf(x) =Ic,+00,-00 f(x)- withoneofthefivesupplementary indications ("motions ofx") for x-~,-+~+0J-+~-0,-+00J-+-00. Withreference to2and3therewillbenodifficulty informulating precisely thedefinitions -intheforma)orb)-whichcorrespond tothedefinitions justdiscussed. Since.asremarked. \\eassume thesematters tobefamiliar tothe reader, inallessentials, wesuppress allelucidations ofdetailandexamples, andonlyemphasize thatthevaluectowhichafunction tends,forinstance forx--..?;,needbearnorelation whatever tothevalueofthefunctIOn at~. Onlyforthiswewillgiveanexample: let((x)bedefined foreveryxby putting f(x)=0ifxisanirrational number, but((x)=~ifxisarationalq number whichinitslowesttermsisoftheform -~-(q>0).Thuse.g.f(~) =l.t(0)=((~)=1 ,(CfID=O.etc. HerewehaveJOTevery?; lim((x)=o. "'-+~ 1Forif8isanarbitrary positive number andmissolargethatm<B.then therearenotmorethanafinitenumber ofrational POlOtswhose(leastposi­ tive)denomlOator is<m.Theseweimagine marked intheinterval?; - 1 ...?;+1.Asthereareonlyafinitenumber ofthem,wecanfindonenearest ofallto?;;(if?;itselfisoncofthesepointsweofcourseshouldnottakeit intoaccount here).Letddenoteits(positive) distance from~. Theneveryx, forwhich iseitherirrational, orarational number who!>eleastpositive denominator q 1 1is>m.Intheonecase,f(x)=0;intheother,= -<-<8.Therefore weqm have,foreveryxin0<Ix-?;I<d, I((x)-0I<8 i.e.,asasserted, Hm((x)=O. "'-+~ Iftherefore?; isinparticular arational number, thenthislimitdiffersdecidedly fromthevalue((?;)itself. Calculations withlimitsarerendered possible bythefollowing theorem: 10Inthefirstofthesethreecaseswesaythat((x)tendsorconverges toc;inthesecond andthirdcases:f(x)tendsordwerges (definitely) to+00 or-00;andinallthree,wespeakofade/mIte behaVIour oralsoofalimit inthe7"ldersense.IfI(x}showsnoneofthesethreemodesofbehaviour, thenwe saythatI(x)diverges indefinitely forthemotIOnofxunderconsideration. §19.Functions ofarealvariable. 161 Theorem 1.If(I(x),f'J(x),...fp(x)arcgivenfunctions (psome determinate positive integer), eachofwhich, foroneandthesame motionofxofthetypesmentioned inDefinition 5a,tendstoafinite lImit,sayt;.(x)-..Cl'.•.,fp(x)-..cp'then a)thefunction l(x)=[fl(x)+-f'J(x)+-...+-(p(x)]-..Cl+-C2+-...+Cp; b)thefunction f(x)=[{I(x).fs(x)...fp(x)]-..Cl'CI)'" Cp; c)inparticular, therefore, thefunctIOnafl(x)-..aCl'(a=arbitrary realnumber) andthefunctionf1(x)-fl)(x)-..Cl-c~; d)thefunctionfle)-+2.-,provided Cl=t=O. 1X Cl Theorem 2.Ifhmf(x) =C(+±(Xl),then{(x)isbounded ina s-+; neighbourhood of~,i.e.twopositive numbers ~andKexist suchthat I{(x)I<K,whenIx-~I<c5, andcorresponding statements holdinthecaseofa(finite)limf(x) forx-+~+-0,~-0,+-(Xl,-(Xl. Definition 6(Continuity atapoint).If~isapointoftheinterval ofdcfil1ltion of{(x),then{(x)issaidtobecontinuous at~if lim{(x) z-+~ existsandcoincides withthevalue ((~)ofthefunction ate: limf(x)=f(~)· x.....; Ifweinclude thedefinition ofliminthisnewdefinition, wemay alsostate: Definition 6a.f(x)issaidtobecontinuous atapoint ~,iffor everysequence ofx,,'softheinterval ofdefinition, whichtcndsto~, thecorresponding valuesofthefunction Yn=.f(xn)-"?f(g)· Definition 6b.f(x)issaidtobecontinuous at~,if,havingchosen anarbitrary e:>0,wecanalwaysassign8=8(e:)>0,suchthatfor everyxoftheintervalofdefinition with Ix-gI<8wehaveI/(x)-Icg)I<e Definition 7(Righthandandlefthandcontmuity). I(x)is saidtobecontinuous ontheright(right-handedly) orontheleft(left-hand­ edly)iflimf(x) existsatleastforx-"?~+0orx-"?~-0respectively, andcoincides with I(~). . Corresponding toTheorem 1wehaveherethe Theorem 3.Iff1(x),f2(x),•••,f1)(x)aregivenfunctions {Pa 162 Chapter V.Powerseries. particular positive integer), allcontinuous atg,thenthefunctions a)fl(x)+f2(x)+...+lp(x), b)11(x).12(x)...Iv(x), c)afl(x)(a=anarbitrary realnumber), fl(x)-f2(x),and d)iffd~)=!=0,also~~-­/1(x) areallcontinuous atg.Corresponding statements hold,whenonlyright handoronlylefthandcontinuity isassumed. Byrepeated application ofthistheorem tothefunction f(x)=x, certainly continuous everywhere (sinceforx_gwehaveprecisely x_g), weatoncededuce: Allrational functions arecontinuous everywhere, withtheexception of(atmostafinitenumber of)pointswherethedenominator =0.In particular: Rational integral functions arecontinuous everywhere. Similarly, thelimiting relations 42,1-3,showedthat:aX ,(a>0)is continuous foreveryrealx;logxiscontinuous foreveryx>0;x"(IX= arbitrary realnumber) iscontinuous foreveryx>O. Definition 8(Continuity inaninterval). Ifafunction iscon­ tinuousateveryindividual pointofaninterval.1, thenwesaythatitis continuous inthisinterval. Continuity atanendpointoftheinterval is heretakentobecontinuity "inwards", i.e.righthanded continuity at thelefthandendpoint,andlefthandedcontinuity attherighthandend­ point.Theseendpointsof./mayor maynot,according tothecircum­ stances, bereckoned asintheinterval. Functions whicharecontinuous inaclosedinterval giverisetoaseriesofimportant theorems, ofwhich wemaymention thefolloWing: Theorem 4.Iff(x)iscontinuous intheclosedinterval a<x<b andiff(a)>0,butf(b)<0,thenthereexists,between aandb,atleast onepointgforwhichf(g)=0. Theorem 4a.Iff(x)iscontinuous intheclosedinterval a~x~b andTJisanyrealnumberbetweenf(a) andf(b), thenthereexists,between aandb,atleastonepointgforwhichf(g)=TJ.Or:Theequationf (x)-=TJ hasatleastonesolution inthatinterval. Theorem 5.Iff(x)iscontinuous intheclosedinterval a<x~b, then,havingchosenanye>0,wecanalwaysassignsomenumber 8>0 sothat,ifx'andx"areanytwopointsoftheintervalinquestion whose distanceIx"-x'Iis<3,thedifference ofthecorresponding valuesof thefunction,If(x")-f(x')I,is<e.(Theproperty, established bythis theorem, ofafunction continuous inaclosedinterval iscalleduniform continuity ofthefunction intheinterval.) Definition 9(Monotony). Afunction defined intheinterval a...hissaidtobemonotone increasing ordecreasing intheinterval, iffor everypairofpoints XlandX2ofthatinterval, withXl<x2,wein· §19.Functions ofarealvariable. 163 variably have((Xl)::::;:((X'J)intheonecase,orinvariably ((Xl)~((x'J)' intheDther.Wealsospeakofstrictly increasing andstrictly de· creasing functions, whentheequality signs,intheinequalities between thevaluesofthefunction justwritten down,areexcluded. Theorem 6.Thepointe,certainly existing underthehypotheses ofTheorems 4and4a,isnecessarily uniqueofitskindifthefunc· tion((x)underconsideration isstrictlymonotone intheinterval a...b. Thusinthatcase,toeach1]between f(a)and((b)corresponds one andonlyoneeforwhichfee)=1].Wesayinthiscase:Theinverse function ofy={(x)iseverywhere existentandone-valued (ory=((x) isreversible) intheinterval. Definition 10(Differentiability). Afunction {(x)defined ata point ~andinacertainneighbourhood of~issaidtobedifferentiable at~ifthelimit cxi~ts.Itsvalueiscalledthe(unique derivative or)differential coefficient of((x)at~andisdenoted byf'(~).Ifthelimitinquestion only existsontheleftorontheright(thatis,onlyforx-.e+0or x-.~-0respectively), thenwespeakofrighthandorlefthand differenHability. differential coetlicient, etc. H afunctIOn isdifferentiable ateachindividual pointofaninter· valJ.thenwesayforbrevity thatthefunction isditlerentiable in thisinterval. Therulesfordifferentiation ofasumorproduct ofaparticular (fixed) number offunctions, ofadIfference orquotient oftwofunctions, offunctions ofafunction, asalsotherulesfordifferentiatIOn oftheelementary functions andoftheircombinations, wereg-ardasknowntothereader. Allmean~necessary totheirconstruction havebeendeveloped inthe above, ifweantiCIpate aknowledge ofthelimitdefined in112andthere determined inaperfectly directmanner. If,forinstance, itisinquired whether aX(a>0and=toI)I~dIfferentiable, and,ifso,whatisitsdIfferential coefficient, atthepoint ~,then,following Defs.10and4,wehavetochoose anullse­ quence(XII)with term~all=to0andtoexamine thesequence ofnumbers a';+:r"_a~ •a'l'''-1 X..=----'-=a~.---. x" x" IfwcwriteY"forthenumerator inthelastfraction, thenby35,3weknow that(YII)isalsoanullsequence, andindeedoncforwhichnoncoftheterms isequaltoO.X..maythenbewritten intheform X"=a';yn·loga J log(1+y,,) Butbince,asremarked, y"isanullsequence, wehaveby112 Jog(1+Y.)-.1 . YJI Sincethesamethenholdsforthereciprocal values, by41,11a,wededuce X"-.a<'.loga.Thefunction aXisthusdifferentiable foreveryxandhasthe differential coefficicnt a".Ioga. 164 Chapter V.Powerseries. Inthesameway,asregards differentiability anddifferential coefficienl oflogxfor~>0,wededuce, byconsideration of log(I+~'!) i. Xn=log(.;+x,,)-log';= .;=-!.-log(1+~'!)"''' x" x" l; l; thatthedifferential coefficient existshereand=-}. Oftheproperties ofdifferentiable functions weshallforthepre­ sentrequire scarcely morethaniscontained inthefollowing simple theorems: Theorem 7.Ifafunction f(x)isdifferentiable inanintervalJ anditsdifferential coefficient isthereconstantly equalto0,thenf(x) isconstant III!,thatistosayis==f(xo),where XoisanypointofJ Iftwofunctionsf1(x)andf'J(x)aredifferentiable in!andtheir differential coefficients constantly coincide there,thenthedIfference of thetwofunctions ISconstant in!,therefore wchave f'J(x)=f1(x)+c=f1(x)+[I;(xo)-f1(xo)] where XoisanypointofJ. Theorem 8.(Firstmeanvaluetheoremofthedifferential calculus.) Iff(x)iscontinuous intheclosedintervala~x<banddifferentiable inatleasttheopeninterval a<x<b,thenthereis,inthelattcr interval, atleastonepoint ~forwhich fJ~)-f(a)=f'(~).b-a (Inwords: Thefinitediffcrence quotient relativetotheendpoints ofthe intervals isequaltothedifferential coefficient atasuitable interiorpoint.) Theorem 9.Iff(x)isdifferentiable atgandf'(~)is>0«0)then f(x)"increases" ("decreases") at~,i.e.thedifference {thesame} .f(x)-f(g)has(h . )SIgnas(to)(x-g),t eOpposIte providedIx-gIbelessthanasuitable number S. Theorem 10.Iff(x)isdifferentiable ataninterior pointgofits interval ofdefinition, thenunlessf'(g)=0thefunctional valuef(g) cannotbe;:::::everyotherfunctional valuef(x)inaneighbourhood ofg oftheformIx-gI<S,i.e.gcannotbea(relative) maximum point. Similarly theconditionf'(g)=0isnecessary forgtobea(relative) minimum point,i.e.suchthatf(~=)isnotgreaterthananyotherfunctional valuef(x),aslongasxremains inasuitable neighbourhood off Definition 11(Differential coefficients ofhigherorders. If f(x)isdifferentiable in/,then(inaccordance withDef.1)f'(x)isagain afunction defined in./*.Ifthisfunction isagaindifferentiable in.l. *andcalledthederivedfunction off(x). §19.Functions ofarealvariable. 165 thenitsdifferential coefficient iscalledtheseconddifferential coefficient of{(X)andisdenoted by{"(x).Correspondingly, weobtainthethird and,generally, thekthdIfferential coefficient of{(x),whichisdenoted by((k)(x).Fortheexistence ofthekthdifferential coefficient atl;it isthusCv.Def.10)necessary thatthe(k-1)thdifferential coefficient shouldexistbothatl;andatallpointsofacertainneighbourhood of1;.­ Thelthdifferential coefficient of{(k)(X)is«k+ll(x), k20,l~O.(As othdIfferential coeHicient of((x)wcthentakethefunction itself) Oftheintegral calculus weshall,inthesequel,require onlythe simplcst concepts andtheorems, exccptinthetwoparagraphs onFourier series,whereratherdeeper material ha,;tobebrought 1O. Definition 12(Indefinite integral). Ifafunction {(x)isgivenin anintervala...bandifadifferentiable function F(x)canbefound suchthat,foranpointsoftheinterval inquestion, F'(x)={(x),then wcsaythatF(x)isanindefinite integral of{(x)inthatinterval. (Be­ side"F(x),thefunctions F(x)+carcthenalsoindefinite integrals of{(x),ifcdenotes anyrealnumber. BeSides these,however, there arenoothers}. Wewrite F(x)=f(x)dx. Inthesimplest cases,indefinite integrals areobtained byinverting the elementary formulae ofthedifferential calculus E.g.from(sinIXx)'=etcosetX fsina:xitfollows thatcosetxdx=~-, andsoonTheseelementary ruleswe assume known Special integrals ofthiskind,excepting theveryslInplest, are liltleusedinthesequel; wementiOn JdxI 1 I 2 x-I--=..log(1+x)--;-log-(1-x+x2)+-=tan-1-,_-'-.1-+x33 ti V3 V3 Jdx{ifx2+x{if+I\"2 -1 --1__=---log--- -~--'-+-(tan(:1:\12-1)+tan(xv'2+1)],1+x' 8 x~-xV'2+I 4 f[1] sin.~cotx-xdx=log-----;;-• Though inindefinite intcgrals, wefindnomorethananewmode ofwriting forformulae ofthedlffercntial calculus, thedefinite integral introduccs anessentially newconcept. Definition 13(Definite integral). Afunction defined inaclosed intcrval a...bandtherebounded issaidtobeintegrable overthis interval Ifitfulfilsthefollowing condition: Dividetheinterval a.•.binanymanner intonequalorun· equalparts(1£:;;::::1,apositive integer), anddenotebyxl'x:l'..., xn-1 thepointsofdivision bctwecn a=Xoandb=xn'Nextineachof thesenparts(inwhichbothendpoints maybereckoned) chooseany 166 Chapter V.Powerseries. point,anddenotethechosenpointsincorresponding orderby~l>~2'•••I ~n'Thenformthesum11 Sn=i(x.-X"-l)f(~.). 11--=-1 LetsuchsumsSnbeevaluated foreachn=1,2,3,...independently (thatistosay,ateachstagex.and ~"maybechosenafresh). But,atthe sametime,l",thelengthofthelongestofthenpartsintowhichthc interval isdivided whenforming Smshalltendto012• IfthesequenceofnumbersSI'S2'...,inwhatever waytheymayhave beenformed, invariably provestobeconvergent andalwaysgivesthesame13 limitS,thenf(x)1villbecalledintegrable inRiemann's senseandthelimit Swillhecalledthe definite integraloff(.x)overa •..b,andwritten bff(x)dx. a xiscalledthevariable ofintegration andmayofcoursebereplaced by anyotherletter.-Insteadoff(~.)wemayalsotake,toformS7llthe lowerbound Cl:.ortheupperboundf3.ofallthefunctional values 14inthe interval X"_l<x<x... Theorem 11(Riemann's testofintegrability). Thenecessary andsuffi­ cientcondition forafunctionf(x),definedintheclosedintervala...b andtherebounded, tobeintegrable overa ...b,isasfollows: Given E>0,achoiceof11andofthepoints Xl'x2,•••IX"_lmustbepossible, forwhich ifi.=Ix,,-X"-lIisthelengthofthevthpartofa•..banda.the oscillation off(x)inthissub-interval. Thiscriterion mayalsobeexpressed asfollows, assuming thenotation chosensothata<b:Afterchoosing E,wemustbeabletoassigntwo "step-functions" (functions constant instretches) suchthatina::':::X=:;;::b wehavealways g(x)<f(x)<G(x) 11IfI(x)>0,a>b,andweconSider aplaneportion Sbounded onthe onesidebytheaXisofabscissae, ontheotherbytheverticals through aandband bythecurvey=I(x),thenSnisanapproximate valueoftheareaofS.This however onlyprovides asatisfactory representation ify=I(x)isacurveinthe intuitive sense. 12Wemaythenalsosaythatthesubdivisions, withincreasing n,become indefinitely closer. • 13Itiseasilyshewnthatifthesequence (S,,)isinvariably convergent italso ipsolactoalwaysgivesthesamelimit. 14InthesecasesS"givestheareaofa("step-") polygon inscribed orcircum­ scribedtotheplaneportionS. aswellas15§19.'Functions ofarealvariable. bJ(C(x)-g(x»dx<r::. "167 Itsufficesinfacttoput,inXV_I::;xSxv, g(x)~lXv,C(x)=f3",v=I,2,...,n, together with g(b)=1Xn>C(b)=-f3n. Fromthiscriterion, thefollowing particular theorems arededuced: Theorem 12.Everyfunction monotone ina::::::x<b,andalsoevery function continuous ina~x:s:::b,isintegrable overab. Theorem 13.Thefunction/(x) isintegrable overab,if,ina...b, itisbounded andhasonlyafinitenumberofdiscontinuities. Riemann's testofintegrability mayalsobegiventhefollowing form: Theorem 14.Thefunctionlex)isintegrable overa .••bif,and onlyif,itisbounded thereandif,twoarbitrary positivenumbers 8andr:: beingassigned, thesubdivision ofa...bintonsu~-intervals described intheorem 11canbesocarriedoutthatthesub-intervals ivinwhichthe oscillation oflex)exceeds 8adduptoatotallengthlessthanz. Theorem 15.Thefunction/(x) iscertainly notintegrable overa ...b ifitisdiscontinuous atc'verypointofthatinterval. Theorem 16.Iflex)isintegrable overab,then/ex) isalsoin- tegrable overeverysub-interval a'•.•b'ofab. Theorem 17.Ifthefunction /(x)isintegrable overa...b,then everyotherfunction/1(x)isintegrable overa...b,andhasthesame integral, whichresultsfrom/(:~)byanarbitrary changeinafinitenumber ofitsvalues. Theorem 18.Iflex)and/1(x)arctwofunctions integrable over a...b,thentheyhavethesameintegral provided thattheycoincide at leastatallpointsofaseteverywhere denseina...b(e.g.allrational points). Forcalculations withintegrals wehavethefollowing simpletheorems, where/(x)denotes afunction integrable overtheintervala...b. u b Theorem 19.WehaveJ/(x)dxc.--J/(x)dxandifaI'a2,as b a arethreearbitrary pointsoftheinterval a.•.b, a. ~ al J/(x)dx+Jf(x)dx+Jf(x)d:\;=O. ".a, Theorem 20.If/(x)andg(x)aretwofunctions integrable overa...b, (a<b),andifina••.bwehaveconstantly /(x)s:;g(.\:),thenwealso have b b 1/(x)dx~Ig(x)dx. " a ~.Itisimmediately obvious fromthefirstformofthecriterion thatastep­ function suchKSG(.\:)-g(.\:)isinte~rable. 168 ChapterV.Powerseries. Theorem 20a.II(x)Iisintegrable withI(x)andwehave,ifa<b, b bIfI(x)d xI<fII(x)Idx. a aTheorem 21.(First'»'leanvaluetheore'»'l oftheintegral calculus.) Wehave b fI(x)dx=Jl,'(b-a) ifJl,isasuitablenumberbetween thelowerbound IXandtheupperbound {1ofI(x)ina...b(IX~P.s::f3).Inparticular wehave bIff(x)dxIs:K.(b--a) ifKdenotesaboundaboveofII(x)Iina...b. Theorem 22.Ifthefunctions/1(X),J2(x),...,J1'(x)areallintegrable overa...b(p=fixedpositive integer), thensoaretheirsumandtheir product andfortheintegralofthesumwehavetheformula b b bf(/1(x)+...-I-11'(x»d:'C=f11(x)d x-I-...-I-f11'(x)dx; a a a i.e.thesumofafixednumberoffunctions maybeintegrated termbyterm. Theorem 22a.IfI(x)isintegrable overa...bandifthelower boundofII(x)Iina...bis>0,thenItx)isalsointegrable overa...b. Theorem 23.IfI(x)isintegrable overa...b,thenthefunction x F(x)=ff(t)dt iscontinuous intheinterval a...handisalsodifferentiable ateverypoint oftheinterval, whereI(x)itselfiscontinuous. IfXoissuchapoint,then F'(xo)=I(xo)there. Theorem 24(Funda'»'lental theore'»'l ofthedifferential andintegral calculus). Iff(x)isintegrable overa•••b,andIfI(x) hasanindefinite integralF(x)inthatinterval, then b ff(x)dx= F(b)-F(a). a Theorem 25(Change ofthevariable ofintegration). If I(x)isintegrable overa...handx=rp(t)isafunction differentiable in IX•••f3,withrp(IX)=aandrp(f3)=h,iffurther,whentvariesfromIXto f3,rp(t)variesmonotonely (inthestrictersense)fromatob,andifrp'(t), thedifferential coefficient ofrp(t),isintegrable 16over IX.•••f3,then b fJ fl(x)d x=ff(tp(t»·ep'(t)dt. a IX 10Thederivative ofadifferentiable function neednotbeintegrable. Examples ofthisfactare,however, notveryeasilyconstructed (cf.e.g.H.Lebesgue, Lec;ons surl'integration, 2ndEdition, Paris1928,pp.93-94). §19.Functions ofarealvariable. 169 Theorem 26(Integration byparts). Hf(x)isintegrable over a...bandF(x)istheindefinite integral off(x),iffurtherg(x)isa function, differentiable ina ...b,whosedifferential coefficient isinte­ grableovera...b,then17 b bff(x)g(x)dx=[F(x).g(x)]:-IF(x).g'(x)dx. a a Thefollowing penetrates considerably further thanalltheabove simpletheorems: Theorem 27(Second 1'neanvaluetheore1'n oftheinte­ gralcalculus). Iff(x)and'P(x)areintegrable overa ...band'P(x) ismonotone inthatinterval, thenanumberg,withas::::g::::::b,canbeso chosenthat b I; b I'P(x).f(x)dX-c'p(a)IJ(x)dx+'P(b)IJ(x)dx. u u I; Here'P(a)mJ.yalsobereplaced bythelimit,certainly existing underthe hypotheses, 'Pa=limq;(x),andsimilarly 'P(b)byCfib=limCfi(x);butin ~a+O ~b+O thiscaseadifferent valuemayhavetobechosenforg. Wemention onlythefollowing oftheapplications oftheconcept of integral aboveconsidered: Theorem 28(Area). Hf(x)isintegrable overa ...b,(a<b)and, letussuppose, alwayspositive intheintcrval18,thentheportionofplane surface bounded bytheaxisofabscissae, theordinates through aandb, andthecurvey0=f(x)-ormoreprecisely, thesetofpoints(x,y)for whicha-:::;;x-sb,andatthesametime,foreachsuchx,0~y::::f(x), b -hasameasurable areaanditsmeasure isIf(x)dx. u Theorem 29(Length). If.\:=Cfi(t)andy=«/J(t)<iretwofunctions differentiable inex:::::t:S[3,andif'P'(t)andifs'(t)themselves arecon­ tinuousin(X•••[3,thenthepathtracedoutbythepointx~-=Cfi(t),y=«/J(t) intheplaneofarectangular coordinate system0x,0y,whentdescribes theinterval fromexto[3,hasameasurable lengthandthisisgivenbythe integral fJIv'cp'(ff-+rp'-(l)2 dt. IX Finallywemaysayafewwordsonthesubjectofso-called improper integrals. 17Ilere[h(x)]~denotes thedifference 1z(h)-Tl(a). 18-whiehmayalwaysbearranged bytheaddition ofasuitable constant. 170 Chapter V.Powerseries. Definition 14.Iff(t)isdefined fort>aandisintegrable over a<t::;:;x,foreveryx,sothatthefunction x F(x)=ff(t)dt a isalsodefinedforeveryx>a,then,iflimF(x)existsand=c,wesay thattheimproper integral x--++oo +00 JJ(t)dt a converges andhasthevaluec. Theorem 30.Iff(t)isconstantly :-::0orconstantly :S"0forevery 00 t>a,thenff(t)d tconverges ifandonlyifthefunction F(x)ofDef.14 a isbounded forx>a.Iff(t)iscapableofbothsignsfort:::::a,thenthe sameintegral converges if,andonlyif,givenanarbitrary E>0,Xo>a canbesodetermined that x"IJJ(t)dtI<E >.' foreveryx'andx"both>Xo' Andquiteanalogously: Definition 15.Iff(x)isdefined, butnotbounded, intheinterval a<t~b,openontheleft,andisintegrable, foreveryxofa<x<b, overtheinterval x<t:'Sh,sothatthefunction b F(x)=ff(t)dt x isdefinedforeachofthesex's,then,iflimF(x)existsand=c,wesay >.-?a+O thattheimproper integral(improper ata) bff(t)dt a isconvergent andhasthevaluec. Exactly analogous conventions aremadeforaninterval openonthe right.Thecaseofaninterval openonbothsidesisreduced tothetwopre­ cedingcasesbydividing itataninteriorpointintotwohalf-open intervals, andthentakingtheorem 19asadefinition. Theorem 31.IfinthecaseofDef.15,wefurtherhavef(t)20every­ whereor<0everywhere, thentheimproper integral inquestion exists if,andonlyif,F(x)remains bounded ina<x~b.Iff(t)assumes both signs,thentheintegral existsif,andonlyif,givene:>0,wecanchoose 8>°sothatx,. Iff(t)dtI<e: x' foreveryx'andx"bothbetween a(excl.)anda+8. §20.Principal properties offunctions represented bypowerseries.171 §20.Principal properties offunctions represented by powerseries. Weinterrupted ourdiscussion ofpowerseriesattheobservation, terminating §18,thatthesumofapowerseries,intheinterior of itsinterval ofconvergence, defines afunction, whichwewillnow denotebyf(x): f(x) E::iall(x-xo)'"Ix-XoI<r. •=0 Weresume itatthatpoint,andagreeinthisconnection, unless specialremark tothecontrary ismade,toleavetheinterval ofcon' vergence openatbothends,evenshouldthepowerseriesconverge at oneorbothoftheendpoints. Nowif,asisthecasehere,aninfiniteseriesdefinesafunction inacertaininterval, thenthemostimportant problem is,ingeneral, todeduce fromtheseriestheprincipal properties ofthefunction re­ pre"ented byit-interpreting theseforlllstance inthesenseofthe summary oftheprecedmg section. Inthecaseofpowerseries,thispresents nogreatdifTiculty. We shallsee,onthewhole,thatafunction represented byapowerseries possesses alltheproperties whichwemayconsider particularly im­ portant andthatthealgebra ofpowerseriesassumes apeculiarly SImpleform.Forthisreason. powerseriesplayaprominent part, anditisprecisely onthisaccount thattheirdiscussion belongs tothe elements ofthetheoryofinfinitesenes. Intheseinvestigations, wemay,without thereby restricting the scopeoftheresults, assume Xo=0,i.e.assume theseriestobeof thesimplified form2:anx".Itsradiusofconvergence isofcourse assumed positive(>0),butmaybe+00,i.e.theseriesmaybe everywhere convergent. -Wethenhave,first,the °Theorem. Thefunctionf(x)defined,initsintervalofconver-96 gence,bythepowerseries1:an(x-xn)",iscontinuous atx=xo; 11=0 thatistosay,wehave limf(x)=limian(x-xor'=ao=f(xo)' X-)oZo X-+Zo,,=0 Proof.If0<(!<',thenbyS3,D, .2;IanIe,,-lconverges with ,,=1 IfwewriteKC>0)forthesumoftheformer, thenwehave,for everyIx-XoI~(!, If(x)-aoI=I(x-xo)·i'all(x-').'0),,-1I<Ix-XoIK.,,=1 172 Chapter V.Powersenes. ITtherefore e>0isarbitrarily givenandif()>0ISlessthanboth (}and~,thenwehave,foreveryIx-Xo1<d, If(x)-anI<e; whichby§19,Def6b,provesallthatwasrequired. Fromthistheorem, weimmediately deduce theextremely far. reaching andveryfrequently applied: 97. 0Identity Theorem forpowerseries.Itthetwopowerseries .J;anxnand1;bnxn n=O n=O havethesamesumillallintervalIxI<eillwhichbothDJthnnconverge 19, thenthetwoseriesareentirelyidentical, thatistosay,foreveryn=U,I,2, •••,wethenhave sidesofProof From (a) ao+a1x+a2x2+...=bu+b1X+b~x2+... itfollows, bythepreceding theorem, letting x-+0onboth theequation, thataO=boo Leaving outthesetermsanddividing byx,wcmferthatfor0<Ixl<(} (b) a1+aJx+a3x2+...=b1+b~x+bax2+"', anequation fromwhichwededuce, inexactlythesameway20,that a1=b1and Proceeding inthismanner, weinfersuccessively (moreprecisely: by complete induction) thatforeverynthestatement isfulfilled. Examp Iesandillustrations. 1.Thisidentity theorem willoftenappear bothinthetheory andin theapplications. Wemay al~ointerpret itthus:ifafunction canbere· presented byapowerseriesintheneighbourhood oftheorig-in,thenthisis onlypossible moneway.Inthisform,thetheorem mayalsobecalledthe theorem ofuntqueness Itofcourseholds,inthecorresponding statement, for thegeneral powerseriesIan(x-xo)n. 2.Sincetheasser!ioninthetheorem culminates inthefactthatthe corresponding coefficients onbothsidesoftheequation (a)areequal,wemay alsospeak,whenapplying thetheorem, ofthemethodofequatmg coefficients. 10Orevenforeveryx=Xvofanullsequence (xv)whosetennsareall=1=O. -Intheproofwehavethentocarryouttheltmlting processes inaccordance with §19,Def.4a. 20Forx=0,equation (b)isnotinthefirstinstance secured, sinceitwas established bymeansofdivision byx.Dutforthelimiting processx....,.0thisis qUIteimmaterial (cf.§19,Def.4). )~a(x-x)"..;....;n 0,,=0§20.Principal properties offunctions represented bypowerseries.173 3.Asimpleexample ofthisformofapplication isthefollowing: We certainly have,foreveryx, (l1-x)k(l+x)k=(l+x)2k or Ifwemultiply outontheleft,by91,Rem.1,andequatethe coefficient~ on bothsides,thenweobtaIn,forinstance, byequating thecoefficients ofx": arelation between thebinomial coefficients whIchwouldnothavebeenso easytoprovebyothermethods. 4.If((x)isdefined forIxI<randwchavc,forallsuchx's, fe-x) =I(x), then((x)iscalledaneve11function. Ifitisrepresentable byapowerseries, thenwcatonceobtainbyequatJl1g' coefficients, a,=a3=a,-~ ...=aH:+1=0 sothatinthepownseri('sof((x),onlyevenpowers ofxcanhavecoefficients diff('rent fromO. .5.Ifontheotherhand,((-x)= -((x),thenthefunction issaidtobe odd.ItsC'xpansion inpowerseriescanthenonlycontain oddpowers ofx.In particular, ((0)=O. "Venowproceed onestepfurtherandproveanumber oftheorems whichmustberegarded asinevery re~pect themostImportant in thetheoryofinfilliteseries: °Theorem 1.If isapowerserieswith(positive) radiusr,thenthefunction((x)thereby rvpresented, for1x-XoI<r,mayalsobeexpanded inapowerseries wtthanyotherpointXloftheinterval ofconvergence ascentre,'we have,infact, where bk=i(nth)an+1<(Xl-XO)", n-0 andtheradiusrofthisnewseriesisatleastequaltothepositive number l'-IXl- XoI· Proof. IfXlliesintheinterval ofconvergence oftheseries,so thatIXl- XoI<1',then f(x)=.2an[(Xl-xo)+(X-xJ]" i.e. ,,=0 (a){(X),,?aan[(Xl-xo)"+(n(Xl-Xo),,-1(X-Xl)+... ...+(:)(x-Xl)"];9S. 174 Chapter V.Powerseries. andallthatwehavetoshowisthatwemayheregrouptogether alltermswiththesamepowerof(x-Xl)'i.e.thatthemainre· arrangement theorem 90maybeapplied. If,however, totestits validity, wereplace, inthelatterseries, everytermbyitsabsolute value,thenweobtain'the series .21an1[IXl-XoI+IX-xtI]n; 11=0 andthisiscertainly stillconvergent, if IXl-XoI+IX-XlI<r,orIX-XlI<"-Ixt-XoI· Iftherefore XisnearertoXlthaneitheroftheendpoints ofthe original interval ofconvergcnce, thentheprojected rearrangement is allowed, andweobtainforr(x),asasserted, arepresentation of theform Ifweproceed indetaIltogroupthetermscontaining (x-Xl)"to· gether,bywriting thetcrmsoftheseries(a)insuccessive rowsone belowtheother,thenthekthcolumn gives bk=(:)ak+(h11 )ak+1(Xl-xo)+...=,,~o(n1k)an+l..(x 1-XO)'I, whichcompletes therequired proof 21. Fromthistheorem wededuce themostdivcrseconsequences. First wehavethe 0Theorem 2.Afunction represented byapowerseries f(x)=1;a,,(x-xo)" 11=0 iscontinuoHs ateverypointXlinterior totheinterval0/convergence. Proof. Bythepreceding theorem, wemaywrite,foracertain neighbourhood ofXl' f(X)=.ian(X-xo)"=ib"(X-Xl)" 11=0 11=0 with bo=ian(Xl-xo)"=f(xl)· ,,=0 Forx-xl' thesecondoftherepresentations off(x),by96,atonce givestherequired relation (v.§19,Def.6): limf(x)=f(xt). Z-+ZJ 0Theorem 3.Afunction represented byapowerseries f(x)=.2an(x-xoY' 11=0 isdifferentiable ateveryinterior point Xl0/the'interval ofconvergmct 21Wethushave,quiteincidentally, afreshproofoftheconvergence, already l'stablished in9~,ofthedifferent seriesobtained forthecoefficients b•• §20.Principal properties offunctions represented bypowerseries.175 (v.§W,Def.10)anditsdifferential coefficient atthatpoint,r'(Xl)' maybeobtained bymeansofterm-by-term differentiation, i.e.wehave f'(Xl)=f;nan(Xl-XO)"-1=1;(n+1)an+!(Xl-Xor· 11=1 11=0 Proof. Since((x)=.iJbn(x-xlt,wehaveforeveryxsuf- 11=0 ficiently nearXl: f(x)-[(Xl)_b+b(x-x)+... X-Xl -1 ':l 1 ' whence forx-+Xl'by96,takingintoaccount themeaning ofbl, weatoncededucetherequired result:f'(Xl)=bl=;Enan(Xl-xo)n-1. Theorem 4.Afunction represented byapowerseries, f(x)=i;an(x-xot, II=U has,ateveryinterior pointXlofitsinterval ofconvergence, differential coefficients ofeveryorderandwehave f(k)(Xl)=k;b"=i(n+1)(n+2)...(n+k)anH(xt-xo)'" II=U Proof. Forevery Xoftheinterval ofconvergence wehave,as wchavejustshown, {'(x)=J;(n+1)an+l(x-xo)'" n=O {'(x)isthusagainafunction represented byapowerseries,-and infactbyonewhich,inaccordance with95,hasthesameinterval ofconvergcnce asthcoriginal series.Hencethesameresultmaybe againapplied tof'(x),giving f"(x)=1;n(n+1)an+l(x-xo)n-l=l'(n+l)(n+2)ant2(x-xot 11=1 n=O ByarepctitlOn ofthissimpleprocess, weobtainforeveryk, «k)(X)=.2(n+1)(n+2).,.(n+k)(a"H(x -xo)'" n~O validforeveryxoftheoriginal interval ofconvergence. Putting in particular X=xl'wetherefore atoncededucetherequired statement. Ifwesubstitute, forthecoefficients bkinthcexpansion oftheorem 1, thevalues:1{(k)(Xl)nowobtained, thenwefinallyinferfromallthe abovetheso-called °Taylor series 22•IfforIx-XoI<r.wehave 99. {(x)=.2an(x-Xo)", n=O andifXlisaninterior pointoftheinterval ofconvergence, thenwe 23BrookTay/or:Methodus incremento rumdirectaetinversa, London 1715. -Cf.A.Prmgsheim, Gcschichtc desTnylorschen Lehrsntzes, Bibl.mnth.(3) Vol.I,p.433. 1900. 176 Chapter V.Powerseries. have,foreveryxfor1vhich 23Ix-XlI<rl-=r-IXl-Xo" f(x)=f(xl)+f'\X1)(x-Xl)+f"i,X1)(X-XI)2+... f(k)(Xl)...+k'(X-Xl)k+... WithTheorem 3forthedifferentiation ofourseries,wecouplethecor­ responding theorem forintegration. Sinceafunction represented bya powerseriesiscontinuous intheinterior ofitsinterval ofconvergence, itisalso,by§19,Theorem 12,integrable overeveryinterval contained, together withitsendpoints, intheinteriorofthisintervalofconvergence. Forthiswehavethe oTheorem 5.Theintegralofthe(continuous) functionf(x)represented <fj byL;an(x-xo)nintheintervalofconvergence, maybeobtained bymeuns n~O oftermbytermintegration, withtheformula provided XlandX2arebothinteriortotheintervalofcom'ergence. Proof. By95,2,thepowerseries ~aF(x)=L;_n_(x-xo)n+l n~On+1 hasthesameinterval ofconvergence asthegivenseries '"f(x)=Ean(x-xo)n. 1l"--=O By98,3,thefirstseriesisanindefinite integralofthesecond. lIenceby §19,Theorem 24,thestatement followsatonce. Thesetheorems onpowerserieswemaycomplete inaspecial direction bythefollowing important addition: Theorem 2onthe continuity ofthefunction represented byapowerserieswas,aswe mayagainexpressly observe, onlyvalidfortheopeninterval ofcon­ vergence. Thus,forinstance, inthecaseofthegeometric seriesL;xn, 23Thenumber 1'1=l'-IX,- XoIofthetextneednotbetheexactradius ofconvergence ofthenewseries.Onthecontrary, thelattermayproveconSiderably larger.Thusforf(x)=Exn=1~xandXl= -~weobtain,byaneasycal- culation, f(x)~E.oo .(2)kH. (x-I-l)k k-"O3 2 andtheradiusofthisseriesisnot=l'-IXl-XoI~=~'butis=~. §20.Principal properties offunctions represented bypowerseries.177 1ofsum1_-;;,wecandeduce fromourconsiderations neitheritscon- tinuityatthepomtx= -1,noritsdiscontinuity atx=+1,by immediate inspection oftheseries. Evenifthepowerseriescon- vergedatoneoftheendpoints oftheintervals (ashere2J:"for x=-1),weshouldnotbeabletoconclude thisfactdirectly. That however, inthislastparticular case,thepresumption is,atleastto someextent,justified, welearnfromthefollowing: Abe1'slimittheorem24•Letthepowerseriesf(x)=ianx"100. n=O haveradiusofconvergence randstdlconverge forx=+,. Then aJlimf(x) existsand=2.'an"'''. ~~r-O n=O ." Ori1totherwords:If2Janx..stillconverges forX=+"then ..~O thefunctionf(x)definedbytheseriesin-,<x<+"isalso continuous onthelettattheendpoint x=+,. Proof. Thereisnorestriction 2.;inassuming r=+1.For jf~a"x"hasradiusr,thentheseriesXa,,'x",inwhichan'=anr", obviously hasradlllS1;andthelatterseriesisconvergent at+1or -1,if,andonlyif,theformerwasat+,or-rrespectively. "Vctherefore infutureassumer=+1.Ourhypothesis is, therefore, thatf(x)=2'anx"hasradIUs1andthat2:an=scon­ verges; andourstatement isthat limf(x)=s, I.e. ",-~1-0 S-f(x)=(1-x)1(s-S,,)x"==(1-x)ir"x". n=O 7l=ONowby91(v.alsolater,102),wehaveforIxI<1, t co co." aJ-_.-Y'ax"= \ 1x")'ax"=Y'sx.. 1-xn~on n~o':;:o n n~On 1 ifbys"wedenote thepartIalsumof~an'Consequently f(x) =(1.-x)2'snxnandsince1=(1-x)2'x",wetherefore deduce, for Ixl<1, (a) "Journal f.d.reineu.angew. Math.Vol1,p.311.1826.cf.233and §62.-Thetheorem hadalready beenstatedandllsedbyGausstD1Squis. genernles circascriem"',1~12;WerkeHI,p143)andinfactprecisely In theformproved further on,thatrn-+0involyes(1-x)_:~:rnxn-+0ifX-+1 fromtheleft(v.eq.(a»).TheproofgivenbyGaussloe.cit.ishowever tn­ correct,asheinterchanged thetwolimiting processes whichcomeundereon­ si,lcration forthistheorem, without atalltesting-whether hewasjustified in sodOIng ••Thisremark holdsingeneral foralldiscussions of(noteverywhere convergent) powerseriesofpositive radiusr. 178 ChapterV.Powerseries. Herewehavewrittens-Sn=r,.,the"remainder" oftheseries; theseremainders, by82,Theorem 2,formanullsequence. Ifnowr::>0isarbitrarily given,thenwefirstchoosemsolargethat, ~ eforeveryn>m,wehaveIrnI<2'Wethenhave,forO::Sx<I, m e: !XlIs-I(x)I<I(1-x)Ernx"1+-,-(1-x)·Exn, 1I~O 2 n--m+l hence,ifpdenotesapositive number greaterthanIroI+Ir1I+..+IrmJ,thisis e Xm+-1>-P.(1-x)+2(l-x)'I-x' Ifwenowwrite8=thesmallerofthetwonumbers have,for1 -0<x<I, e:eIs-I(x)I<"2-+2=e.Iand-~thenwe2p. which,by§19,Def.5,provestherequired statement "f(x)~sforx~1 -0". Wehaveofcourse,quitesimilarly, Abel'slimittheorem fortheleft endpoint oftheintervalofconvergence: XJ IfEa"xnstillconverge forx= -r,then n~O co limf(x) existsand=E(-I)nanrn. X~-T+U 1I-U Thecontinuity theorem 98,2andAbcl'stheorem 100together assert that 101. lim(Eanxn)=Ean;n x~~ iftheseriesontherightconverges andxtendstoefromthesideonwhichlies theorigin. IftheseriesEanendiverges, wecannotassertanything, without furtherassumptions, astothebehaviour ofEanxnwhenx~g.We havehowever inthisconnection thefollowing somewhat moredefinite: Theorem IfEanisadivergent seriesofpositiveterms,andEanxn hasradiusI,then coI(x)=Eanxn_+00 n=O whenxtendstowards+1fromtheorigin. Proof. Adivergent seriesofpositive termscanonlydivergeto+00.Iftherefore G>0isarbitrarily given,wecanchoosemsolarge thatao+a1+...+am>G+1,andthenby§19,Theorem 3,choose 8<1sosmallthatforevery1>x>1-0,wecontinue tohave ao+a1x+...+amxm>G. §21.Thealgebraofpowerseries. Butthenwehave,afortiori, er, f(x)=};a"x">G, 11-0179 whichisallthatrequired proof. Remarks andexamples forthetheorems ofthepresent paragraph willbegivenindetailinthenextchapter. §21.Thealgebra ofpowerseries. Beforewemakeuseofthefar-reaching theorems ofthepreceding section(§20),whichleadtotheverycentreofthewidefieldofapplication ofthetheoryofinfiniteseries,wewillenterintoafewquestions whose solution shouldfacilitate ouroperations onpowerseries. Thatpowerseries,aslongastheyconverge, maybeaddedandsub­ tractedtermbytermalreadyfollowsfrom83,3and4.Thatwemay immediately multiply outtermby term, intheproduct oftwopower series,provided weremainintheinterioroftheintervals ofconvergence, followsatoncefrom91,sincepowerseriesalwaysconverge absolutely intheinterioroftheirintervals ofconvergence. Wetherefore have,with };anx"±};b"x"=};(an±bn)x" 00 00 00 also};a"xn.};bnxn=};(aDbn+a1bn-1+...+anbo).~n, nU 110 11--0 provided xisinteriortotheintervals ofconvergence ofbothseries 26. Theformulae 91,Rem.2and3werethemselves afirstapplication ofthistheorem.Ifthesecondseriesis,inparticular, thegeometric series, thenwcfind I.e. orrr; 00 !1:' Eanxn.Exn=Esnxn, n-0 n:::=O 11:.--0 1 y, ere 1 _xnEoanxn=n~uS"xn 00 00 Eanxn=(I--x) EsnxnJ 71---U 11c:.oU102. whereSn=aD+a1+...+amandIxI<1andalsolessthanthe radiusof};a"x". Weinferinassimpleamannerthateveryseriesmaybemultiplied -andinfact,arbitrarily often-byitself.Thus (00)2 00Ea"x"=E(aDan+a1a"_1+...+anaD).~n; nU 11~O andgenerally, foreverypositive integralexponent h, (iaX,,)k=Ea(k)xn 103." " n=() ,,~() 20Hereweseetheparticular importance ofCa/lchy's product (v.91,1). 180 Chapter V.Powerseries 104.wherethecoefficients an'kJareconstructed fromthecoefficients aina "perfectly determinate manner -eventhoughnotanextremely obvious one27forlargerk's.Andtheseseriesareallabsolutely convergent, solongasZanx"itselfis. Thisresultmakesitseemprobable thatwe"may"alsodivide bypowerseries,-thatforinstance wemayalsowrite 1 =C+Cx+Cx'J+...ao+atx+a2x2+... 0 J 'J andthatthecoefficients cnmayagainbeconstructed inaperfectly determinate manner fromthecoefficients an'Forwemayfirst, writing-an=an',forn=1,2,3,.."replacethelefthandratiobyao 1 1 lJu•1-(a,'x+a/x2+...) andthenby ;[1+(a/x+a'J'xll+...)+(at'x+...)'J+(at'x+...)3+...] o whichmustactually resultinapowerseriesoftheform:Eenx",if thepowersareexpanded by103andlikepowers ofxthengrouped together. Ourjustification forwriting theabovemayatoncebetested fromasomewhat moregeneral pointofview: Wesuppose givenapowerseries2'anx"(intheabove,the seriesian'xn),whosesumwedenotebyf(x)ormoreshortlybyy. n=O Wefurther suppose givenapowerseriesiny.forinstance g(y)=2'bny"(intheabove,thegeometric series2'yn)andinthis wesubstitute forytheformerpowerseries: Underwhatconditions dowe,byexpanding allthepowers, in accordance with103,andgrouping likepowers ofxtogether, ob­ tainanewpowerseries Co+ClX+cllx2+...whichconverges and hasforsumthevalueofthefunction ofafunction g(f(x))? We assertthe °Theorem. Thiscertainly holdsloreveryxlorwhtch1;IanxnI n=O converges andhasasumlessthantheradius01~bny". 27Recurrence formulae fortheevoluation ofa~)aretobefoundin J.W.LGlalsher, NoteonSylvester's paper:Development ofanideaofEisen stein(Quarterly Journal, Vot14,p.79-84, 1875),wherefurtherreferences to thebibliography mayalsobeobtained. SeealsoB.lIansted, Tid~krift for Mathematik, (4)Vot5,pp.12-16,1881. §21.Thealgebra ofpowerseries. Ull Proof.\Vehaveobviously hereacaseofthemainrearrangement theorem 90,andwehaveonlytoverifythatthehypotheses ofthat theorem arefulfilled. Ifwefirstwrite yk=(ao+a1x+...)k=ao(k)+a/klx+a~(k)x'3+"., forming thepowersby103,andalsosuppose thisnotation 28adopted fork=0andk=1,thenwehave,in bo=bo(aci°)+aiD}x+.,.-I-a~O)x"-I-) blY=b1(acil )+a?1x+...-I-a~l)x"+ ) (A) bT,b(Ik)+(k)+-I-Ik).. )kY=kaoalx•••anx-1-... (foreverydefinite n=0,1,2,...)theseries Z(k)occurring inthetheorem 90.1£wenowtake,instead ofy=Lanx",theseries 'YJ=L!anxnI,and,writingIxI=~,form, quitesImilarly, 1IboI=IboI(aci°)+aID)~+ + a~O)tl+...) Ib1I'YJ=Ib1I(agl)+ail}~+-+a~I);n-1-"') [I;.i";.I;.i(~"~:.l";~.:.~~~'i;~.:.): thenallthenumbers inthisarray(A')are~0andsincefurthermore 2,'IbkI'YJkwasassumed toconverge, themainrearrangement theorem ISapplicable to(A'),Butobviously everynumber ofthearrayAis inabsolute value :::;::thecorrespondIng number in(A');henceour theorem isafortioriapplicable to(A)(cf.90,Rem.3).Inparticular, therefore, thecoefficients standing vertically onebelowtheotherin (A)alwaysform(absolutely) convergent series en 2-'bkan1k)=cnk=O andthepowerseriesformed withthesenumbers ascoefficients, i.e.(A') 00 2Jc..x" ,,~O ISagain,fortheconsidered valuesofx,(absolutely) convergent and hasthesamesumas2:bny".Wetherefore have,asasserted. g(((x)=i'cnx" ,,~o withtheindicated meaning ofc.., Remarks andExamples. 105. 1.Ifthe"outer" seriesg(:v)=Ibkxi'converges everywhere, tbenour theorem evidently boldsforeveryzforwhichIa"z"converges absolutely. a.Wehavetherefore towrite a~O)=1,ala)=aJO)=...=0,anda~l)=a", thelatterforn=0,I,2,.... 1 (051) 182 Chapter V.Powerseries. Ifbothseries('onverge everywhere, thenthetheorem holdswithout restriction foreveryx. 2.Ifao=0andbothserieshaveapositive radius, thenthetheorem celtainly holdsforevery"sufficiently" smallx,thatistosay,thereisthen certainly apositive numbere,suchthatthetheorem holdsforeveryIxI<e. Forify=alx+allx2+.."then'7=Iat',1xI+Ia,1·1XliI+...;andsincefor x-+0,wenowabo,by96,have'7-+0,'7iscertall1ly lessthantheradius ofIbkykforallxwhoseabsolute valueislessthanasuitable numbere. yn 3.Intheseries2JnJ'we"may" forinstance substitute y=Ixnfor x"Ix1<1,ory=2J-foreveryn,andthenrearrange inpowers ofx.n! 4.Towrite,aswedidabove: 1 ' -------1I~ =Co+ctX+clIx2+...ao+alx+allX+... is,wenowsee,certainly allowedifao=f0andfurtherxisinabsolute value sosmallthat '7=I::xI+I::XliI+...<1I whichbyRem.2iscertalDly thecaseforeveryIxI<ewithasuitable choiceofe.Wemaytherefore say:We"may" diVidebyapowerseriesof posltwe radmsItItsconstant term15=f0andprovided were~trlct ourselves to sufficiently small 29valuesofx. Todetermine thecoefficients cnbythegeneral method usedtoprove theirexistence, would, -evenforthefirstfewindices, -beanextremely laborious process. Butoncewehaveestablished theposslblltty oftheexpansion -whichisatthesallletimenecessanly uniqueby97,-wemaydetermine thecn'smorerapidly byremarking that :Eanxn•ICIIxn=1, sothatwehavesuccessively aoCo=1 aoCl+atCo=0 aoclI+atCl+allCo=0 aoca+atca+allCl+a3Co=0 Fromtheserelations, sinceao*0,thesuccessive coefficients co,ChC2,•••maybe uniquely determmed, thesimplest method bemgwiththeaidofdetermmants, by Cramer's Rule,whichimmediately yieldsaclosedexpression 3uforcnmtermsof (l+x+~~+~~+ ...)-1xformanysubsequent investigations or ExpandaUtall·..tan' 5.Asaparticularly important example wemaysetthefollowing question 31: 1 2.Howsmallxhastobe,isusuallyimmaterial. Butwhatisessential, isthat somepositive radiuseexists,suchthattherelation holdsforeveryIxI<g. -Thedetermination ofthepreciseregionofvalidity requires deepermethods of function theory. 80Explicit formulae forthecoefficients oftheexpansion, inthecaseofthe quotient oftwopowerseries,maybefounde.g.inJ.Hagen,Ondivision ofseries, Americ. Journ.ofMath.,Vol.0,p.236,1883. 11Euler:Instituttones calc.ditT.,Vol.2,§122.1755. §21.Thealgebra otpowerseCles. 183 UJpowers0/z.Herethedetermination ofthenewcoefficients becomes Peculiarly elegant ifwedenotethem,notbyCn,butby:"'n orasweshallnI' do,forhistoric reasons, by~,;.Thentheaboveequation is (Xxg ) ( BtBg0 )11+-ifi+3i+...BO+-fix+21X"+...== andtheequations fordetermining Bnare,insuccession, !!l--011-, and,ingeneral, forn=2,3,.•., I.Bo+_l__.B,+__I__B.+...+.!.._~n.=1-_0 n!0'(n-I)l I!(n-2)121 11(n-1)1- • Ifwcmultiply byn!,wcmaywritethismoreconcisely: (~)Bo+(~)Bl+(;)BJ+...+(n~1)B"_l=O. NowIfweherehadBVinplaceofIJv,toreachv,thenwecouldwritein,tead (B+l)"-B"=O; 106. andtherecurring formula underconsIderation alsomayboborneinmindunder thisconvenient form,asasymbolic equation, i.e_oncwhichisnotintended to beinterpreted literally, butonlybecomes valid'vlthapartIcular convention, ­ heretheconvention tbatafterexpanding thentbpoweroftbebinomial (13+I), wcreplace eachBVbyBv•Ourformula nowyields,forn=2,3,4,5, ••• successively, theequations 2 BI+1=0, 3BJ+3BI+1=0 , 4B3+6B,+4BI+I=0, 5B.+10Bs+10IJ.+5BI-I-1=0, fromwhichwcdeduce 1 Bl=-2' andthen1 Bg={f' B--.!. •-30 and 691 BIS=-2730' ThesearecalledBernoulli's nU1nbers andwillbementioned repeatedly lateron(§24,4;§32,4;§55,IV;§64).Fortbemoment, weareabletoinfer onlythatthenumbers Bnaredefinite ratIOnal numbers. Theydonot,however, conform toanyapparent orsuperficial law,andhaveformedthesubjectof manyelaborate discussions sa. s.Bernoulli's numbers arefrequently indexed somewhat differently, Bo' BpBsB6>B7,...beingomitted and(-I)k-IB kwritten instead ofBgk,for k=I,2,'"Atableofthenumbers B2,B•••..,toB,••maybefound 10 J.C.Adam;,Journ.f.d.leine 11.angew. Math.,Vol.85,1878 Welllaymention inpas.mg thatBl20ha.fornumerator anumber With113digIt.,andforde­ nominator tbenumber 23582559aO;whileBus h~\sthedenominator 6and, 184 ChapterV.Powerseries. Finallywewillproveonemoregeneraltheorem onpowersenes: ~ Giventhepowerseriesy=:Ean(x-xo)",convergent forIx-XoI<r, 1I~--'O wehave,foreveryxintheneighbourhood ofxo, adeterminate corre­ sponding valueofy,inparticular forx=-=Xothevaluey=ao,whichwe willaccordingly denotebyyo'Thenwehave y-Yo=al(x-xo)+a2(x--xo)2+.... Because ofthecontinuity ofthefunction, toeveryxnearXoalsocorre­ spondsavalueofynearyo'WewouldnowenquirewhetherorhozlJfar everyvalueofynearYoisobtained andwhetheritisobtained onceonly.If thelatterwasthecase,notmerelyywouldbedetermined byx,butcon­ verselyxwouldbedetermined byy,andtherefore xwouldbeafunction ofy.Thegivenfunctiony=f(x)would,aswesayforbrevity, be reversible intheneighbourhood ofXo(cf.§19,Theorem ti).The question ofreversibility isdealtwithby: 107. 0Reversion theorem forpowerseries. Giventheexpansion y-Yo=al(x-xo)-I-(l2(x-xo)2-I-..., convergent forIx---XoI<r,thefunction y=f(x)therebydetermined is reversible intheneighbourhood ofxo,underthesolehypothesis thatal'*0; i.e.therethenexistsoneandonlyonefunction x=ep(y)whichisexpressible byapowerseries,convergent inacertainneighbourhood of)'0'oftheform x-xo=bl(y--Yo)-I-b2(y-YO)2+... andforwhich,inthatneighbourhood, wehave(inthesenseof104) f(ep(y»==y. Moreo'ver bl=1:al' Proof. Aswehavealreadydonemorethanonce,weassumein theproofthatxoandYoare=0,-whichimpliesnorestriction 33.But wewillthenfurtherassumethatal=1,sothattheexpansion (a) y=x+a2x2+a3x3+... istheonetobereversed. Thattooimpliesnorestriction, forsinceal=l=0, byhypothesis, wecanwritealx+a2x2+...intheform (alx)+a22(alX)2+~3(alX)3+....al al inthenumerator, anumber With107digits.Thenumbers B.,Bh••• ,toB•• hadprevIOusly beencalculated byOhm,ibid.,Vo!.20,p.Ill,1840.-The numbers B'JJfirstoccurinJames Bernoulli, Arsconjectandi, 1713,p.96.- Acom­ prehenSive account isgivenbyL.Saalschutz, "Vorlesungen uberdieBernoulllschen Zahlen", Berlin(J.Springer) 1893,andbyN.E.Niirlund, "Vorlesungen uber Dlfferenzenrechnung", Berlrn(1.Springer) 1924.Newinvestigations, whichchiefly concern thearithmetical partofthetheory,aregivenbyG.Frobenius, Sitzgsber. d.Berl.Ak.,1910,p.809-847. 33Or:wewriteforbrevityx-Xo=x'andy-Yo~-y'andthen,forsim­ plicity's sake,omittheaccents. §21.Thealgebra ofpowersenes. Ifwewriteforbrevity a1x=x'and,forn:?::2,185 an ,an=an' 1 andsubsequently, forsimplicity's sake,omittheaccents, thenwe obtainprecisely theaboveformofexpansion. Itsuffices therefore to consider this.Butwecanthenshowthatapowerseries,convergent inacertaininterval, oftheform (b) x=Y+b2y2+bay3+... existswhichrepresents theinverse function oftheformer, sothat (c)(y+b!ay!a+...)+a2(y+b2y2+...)2+aa(y+b2y2+"')3+... isidentically =y,ifthisseriesisarranged inpowers ofy,inaccor­ dancewith104,-i.e.allthecoefficients mustbe=0exceptthat ofyt,whichis=1. Sincewehavewritten, forbrevity, xinstead ofalx,weseethat thesenesontherighthandsideof(b)hasstilltobedivided byal torepresent theinverse oftheseriesalx+a'Jx2--1-••"wherea1has nospecialised value.Inthisgeneral caseweshalltherefore have bl=.!--ascoefficient ofyl.a1 Ifweassume, provisionally, thatthestatement (b)iscorrect, thenthecoefficients b..arequiteuniquely determined bythecondition thatthecoefficients ofy2,y3,...in(c)aftertherearrangement, have alltobe=O.Infact,thiSstipulation givestheequations b2+a2=0 (d) ba+2b2a2+a3=0 b.+(b./+2ba)a'J+3b2aa+a.=0 fromwhich,asisimmediately evident, thecoefficients b..maybe determined insuccession, without anyambiguity. Thusweobtain,the values (e)b2=-a2 ba=-2b2a2-a;J=2a22-aa b,=-(b22+2ba)a2-iJb2aa-a, bo=... butthecalculation soonbecomes toocomplicated toconveyanyclear ideaofthewhole. Nevertheless, theequations wehavewrittendown showthatifthereexistsatallaninverse function ofy=((x), capable ofexpansion informofapowerseries,thenthereexists onlyone. Nowthecalculation justindicated showsthatwhatever mayhave beentheoriginal givenseries(a),wecaninvariably obtainperfectly 186 Chapter V.Powerseries. ueterrninate v,lluesb.,<,0thatwecaninvariably construct apowel seriesy+b~y'.l+...whichatleastformally satisfies thecondItions oftheproblem, theseries(c)becoming identically =y.Itonly remains tobeseenwhether thepowerserieshasapositive radius ofconvergence. Ifthatcanbeproved, thenthereversion iscompletely carried out. Therequired verification may,asCauchy firstshowed, actually beattained, inthegeneral case,asfollows: Choose anypositive numbers"nforwhichwehave Ia"I<". and2:"..x..hasapositive radiusofconvergence. Proceeding IDthe abovemanner, fortheseries: y=x-a'.!x2-a3x3-+... whoseinverse is,then,say, x=y+fi~y2+fJ8y3+... weobtain,forthecoefficients fi..,theequations fJ2="2 fJs=2fJ'.!a2+"8 fJ,=(fJ2'.l-I-2fJa)a'.!+3fi'.!as+a, inwhichallthetermsarenowpositive. Thusforevery". fJ..2.Ib.l· If,therefore, itispossible sotochoose theathattheseries2:fiv' hasapositive radiusofconvergence, itwouldfollowthat2:b..y";l~o hadapositive radiusandourproofwouldbecomplete. Wechoosethea,,'sasfollows: Thereiscertainly apositive number(2,forwhichtheoriginal seriesx+a'.!x2-j-...converges absolutely. Apositive number /{must,however, thenexist(byS2, Theorem 1and10,11)suchthatwehave,forevery 'J'=2,3,..., la..I(2"~/{ orla..I~::'. Wethenchoose, for"=2,3,..., J( "=-n(!'J" sothatweareconcerned withreversing theseries,convergent for Ixl«h x9 ( X X9) ](·x9y=x-/{'-' 1+-+--:-+ ..·=x- .(!2 (! (!2 (!((!_x) Butthisfunction isimmediately reversible. Forwemayatoncesee bydifferentiation -wearedealing, infact,withasimplehyperbola, ofwhichthestudent shoulddrawagraphforhimself-,thatin -oo<x<x =n(1_1/~k-) 1 <::"VK+l?' §21.Thealgebra ofpowerseries. 187 thefunction increases monotonely (inthestricter sense)from-00 tothevalue Yl=2K+e-2VK(K+e5 andtherefore possesses, fory<Yl'auniquelv determined inverse whosevaluesare<Xl'Forthis,since K·x·y=x-eCe-x)or(K+e)x·J-e(e+y)x+e'!.y=O, wehave,uniquely, X=2(Ke+e)[e+Y-v'y2-2(2K+e)y+e2]. Further y'!.-2(2K+e)y+e'!.~(y-Yl)(y--y~), ifwewriteforbrevity, withtheabovedefined valueofYl' Y!l=2K+f!+2VK(K+e), andbothYlandYJare>0,sincethesecond isandthetwohave product=e2.But x=2(:~e)[1+;- (1-~)~'(1-;J~J. Inthefollowing chapter weshallseethat,forIzI<1,thepower (1-z)~-canactually beexpanded inapowerseries-beginning with1 - -~-+..'.Assuming thisresult,itfollows immediately that xalsomaybeexpanded inapowerseries,convergent atleastfor lyl<Yl: x=eH[1+2:-(1--!-+...)(1-~-+...)J2(K+e) e 2y, ~YH =Y+fJ~Y':l+.... Byourfirstremarks theproofisherebyentirely completed. Theactualconstruction oftheseries y+b2y':l+... fromtheseries X+a':lx':l+... herealsoinvolves ingeneral considerable difficulties andnecessitates theuseofspecial artifices ineachparticular case34.Examples of thiswilloccur 111§§26, 27. Weonlynotefurther, afactwhichwillbeofuselateron,-that if(b)istheinverse of(a),thentheinverse oftheseries (a') y=x-alax'!.+a3x3-+-... wherethesignsarealternated, isobtained from(b)bysimilarly alternating thesigns,i.e. (b') x=Y-b':ly2+b3y3-+-... 3<1Theg-eneral valuesofthecoefficients ofexpansion b,.areworked out asfarasb13byC.E.vanOrstrand, Reversion ofpowerseries,Plnlos.Magazine (6), Vol.19.p.366,1910. 188 Chapter V.Powerseries. in(c), (c) isnecessarilyThisisatonceevident, ifwefirstactually expandthepowersof (y+b2y2+...) obtaining, say, (y+b2y2+...)+a2(y2+b3(2)y3+ ) -I-a3(y3-I-b4(3)y4+ )+... Underthenewassumption, thesameprocess, sincetheproduct oftwo serieswithalternating coefficients isagainaserieswithalternating co­ efficients, gives (c') (y-b2y2+...)-a2(y2-bp>y3+ )+a3(y3-bp>y4+ )-... Andfromthisweimmediately inferthatonequating tozerothccocffi­ cientsofy2,y3,.•.,wemustobtainthcidentical equations (d),thusde­ ducingforbvprecisely thesamevaluesasbefore. Theexactanalogue holdsgoodwhenthetwopowerseriescontain, fromthefirst,onlyoddpowersofx.Thus,iftheinverseseriesof y=x+a3x3-I-a5x5-1-••• IS X=Y+b3y3+b5y5+-..., thentheinverseseriesof y=x-aax3+a5x5--I- x=y-b3y3+b5y5--1-... Exercises onChapter V. 64.Determine theradiusofconvergence ofthepowerseriesEanxn,when anhas,fromsomepointonwards, thevaluesgivenInEx.34or45. 65.Determine theradIiofthepowerseries 1:'i!tn'·xn. 0<'i!t<1;.EC.[)1.'.2.(2~;~1j)\nj .En!n.En!." >1.E(n!)3."nnx,an'x,a;(3n)!x • 66.Denoting byKandf'thelowerandupperlimitsof1-an-I,theradius an+1I rofthepowerseriesEa."xninvariably satisfies therelation K~r;'i;1'-.Inpar- tIcular:Iflim 1 1an-Iexists,ithasforvaluetheradiusofEanxn. an+1 67.EanxnhasradIUsr,1:an'.\:nradIUsr'.WhatmaybesaidoftheradIUs ofthepowerseries E(an±an')x"', 67a.WhatistheradiusofEa."x'"If0<t,-rTi"Ian1<+oo? 00 68.ThepowerseriesE--lE."-xn,whereE."hasthesamevalueasinEx.47, n~2nogn converges atbothendsoftheinterval ofconvergence, butineithercaseonlycon­ ditionally. 69.Prove,wIthreference to97,example :I,that 1:(~)'-o(_l)nE(-1)"(2n)".(2:1). v=o I v=o V §22.Therationalfunctions. 189 70.Asacomplement toAbel'stheorem 100,itmaybeshewnthatInevery caseinwhichEanxnhasaradius T~1,wehave hmsn;;:::;hm(1:UnX");::;;;hms" x_1-0 n=O (sn-ao+a,-+•.•+an)' 71.Theconverse ofAbel'stheorem 100,notingeneraltrue,holds,howevcr, ifthecoeffiCients anarc~0;iftherefore, mthatcase, limEanx"x_r-O eXists,thenEa",nconverges anditssumisequaltothatlimit. '" '"72.Let 1:a"x"=f(x)and1:b",,"=g(x), 1l=--=1 12=1 bothseriesconverging forIxI<e.Wethenhave(forwhatvaluesofx?) ~ 00 1:b"f(xn)=1:ung(x"). n·-1 U-=1 (Byspecialising thecoefficients manyinteresting identities maybeobtained. Write 1e.g.bn==1,(_I)n-"n'etc.) 73.Whatarethefirsttermsoftheseries,obtained bydivision, for 1 I? x2x" ' Xx2 1 -2!+4!-+... 1+'2+3-+... (Further exercises onpo\\<erserieswillbefoundinthefollowmg Chapter.) Chapter VI. Theexpansions oftheso-called elementary functions. Thetheorems ofthetwopreceding sections (§§20,21)affordusthe meansofmastering completely alargenumber ofseries.Wcproceed to explainthisinthemostimportant cases. Acertain-notverylarge-number ofpowerseries,orfunctions represented thereby, haveaconsiderable bearingonthewholeofAnalysis andaretherefore frequently referred toastheelementary functions. Thesewilloccupyusfirstofall. §22.Therational functions. Fromthegeometric series '" 1l+x+x2+...=1:x"=1---' Ixl<1, 1I~0 -x whichformsthegroundwork formanyofthefollowing specialinvesti­ gations, wededuce,byrepeated differentiation, inaccordance with98,4: 00 1 '"(n-/-2) 11:(n+1)x"=(-1'--=--)"> 1:2xn=('1---)"••• n~O X 11=0 X andgenerally, foranypositivep: 1I~0(n,~P)xn=-~(I_~)P+l' IxI<1.10S. 7' (o51) 109.190ChApter VI.Theexpansions oftheso-called elementary functions. Ifwemultiply thisequation oncemore,inaccordance with9J,by :Ex"=-II ,weobtain,by91and108:-xi[(P)+(P+I)+...+(p+n)]xn=J;(n+Pi1)xn. n=OP P P n=OP+ Bycomparing coefficients (inaccordance with97),wededucefronl thisthat (:)+(P;1)+...+(P;n)=(n;~i1), whichmayofcoursebeproved quiteeasilydirectly (byinduction). Ifwedothis,wemayalsodeduce theequalIty108byrepeated mul- tiplication of:Exn=-11withitself,by103.-x Sincewehave (n;p)=(n:p)=(_lr(-Pn-I) wcobtninfromlOS,ifwetherewnte-xforxand-kforp+1, theformula validforIxI<1andnegative integralk.Thisformula isevidently anextension ofthebinomial theorem (29,4)tonegative integral exponents; forthistheorem mayforpositive integral k(orfork=0), alsobewrittenintheform109,asthetermsoftheseriesforn>k areinthatcaseall=O. Formulae suchasthosewehavejustdeduced have-aswemayobscrve immediately, andonceforall-atwo-fold meaning; ifwereadthemfrom lefttoright,theygivetheexpansion orrepresentation ofafunction bya powerseries; ifwereadthemf.-omrighttoleft,theygiveusaclosedex­ pression forthesumofaninfiniteseries.Accordmg tocircumstanccs, theone interpretation ortheothermayoccupy theforemost placeinourattention. Bymeansofthesesimpleformulae wemayoftensucceed in expandmg, inapowerseries,anarbitrary givenrational function f(x)=~o_La,x+ + am~~,bo+b,x+ +bkxk namely whenever f(x)maybesplitupintopartialfractions, i.e.ex­ pressed asasumoffractions oftheform A (x-a)P Everyseparate fraction ofthiskind,andtherefore thegivenfunc­ tionalso,canbeexpanded inapowerseriesby10S.Andinfact thisexpansion canbecarriedoutfortheneighbourhood ofeverypoint xl!distinct froma.Weonlyhavetowrite (I)"1 ( 1 )1' x-a =(Xo-=-~),,' 1-(~-=::) §23.Theexponential tunctlOD. 191 andthenexpand thelastfraction by10S.Bythismean~wesce, atthesametime,thattheexpansion Willconverge forIx-XoI<Ia-XoI andonlyforthesevaluesofx. Thismethod, however, onlyassumes fundamental importance when wecometousecomplex numbers. Examples. 110. §23.Theexponential function. 1.Besides thegeometric senes,thesocalledexponential senes 00xn x'Jx3 xn n~~n!_1+x+-2-'+-3-;+-...+-n!+... playsaspeCldlly fundamentdl partinthesequel. Weproceed now toexamme inmoredetailthefunction whichItrepre~ents. This so-called exponential fU'lIction wedenoteprovisionally byE(x).As theseliesconverges everywhere by92,2,E(x)IScertainly, byOS, defllled, continuous anddifferentiable anynumber oftimes,foreveryx. Foritsderived functlOIl, weatoncefll1d E'(x)=E(x), soth.1tforallderived functions ofhigherorderwemustalsohave E(')(x)=E(x). Weshallattempt todeduce allfurtherproperties fromtheseries ftsell.Wehavealready shownin91,3thatifXlandx~arcanytwo realnumbers, wehaveIIIallcases (a) E(Xl+X~)=E(xt}.E(x~). Thisfundamental formula isreferred tobrieflyastheaddition theorem lortheexponential function!. Itgivesfurther E(Xl+x2+Xs)=E(Xl+-X2)"E(xa)=~E(Xl)'E(x2)·E(xs) andbyrepetitIOn ofthisprocess, wefindthatforanynumber ofreal nurnbers xl'x2'•••,xk' (b) E(Xl+-X2+-...-+Xk)=E(x1)·E(x2)·••E(xk). 1Alternative proof. TheTaylor's series99forE(x)is E'(x,) E(x)=E(Xl)+-1-!-(x-Xl)+..., validforallvaluesofXandXl"Ifweobserve thatEl')(Xl)=E(Xl),thenit atoncefollows, replacing XbyXI+x21that E(xl+Xg)=E(Xl)'[1+(I+~.;+...J=E(xl)·E(x2), q.e.d. 192Chapter VI.Theexpansions oftheso-called elementary functions. Ifweherewritex.=1foreach",wededuceinparticular that E(k)=[E(1)1/< holdsforeverypositive integerk.SinceE(0)=1,italsoholdsfor k=O.Ifwenowwrite,in(b),X"=;foreach",denoting bym asecondinteger:?0,thenitfollows that or,--sinceE(m)=[E(l)]m, -that m E(:)=[E(1)]k. IfwewriteforbrevityE(1)=E,wehavethusshewnthatthe equation (c) E(x)=EX holdsforeveryrational X2:O. If~isanypositive irrational number, thenwecaninany number ofwaysformasequence (xJ,ofpositive rational terms,con­ verging to~.Foreachn,wehave,bytheabove, E(xn)=EXn. Whenn_+00,thelefthandside,by9S,2,tendstoE(~),andthe righthandside,by42,1,toEE,sothatweobtain Em=E~. Thusequation (c)isprovedforeveryrealx2:o. But,finally,(a)gives E(-x).E(x)=E(x-x)=E(0)=1, whence wefirstconclude thatE(x)=0cannothold2foranyrealx andthatforx2:0 E(-x)=_1_=~=E-x• E(x)EX Butthisimplies thatequation (c)isalsovalidforeverynegative realx. Wehavethusprovedthattheequation holdsforeveryrealX; andatthesametimethefunction E(x)hasjustified itsdesignation ofexponential function; E(x)isthexthpowerofafixedbase, namely of 1 1 1 1E=E(1)=1+-+- +- +...+- +...112131 nl ~Thismayofcourse, forx>0,bededuced immediately fromtheseries, byinspection, sincethisisaseriesofpositive termswhosetermofrankois=1. §23.Theexponential function. 193 2.Itwillnextberequired toobtainsomefurtherinformation aboutthisbase.Weshallshowthatitisidentical withthenumber e alreadymetin46a,sothat3 lim(1+-.!.-)n=i'-.!.-. 'It ~=OvI Theproofmaybemadesomewhat morecomprehensive, byat onceestablishing thefollowing theorem, andthuscompleting theinvesti­ gationof46,a: oTheorem. Foreveryrealx, Ill. lim(1-I-:)"existsandisequaltothesum4oftheseriesEx~. n-+-oc ,,·ov. Proof. Wewriteforbrevity (1+~)"=X" andi'xv~=s(x)=s.,n ,=0 Itthensuffices toprovethat(s-xn)_0.Nowif,-given,first,a definitevalueforx,-eischosen>0,wecanassume 1>solarge thattheremainder Further, forn>2, (n)x(n)xk(n)x·x"=1-+-1-n+...+knk+...+nn" =1+x+~(1-~) x2+.1(1!)(1-~)x3+21 n JI " n ...+k~[(1-~)(1-~)...(1-k~1)]xt+.... aserieswhichterminates ofitselfatthenthterm.Thetemlinxk, k=0,1,...,evidently hasacoefficient ~0,butnotgreaterthan thecoefficient 1{kIofthecorresponding termoftheexponential series. Thesameisalsotrue,therefore, ofthedifference oftheformerand thelatterterm.Accordingly wehave,forn>p-fromthemannerin whichpwaschosen 5- Is-x..1<2\[1-(1-~)IIxl!l+... '''+;1[1-(1- ~)"'(l-P~I)llxl"+ ;. Everyindividual termofthe(p-1)firsttermsontherighthandside •Wehavehere,therefore, asignificant example ofproblem B.Cf.intro· duction to§9. •Firstproved-ifnotinanentirely irreproachable manner -byEulcr, Introductio inanalysin infinitorum, Lausanne 1748,p.86.-Theexponential seriesanditssumeO:werealready knowntoNewton(1669)andLeibnit: (1676). ~Weassumep>2fromthefirst. 194ChapterVJ.Theexp:msions oftheso-called elementary functions. their5t1111 maychoose Butofanullsequence R;hence alsotendsto0,andwe I!sumremains<2foreveryn>no'isnowobviously thenthterm forpisafixednumber no>psolargethatthis wethenhave,forevery whichprovesourstatement 7.-Forx==1,wededuce inparticular 001(1)" E=.2--=lim1+----=e; ~=oyI"-+00 1& andmoregenerally, foreveryrealx, 00x"E(x)=2:l=c<Il o,-=0v Thenewrepresentation thusobtained forthenumber e,bythe exponential series,isaverymuchmoreconvenient oneforthefurther discussion ofthisnumber. Inthefirstplace,wecan,bythismeans, easilyobtainagoodapproximation toe.For,sinceallthetermsofthe seriesarepositive, weevidently have,foreveryn, 1 1 1 sn<e<sn+-(n+~i)l+fn+i)T(n-+f5+en+I)l(n+1)9-+.0. or i.e. (a) eWehave(1-~)-lt(1-:)-1,"',(I-P:1)_I,anctsotheir product (by41,10), also_1,or[1-(I-~)...(I-P:l)J_O; so,aS:l: andparefixednumbers, theproduct ofthislastexpression by]-1xIPpI also_0;andsinlllnrly forthcothcrtcrms.-Wecanalsoinferthercsult directly from41,12 ?Theartifice hereadoptcd isnotoneimagined adhoc,butonewhich isfrequently used:Thetermsofasequcnce arerepresented asasum XII=xo(ll)+Xl(1l1+...+xkll"wherethetermssummed notonlydepend in-,. dividuaUy onn,butalsoincrease znnumber withn:kn_00.Ifweknow howeachindividual termbehaves forn-+00Iasforinstance, thatx,,(ll)for f1xedvtendsto;",thenwemayoftenattainourcndbyseparating outafixed number ofterms,sayXo(11)+Xl(11)+...+xp(11)withfixedp;thistends,when n-+<Xl,to~o+~1+...+~p.by41,9.Theremaining terms, X~ll~l+...+xfc:) wethenendeavour toestimate inthebulkdirectly, byfinding boundsabove andbelowforthem,whichoftenpresents nodifficulties, provided pwns suitably chosen. §23.Theexponential function. 195 Ifwemultiply thisinequality byql, wewilldenoteforthemoment byg,\\neresndenotes apartialsumofthenewseriesfore.Ifwecal­ culatethesesimplevaluese.g.forn=9(v.p.251)thenwefind 2·718281<e<2·718282, whichalready givesusagoodideaofthevalue 8ofthenumber e. -Fromtheformula (a)wemay,however, drawfurther important infcIences. Anumber isnotcompletely beforeusunlessitisratIOnal andiswritten intheformP....Iseperhaps arational number? Theq inequalities (a)showquiteeasilythatthisisunfortunately notthecase. Forifwehade=P..., thenforn=q, formula (a)wouldgive:q s<.£..<s +-~ '1q qq!q 1 1 wheres'l=2+21+'" +q1' thenqISqISaninteger, which anditfollows that 1g<p.(q-1)I<g+q<g-t-l. Butthisisimpossible; forbetween thetwoconsecutive integers gand g+1therecannotbeanother integerp.(q-1)IdIstinct fromeither: eisanirrational number. 3.Theaboveinvestigations giveusalltheinformation, withregard tothehmitof(1+:)n,whichwe,inthefirstinstance, require; the twoproblems AandB(§9)arebothsatisfactorily solved. Inspite ofthis,wepropose, invIewofthefundamental importance ofthese matters, todetermine thesamelimitagamandinadIfferent way,­ entircly independent ofthepreceding. Weuseonlythefact,previously established, that(1+:r-..e . Thiswewillfirstextendbyshowing that (1+.!._)"n-..e Yn also,when(y,,)isanysequence otpositive numbers tending to+oc. WhenYn=apositive integer, foreveryn,thisisanimmediate con­ sequence oftheprevious result 9. 8Thenumber enasbeencalculated to346placesofdecimals byJ.M.Boormann (Math.magazme, Vo!.I,No.12,p.204,lilil4). oForif6isgiven>0,and 110isdetermined, by46ft,sothat 1(1+~)"-eIremains<8forevery 11>110,thenweshallalsohave 1(1+-tJ1I"_sl<6foreverytI>n1,provided n1issochosenthatforevery 11>111wehaveYn>tlo' 196Chapter VI.Theexpansions oftheso-called elementary functions. Ifthenumbers y"arenotintegers, therewillstillbeforeach11 one(andonlyone)integerk"suchthat k"<y"<k"+1, andthesequence oftheseintegers k"mustevidently alsotendto+00. Now,however, ifk,,~1, (1+_1..)kn<(1-I-,,!-)lIn<(1+..!-)kn+l. kn+1 YII kn Andsincethenumbers k l1areintegers, thesequence (1+L)"n+1=(1-I-iJkn.(l+iJ andthesequence (1)kn(1)kn+11 1+kll+1=1+kn+1 •1+_1 _ k,,+1 bothtendtoe,byourfirstremark. Hence,by41,8,wealsohave (1-I-_~)lJ"_e. Y" Wemaynextshowthatwheny"'--00,wealsohave (1)II~1+Y~-e, or,otherwise, thatwheny"_+00,wehave (1-:J-II"-+e. Allthenumbers y"'must,however, beassumed< -1,i.e. y..>1,sothatthebaseofthepowerdoesnotreduceto0ora negative value;thiscanalwaysbebrought aboutby"afinitenumber ofalterations". Since (1-..!-)-II"= (Y~)lIn=(1+~)lIn-l. (1+~), Y" Y" 1 Y"1 y"1 andsince,withy",y..-1also-++00,thestatement tobeproved ISanimmediate consequence ofthepreceding one. Writing..!-=Z,wemaycouplethetworesultsthus:Yn.. 1 (1+%")Z;-+e provided (z")isanynullsequence withonlypositive oronlynegative terms,-thetermsinthelattercasebeingall> -1.Fromthis wcfinallyobtainthetheorem, including alltheaboveresults: 112. Theorem: If(x..)isanarbitrary nullsequence whosetermsare different from0and> -1fromthefirst10,then11 I (a) Hm(1+xn)"'n=e•..-.", 10Thelattermayalwaysbeeffected by"afinitenumber ofalterations' (cf.3S,6). 11Cauchy: Resume desleconssurlecalculinfinit.,Paris1823,p.81. §23.Theexponential function. 197 Proof. Sinceallthe:I:,,'S=f:0,thesequence (:1:,,)maybedivided intotwosub-sequences', onewithonlypositive andonewithonlyne· gativeterms.Since,forbothsub-sequences, thelimitinquestion, as wehaveproved,exists 12and=e,itfollowsby41,5thatthegivensequence alsoconverges, withlimite. By42,2,theresultthusobtained mayalsobeexpressed inthe form (b) quiteindependently, as and2.-,thatwhichwillfrequently beused. By§19,Def.4,theresultalsosignifies that,invariably: 1 lim(l+x)'"=e. "'~o Fromtheseresults, itagainfollows, weannounced, ofourinvestigations of1. (1+~r-e'"; for(~)iscertainly anullsequence18,sothatwehave,bythepre· cedingtheorem, " (1+~r'-eandtherefore whichwaswhatwereqUIred H. 4.Ifa>0,andxisanarbitrary realnumber, then,denoting bylog thenaturallogarithm (v.p.211), a'"=e"'loga=1+~()ffx+~:t)2x2+(10ft!?x3+... isanexpansion inpowerseriesofanarbitrary power. \Vededucethe limiting relation 15 aZ-l----loga forx_O, Xa>O.113 1llIfoneofthetwosub-series breaksoffafterafinitenumber ofterms, thenwecan,byafiniten;Jmber ofalterations, leaveitoutofaccount. 13Weconsider thisnullsequence forn>IxIonly,sothatwemayal· wayshave~> -1.nl'Combining thiswiththeresultdeduced in2.,thattheabovelimithas thesamevalueasthesumoftheexponential series,wehaveasecondproof ofthefactthatthesumoftheexponential seriesis=tJ"'. 13Directproof:Ifthex;sformanullsequence, thenby3:>,3,so dothenumbers :1'..=aXIl-1;andconsequently, by112(b)I aZ"_I= Y,,·log-a log-a1 x"log(1+:1'..)-+-1-=oga. 198Chapter VI.Theexpansions oftheso-called elementary function~. Thisformula provides uswithafirstmeansofcalculatmg loga· rithms,whichisalready toacertainextentpracticable. Foritgives, e.g.(cf.§9,p.7R) loga=limn(Ya -1) n....oo 2k_ =lim2kCVa-1). .1:....'" Asrootswhoseexponent isapowerof2canbecalculated directly byrepeated takingofsquareroots,wehaveinthisameans(though stillaprimitive one)fortheevaluation oflogarithms. 5.Wehavealready notedthate'"iseverywhere continuous and differentiable uptoanyorder,-witheet:=(ex),=(e"')"= .".Italso shareswiththegeneral powera"',ofbasea>1,theproperty of beingeverywhere positiveandmonotone increasing withx. Morenoteworthy thanthesearetheproperties expressed bya seriesofsimpleinequalities, ofwhichweshallmakeuserepeatl'dly inthesequel,andwhicharemostlyobtained bycomparison ofthe exponential withthegeometric series.Theproofswewillleaveto thereader. 114. a)Forevery 16x,e'"> ]+x, )'"1fJforx<1,e<I _x' x1 et:,,)forx>-1,-1-<-c-<x,+x ~)forx<+1,x<eX_1<_x_l-x' e)forx>-1, e)forx>0,X 1-I-x>eHz, XxP e>P1'(P=O,1,2, ...), (x)1/ .3'-'LfJ)forx>Oandy>O, e"'>1+y>cx+II, (})foreveryx=!=O,le'"-11<el2:1-1<IxIcl"'l. §24.Thetrigonometrical functions. Wearenowinaposition tointroduce thecIrcular functions rigorously, i.e.employing purelyarithmetical methods. Forthispur­ pose,weconsider theseries,everywhere convergent by92,2: x2X' kx2kC(x)=1 -2i+4i-+...-I-(-1)(2k)!+... '6Onlyforx=0dotheseandthefollowing inequalities reducetoequa· lities.-Thereadershouldillustrate themeaning oftheinequalities onthl!: relative curves. §24.Thetrigonometrical functions. 199 5 (-x)= -5(x). C(-X)=C(x),and x3x~ kx2t+1 Sex) ~~.x-3-'+5i-+..'1-(-1)(~k-+-I)I-l-"" Eachoftheseseriesrepresents afunction everywhere continuous and differentiable anynumber oftimesinsuccession. Theproperties of thesefunctions willbeestablished, takingasslarting pointtheirex· pansions inseriesform,anditwillbeseenfinallythattheycoincide withthefunctions cosxandsinxwithwhichwearefamiliar from elementary studies. 1.Wefirstfind,by98,3,thattheirderived functions havethe following values: C'= -5 ,CIf= -C,C",=5 , C''''=C; 5'==C,5"= -5,5'"= -C,5""=5 ; relations validforeveryx(which symbol isforbrevityomitted). Since,here,the4thderived functions areseentocoincide withthe original functions, thesameseriesofvaluesrepeatsitself,inthesame order,fromthatpointonwards inthesuccession ofdifferentiations. Further, weseeatoncethatC(x)isaneven,andSex)anodd. function: or (b)Thesefunctions also,liketheexponential function, satisfysimplead ditiontheorems, bymeansofwhichtheycanthenbefurtherexamined. Theyaremosteasilyobtamed byTaylor's expansion (cf.p.191,foot­ note1).Thisgives,foranytwovaluesXlandx!!'-smcethetwo seriesconverge everywhere (absolutely), - C'(x,) C"(x,) !!C(x+x)=C(x)+----x-+---x+...I 2 I 11 2 ~[:) , andasthisseriesconverges absolutely, wemay,by89,4,rearrange itinanyorderweplease, mpatttcularwemaygrouptogether all thosetermsforwhichthederived functions whichtheycontain have thesamevalue.Thisgives C(Xl+x2)=C(xl)[1-~,!g+~,!4-+...J r X.,3x.~ ,- 5(Xl)LX:!-;)"j-j---7:1-+"'j (a) C(xI+x!!)=C(xl)C(x:!) -5(Xl)S(X:J); andwefind17quitesimilarly 5(Xl+X2)=5(Xl)C(X!!)+C(Xl)5(X!!). 17Second proof. Bymultiplying outandrearranging inseriesform, weobtainfrom C(x,)C(Xg)-5(x,)5(xg) theseriesC(x,+xg),-asin91,3fortheexponential series. T h ir dproof. Thederived function off(x)= [C(x,+x)-C(x,)C(x)+5(x,)5(x))g+[5(x,+x)-5(x,)C(x)-C(x,)5(X))I is,asmayatoncebeseen,==O.Consequently (by§19,theorem 7),f(x)=reO)=O. Henceeachofthesquarebrackets mustbeseparately ==0,whichatoncegives boththeaddition theorems. (c) (d)200Chapter VI.Theexpansions oftheso-called elementary functions. Fromthesetheorems, -whoseformcoincides WIththatofthe addition theorems, withwhichwearealready acquainted fromanele­ mentary standpoint, forthefunctions cosandsin,-iteasilyfollows thatourfunctions CandSalsosatisfyalltheothersocalledpurely goniometrical formulae. Wenote,inparticular: From(a),writing xI!= -xl'wededuce that,foreveryx, C2(x)+S2(x)=1; from(a)and(b),replacing bothXlandxI!byx: C(2x)=C2(x)-S2(x) S(2x)=2 C(x)S(x). 2.Itisalittlemoretroublesome toinfertheproperties of periodicity directly fromtheseries. Thismaybedoneasfollows: Wehave C(O)=I>O. Ontheotherhand,C(2)<0;for 222&(2626 )(210212 )C(2)=1 -2l+4i-61-8!-lOt-T2!-... wheretheexpressions inbrackets areallpositive, -sinceforn::::::2, 2"2"+· n!-(n+~)!>0 d hfC()4+161. .I .antereore2<1 -"224= -3"'I.e.certamynegatIve. By §19,Theorem 4,thefunction C(x)therefore vanishes atlea'>tonce between 0and2.Sincefurther, asmaybeagaineasilyverified, Sex)=x(1-t:J+~:(1-6~~)+... ispOSItIve forallvalues ofxbetween 0and2,andtherefore C'(x)= -S(x)constantly negative there,-itfollows thatG(x)is (strictly) monotone decreasing inthisinterval andcanonlyvanishat onesinglepoint ~inthatinterval. Theleastpositive zero0/C(x), i.e.~,isaccordmgly awell·defined realnumber. Weshallimme­ diatelyseethatitisequaltoaquarter oftheperimeter ofaCIrcle ofradius1andweaccordingly atoncedenoteit18by;: ~=i, C(i)=O. From(c),itthenfollows thatSI!(~)=1,i.e.sinceS(x)wasseen tobepositive between 0and2,that S(i)=1. 18Thesituation isthusthatnistostandforthemoment asamereab­ breviation for2~;onlysubsequently shallwesh(lwthatthisnumber nhas thefamiliar meaning forthecircle. §24.Thetrigonometrical functtons. 201 S(2n)=O. addition theorems that,foreveryz,Theformulae (d)showfurther that C(n)=-1, andbyasecond application, that C(2n)=1, Itthenfinallyfollows fromtheS(n)=0, +yC(x+i)= -S(x), C(x-1-n)=-C(x), C(n-x)=-C(x),S(x+i)=C(x), S(x+n)=-S(x), S(n-x)=Sex), C(x+2:'r)=O(x),S(x+2:'r)=S(x). Ourtwofunctions thuspossess 19theperiod21T. 3.Ittherefore onlyremains toshowthatthenumber n,intro­ ducedbyusinapurelyarithmetical way,hasthefamiliar geometrical SIgnificance forthecircle.Thereby weshallhavealsoestablished the complete identity ofourfunctions C(x)andS(x)withthefunctions cosxanddnxrespectively. LetapointP(fig.3)oftheplaneofarectangular coordinate systemOXY,beassumed tomoveinsuchamanner that,atthe timet,itstwocoordinates aregivenby x=C(t)andy=Set); thenitsdIstance lOPI=11x~+y~fromtheoriginofcoordinates is constantly =1,by(c).ThepointPtherefore movesalongtheperi­ meterofacircleofradius1andcentreO. If,inparticular, tincreases from0to2n, thenthepointPstartsfromthepointAof thepositive x-axisanddescnbes theperi­ metcrofthecircleexactlyonce,inthemathe· matically positive (i.e.anticlockwise) sense. Infact,astincreases from0ton,x=C(t) dccreases, asisnowevident, from-1-1 to-1,monotonely, andtheabscissa ofP thusaSS\lmeS eachofthevaluesbetween+1 and-1,exactly once.Atthesametime, S(t)remains constantly posItive; thistherefore impliesthatPdescribes theupperhalfofthecirclefromAtoBsteadily, andpassesthrough(c) '92nisalsoaso-called primitive periodofourfunctions, i.e.aperiod, no(proper) fraction ofwhichisitselfaperiod. Fortheformulae (e)showthat 2n=niscertainly notaperiod. Andafraction 2n,withm>2,cannotbe2 m aperiod, asthene.g.S(~)=S(0)=0,whichisimpossible sinceS(x)was seentobepositive between 0and2andinfact,asS(n-x)=S(x),ispositive !)etween 0andn.Similarly forC(x),2n(m>1)cannotbeaperiod.- 202Chapter VI.Theexpansions oftheso-called elementary function". eachofitspointsexactly once.Theformulae (e)thenshowfurther thatwhentincreases from:reto2:re,thelowersemi-circle isdesCribed inexactly thesamewayfromBtoA.Theseconsiderations provide usfirstwiththe Theorem. Itxandyareanytworealnumbers torwhichx!l+y'!=1, thenthereexistsoneandonlyonenumber tbetween 0(incl.)and2:re (excl.),torwhich,simultaneously, C(t)=xandS(t)=y. Ifwenextrequire thelengthofthepathdescribed hyPwhen thasincreased from0toavalueto'theformula of§19,Theorem 2D givesatonce,forthis,thevalue t,) enJVC/:J-+ S'~dt=Jdt=to' o 0 Inparticular, thecomplete perimeter ofthecircle IS 2n 2n =JVCi2+S'"Jdt=Jdt=2:re. o 0 Theconnection whichwehadinviewbetween ouroriginal conside­ rat:onsandthegeometry ofthecircle,isthuscompletely establi-,hed: C(t),asabscissa ofthepointPforwhichthearcAP=t,coincides withthecosineofthatarc,orofthecorresponding angleatthecentre, andS(t),asordinate ofP,coincides withthesineofthat,mgle.From nowonwemaytherefore writecostforC(t)amisintforS(t).­ Ourmodeoftreatment differsfromtheelementary onechieflyinthat thelattermtroduces thetwofunctions fromgeometrical considerations, making usenaively, aswemightsay,ofmeasurements oflength, angle, arcandarea,andfromthIStheexpansion ofthefunctions inpower seriesisonlyreached astheultimate rc'mlt,Wc,onthecontrary, startedfromtheseseries,examined thefunctions definedbythem,and finallyestablished -usingaconcept oflengthelucidated bythein­ tegralcalculus -thefamiliar interpretation intermsofthecircle. 4.Thefunctions cotxandtanxaredefined asusualbytheratios cosx sinxcotx=sinx'tanx=cosx·; asfunctions, theytherefore represent nothing essentially new. Theexpansions inpowerseriesforthesefunctions arehowever notsosimple. Afewofthecoefficients oftheexpansions couldof course eaSIlybeobtained bytheprocess ofdivision described in105,4. Butthisgivesusnoinsight intoanyrelationships. Weproceed as follows: In105,5,webecame acquainted withtheexpansion"Jo 20Theexpression onthelefthandsideisdefinedinaneighbourhood ofoexclusive ofthispoint;therighthandsideisalsodefined insuchaneigh­ bourhood, butincluswe of0,andmoreover tSconltnuous loYx=().fnsuchcase weusuallymake 110"pedalmentIon ofthefactthatwedefinethelefthand sideforx=0bythevalueoftherighthandsideatthepoint. §24.Thetrigonometrical functions. ~ ""./''I'" 'I'1:.•.1:2n,\X3 ~I'",,1'+- + . + ---=--=---- -- ...eoX'_l "..,.0vI- 22! 3!203 wheretheBernoulli's numbers B"are,itistrue,notexplicitly known, butstillareeasilyobtainable bytheverylucidrecurrence formula106. Thesenumbers wemay,andaccordingly will,infuture,regardas entirely known ~l.Wehavetherefore, -forevery"sufficiently" smallx (cf,105,2,4) Thefunction onthelefthandsideishowever equalto x x andfromthisweseethatitisanevenfunction. Bernou/li's numbers B3, B5,B7aretherefore, by97,4,all=0,asalready seenin106,andwe have,usingtheexponential seriesfore~andwritingforbrevity ~~z: z~z41+--+--+'"2!4! +B.(2)~+B,(2)4+Z'--'z3 z~ =1-2',z4Tz.•• z+:n+5T+'" Ifonthelefthandside,wehadthesigns+and-occurring alter­ nately,bothinthenumerator anddenominator, weshouldhavepre­ ciselythefunction zcotz.Dividing outonthelefthandsidebythe factorz,sothatonlyevenpowers ofzoccur,maywethendeduce straight awaythattherelation z·z41--+--+ ... %cotz =__~'-~!_-- =1 -!!~(2z)'J+B,(2Z)4-+...z'z4 2! 4!1-3!+51-+... obtained fromourequality byalternating thesignsthroughout, isalso valid?Clearlywemay.Forif,totakethegeneral case,wehavefor everysufficiently smallz: 1+asZ3+a•.14+...+ 'J+'+r=tb•z'J+~Z'I+...=1c~zc,z..., thesamerelation holdsgoodwhenthe+signsthroughout arere­ placedbyalternate+and-signs.Ineithercase,infact,thecoeffi­ cients C2"areobtained, according to10:S,4,fromtheequations: ca+b.=aJ;c,+cgb~+b,=a,;clI+C,b'.l+c'Jb,+blJ=all;.••• C2"+Ch-2b'J+...+c'Jb~"-2+b2"=a2,,; ..Asappcars frolDthcdefinition. theyarecertainly allrational. and 116.(a)204Chapter VI.Theexpansions oftheso-called elementary functions. Wetherefore, aspresumed, -nowwritingzforz,-havethe formula 22: 22R 24R 21!k/llI:i.xcotX=1-~X2+-_4X4_+ ...+(_1)k__2_kx2k+..•2! 41 (21.)1 121.2618=1 -"3x-45x-945x-4725x-... Theexpansion fortanxisnowmostsimplyobtained bymeans oftheaddItion theorem cos'x-sin'x2cot2x=-----. -=cotz-tanx,cosX·SInX fromwhichwededuce tanx=cotz-2cot2x therefore 28 <Zl 22k(22k-1)RdtanX=I(_l)k-t ~kX2k-1 k=1 (2k)! =x+..!..ZS+.!x~+~x'+....3 15 315 Fromthetwoexpansions, withthehelpoftheformula x1cotx+tan2'=sinx weobtainfurther <Zl (22k_2)R(b) XI(-l)k-t 2kx2ksinx=k=U (2k)1 _ + 12+7•+31. 6+'s- 16"x360x15120-x... (Anexpansion forl/cosx willbefoundonp.239.) -Theseex­ pansions, atthepresent point,arestillunsatisfactory, astheirinterval ofvalidity cannotbeassigned; weonlyknowthattheserieshavea positive radiusofconvergence, not,however, whatitsvalueis. 5.Fromanother quitedifferent starting point,Eulerobtained an interesting expansion forthecotangent whichweproceed todeduce, especially asitisofgreatimportance formanyproblems inseries ~14. Atthesametime,itwillgiveustheradiusofconvergence ofthe seriesIUiand116(v.241). ltJThisandthefollowing expansions arealmostallduetoEule1'andare foundinthe91handlOthchapters ofhisInt1'oducttO Inanalysm m!mlt01'Um, Lausanne 1748. UWeshallafterwards seethatB~khasthesign(_I)k-l (v.136),so thattheexpansion ofxcotx,aftertheinitialterm1,hasonlynegative coeffi- cients,thoseoftanxand-.~onlypositive coefficients. SlDx ..Thefollowing considerable simplification ofEuleysmethod forobtaining theexpansion i'lduetoSch1'lJte1' (Ableitung derPartialbruch- undProdukt­ entwicklungen tUrdietrigonometrischen Funktionen. Zeil~chrift fUrMath.u. Phys.•Vol.13,p.254.1868). or (*)S:1:'1.Ittetrigonometrical lunctlons. \Vehave,aswasjustshewn, coty=;(cot-~--tan~) 1{ 7<X 7«X±l)}cotnx=-cot---'-cot-- -2 2 12'205 aformula inwhichwemay,ontheright,takeeitherofthesigns±. Letxbeanarbitrary realnumber distinct from0,+1,±2,...• whosevaluewillremamfixedinwhatfollows. Then :n:xcotnx =~a:{cot~ +cot7<(x-J-:22}222 anclapplying theformula(*)oncemoretobothfunctions ontheright handside,takingforthefirstthe+andforthesecond, the-sign, weobtain nxcot:n:x=~{cot~~-+[cotJrJ:~±--_n+cot7<(x-l)J+cot7<(:.±2)I4 4 4 4 41" Athirdsimilarstepgives,fornxcot:n:x,thevalue 1+cot7<(x+1)+cot!:J~+2)+cot1t(x+3)1 8881tX 7<X 7«x+4) -8cot-8- e +cot-8--'I'+cot!'JX-1)+cot7<(x-2)+cot~Jx:-3) 888 sincehereeachpairoftermswhichoccupy symmetrical positions relatively tothecentre(e)oftheaggregate inthecurlybrackets give, exceptforafactor~, atermofthepreceding aggregate, inaccordance Withtheformula(*).Ifweproceed thusthrough nstages,weobtain forn>1 Jlx{ JlX2,,-1_1[7«x+,,) 7«x-1')] 7<X}(t):n:xcot:n:x=- cot-+;E cot--+cot--- -tan-2" 2" v=1 2" 2" 2" Nowby115. limzcotz=1 z~o andhenceforeachet=l=0 I, 1 et:1 1111-2-cot-2- et:'n-+oo n n ifintheaboveexpression weletn-+00and,atfirsttentatively, carry outthelimiting process foreachtermseparately, weobtaintheex­ pansion nxcot:n:x=1+x1;(_1_+_1_)-0=1+2x!li~. ,'~1X+Vx-., ,'=1X-» Weproceed toshowthatthisingeneral faultymodeofpassage tothe limithas.however, ledinthiscasetoarightresult. Wefirstnotethattheseriesconverges absolutely forevery :z::t=±1,±2,...•by70,4,sincetheabsolute valuesofitsterms 206Chapter VI.Theexpansions oftheso-called elementary functions. areasymptotically equaltothoseoftheseries.2 1,.Nowchooseanv arbitrary integer k>6IxI,tobekeptprovisionally fixed.Ifnisso largethatthenumber 2"-1-1,whichwewilldenoteforshortbym, is>k,wethensplituptheexpression (t)fornxcotx, asfollows~ll: JrX{JrX :n:xk[]}"'x{m]} nxcotnX-.cot.-tan-~+'"...+--.2[...- 2n2n2n;:;;1 2n"=k+l . (Inthesquarebrackets wehaveofcoursetoinsertthesameexpression asoccursin(t).)Thetwopartsofthisexpression wedenote by AnandBn.SinceAnconsists ofafinitenumber ofterms,thepassage tothelimittermbytermiscertainly allowed there,by41,9,and wehave • .2~1hmA=1+2x..1-,,--,.n X;';'-1'"-)-00 ,,=1 AlsoBnisprecisely 1lxcot1lx-An'hencelimB..certainly exists. Let'kdenoteitsvalue,depending asitdocsuponthcchosen valuek; thus limBn='k=1lXcot1lX -fl+2x2;;;--_I_.J. fI-JI.ao - '1,=1x'J-V" Bounds aboveforthenumbers Bn,fortheirlimitTkandsofinallyfor thedifference ontherighthandside,maynowquiteeasilybeesti­ mated: Wehave - 2COlacot(a+b)+cot(a-b)=-.-.---smJb------1sin'a andhence cot:n:(:l:_+V)+cot~(J.:-v)_ =-2cota 2n2nsin'fJ'---1sm"a nx nvwriting forthemoment2"=aand2"=fJ,forshort. As2n>k>61xI,wecertainly haveIaI=I~-~I<1andso26 IsinaI=Ia-;;+.../<IaI(1+~i+iI+...)<2IaI. Since,further, 0<fJ<.;-<2,wehave 27 .((/e) P6(P) fJsmfJ=fJ1 -2~-B+"5i1 -6=-7+...>":f 16Cf.Footnote 7,p.194. IIIForthesakeoflaterapplications wemaketheseestimates intheaboVf roughform. 27Cf.p.200. §24.Thetrigonometrical functions. 207 Hence thelatter, andhenceIsin{JI{J v SInet>~=6TXT>1, because">k>61xI.Ittherefore followsthat(for">k) 2\cotJtXI 'Icot~~x+v)+cot~(x=-~11~ 2"2" 2"-v· 36x"-1 Thefactoroutside -certainly<3; Izcotz1= Accordingly,thesignofsummation is-quiteroughly estimated for\~-~Iwas<1,andforIzI<1wchave28 Z2z4 I 11-21-+-41-+", 1+2'+-4--+'". !--Z2~--T---I-i--<3. 1-3!+"51-+... 1-31--51-... m1 '"1IB..I<216x'.l·2J-~-'){'•<216x'.l•~'"~.,'=k+l" -.,uX "=A+lv-."x ButthisISanumber quiteindependent ofn,sothatwemayalso write IEmB..I=ITkI<216x'.l•i:2_1 36----:.!· ~=k+lvx Buttheboundabovewhichwehavethusobtained forTkiscqu,ll totheremainder, afterthekthterm,ofaconvergent series 28.Hence Tk-)-0ask-++CIJ.Ifwereferbacktothemeaning ofTk,wcseethat thisimplies or,asasserted, '"17txcot7tx~1+2x2E., "• "_IX'-v· - aformulawhichisthusprovedvalidforevery.\'eFn,±1,±2,... 6.Weshallinthenextchapterbutonemakeimportant applications (p.236seqq.)ofthismostremarkable expansion inpartialfractions, asitis called,ofthefunction cot.Wecanofcourseeasilydeducemanyfurther suchexpressions fromit;wemakenotcofthefollowing: 18Theconvergence isobtained justassimplyas,previously, thatoftheseries 1.EIi'x-v117. 208ChapterVI.Theexpansions oftheso·called elementary functions. Theformula x=F±1,±3,±5•.•.first~x nxncot2-27lcot:Tlx=:Tltan2 ofallgives 29 tnx{',4xnan--=,.;;..2.-=0(2v+t)2_x2' a:>(11) =2~~-(2v+l)-x -(2-v+l)+x • Theformula z1cotZ+tan-2-=-,­sm, liS.thengivesfurther, forx=F0,±1,±2,.•• n 1+2x 2;1'+ sio1r:x=;;-I:! 2-.)~ ~-•••-x",-x =~+(~ 1 )_(1___1_)+_.... xI-xl+x 2-x2+x Finallyifweherereplacexby~-x,wededuce ~2(22)(22cosJZx=1-2x+ 1+2x-3-2x -3+2--;;-5--=2;)+--···· ByS3,2,Supplementary theorem, thebrackets mayhelebe omitted. Butifwethentakethetermstogether againinpairs,startmg fhb" b . 'dd--I- 1 3 5romt eegmnmg, we0tam,-pravlex,.±2'±2'±2'•••• 1t 1 3 5~ =(i)2-x2(~)2-x2+U)J_x2-+ Withtheseexpansions inpartialfractions forthefunctions cot,tan, 1 d 1 '11' d" fh ' sinanC<iS'weWItermmateourlSCUSSlOn 0t etrigonometrical functions. §25.Thebinomial series. Wehavealready, in§22,seenthatthebinomial theorem for positive integral exponents, ifwrittenintheform (1+X)k=l'C)x", "=0 remains unaltered inthecase30ofanegative integralk.Butwehave thentostipulateIxI<1.Wewillnowshowthatwiththisrestriction 19Theformulafirstfollowsonlyforx=!=0,±1,±2,...butcanthen beverifiedwithoutanydifficulty forX=0,±2,±4I••••(Theserieshas thesum0,asismosteasilyseenfromthesecondexpression, foraneven integral x.) 10Intheformercasetheseriesisinfiniteonlyinform,inthelatterit isactuallyso. §25.Thebinomial senes. 209 119. andshewnumber.(1+X)a=1:(Cl)Xn n=Ontheorem holds 31evenforanyrealexponent cc,i.e. {lx1<1 Clanyreal fromtheseriesthe NowAsinthepreceding cases,wewillstart thatitrepresents thefunction inquestion Theconvergence oftheseriesforIxI<1maybeatonceestablished; fortheabsolute valueoftheratioofthe(n+1)thtothenthtermis =I::=~xIandtherefore-IxI, whichby76,2provesthattheexactradiusofconvergence ofthe binomial seriesis1.Itisnotquitesoeasytoseethatitssumis equaltothe-ofcoursepositive -valueof(1+x)a.Ifwedenote provisionally byfa(x)thefunction represented bytheseriesforixI<1, theproofmaybecarriedoutasfollows. Since.2(:)x"converges absolutely forIxI<1,whatever maybe thevalueofCl,itfollows, by91,Rem.1,thatforanyCland{J,and everyIx1<1,wehave J;(:)x".1;(~)x"=~";[(~)(~)+(~)(n~1)+...+(:)(~)]xn.,,=0 ,,=0 ,,=0 (~)(~)+(~)(n~l)+...+(:)(~)=(at~) asmayquiteeasilybeverified e.g.byinduction 32.Hence-for 11Thesymbol(:)isdefined foranarbitrary realexandintegraln;;;;0 bythetwoconventions (ex)=1, (~)\=a~<x-_I)~,:--=-~n+ 1)forn;;;;1o 111·2·... n andforeveryrealccandeveryn2::1,itsatisfies therelation. whichmayat oncebeverified bycalculation. (a-1)+(a-1)=(a).n-I n n IIIForthis,givethestatement, bymultiplymg byni,theform (~).~i)-:'-:(jJ-=-n:t 1)+... +(~)ex(ex-I)...(a-h+I)·fJ(fJ-I) ...(fJ-n+h+1)+..• +(:)"(ex--I~a--n+-i) =(ex+{J)(ex+fJ-I)...(cc+fJ-n+1). Thenmultiply eachofthe(n+1)termsonthelefthandsidefirstbythecorres­ ponding termof a,(a-I),"',(a-h),...,(a-n), thenbythecorre.ponding termof (fJ-n),(fJ-n+I), ''',(fJ-nth),....fJ andadd,sothatinallwemultiply by(a+fJ-»);grouping together thesimilar termsontheleft,weobtainprecisely theasserted equality, wherenisreplaced byn+1.-Theaboveformula isusuallycalledtheaddition theorem lortIlt binomial coelt,c,ent •. 210Chapter VI.Theexpansions oftheso-called elementary functions. fixedIxI<1,-wehave,foranyaandp, fa·(p=fa+p, Byprecisely thesamemethod asweusedtodeduce fromthead~t~ theorem oftheexponential function, -E(Xl)'E(X"J)=E(Xl+X"J)' ~, thatforeveryrealXwehad(E(x)=(E(I)/"-sowecouldhe conclude thatforeverya, fa=(f1)a, ifweknewherealsothatfawasforeveryreala(withfixedx)a continuous function ofa.Asf1=1+x,theequality fa=(1+x)a wouldthenbeestablished generally forthestatedvaluesofx. Theproofofthecontinuity results quitesimplyfromthemain rearrangement theorem90:Ifwewritetheseriesforfainthemore explicit form (a) fa=1+ax+(~~ __;)X2+(~3 _~I+_~_)X3+... andthenreplaceeachtermbyitsabsolute value,weobtainthesenes n~~PI Cl~~1)..~~ClClI+nn-1)IxIn=n~(IClI~n-1)IxIn, alsoconvergent forIxI<1bytheratiotest.Wemayaccordingly rearrange theaboveseries(a)inpowers ofet,obtaining (XIx3 (-1)"-1 n )(b)fa=1+x-2+3 - +...+-n-x+...a+·.·, i.e.certainly apowerseriesina.Sincethis-stillforfixedxin IxI<1 -converges, bythemanner inwhichitwasobtained, for everya,wehaveaneverywhere convergent powerseriesina,hence certainly acontinuous function ofIX. Th1scompletes theproof 33ofthevalidityoftheexpansion 119and atthesametimefillsthegapleftintheproofofthereversion theorem §21. 83Analternative proof,perhaps stilleasierthantheabove,butusingthe differential calculus, isasfollows: Fromfa(X)=.i;(:)x"itfollowsthat n=O f~(x) _'0in(:)X,,-l~.§(n+1)(n~1)x". Sincehowever (n+1)(:~1)=Cl(Cl:i)~itfollows furtherthat f~(x)=Cl·fa_l(x). But §26.Thelogarithmic series. 211 Thuse.g. v'2=21=Thebinomial seriesprovides, liketheexponential series,anexpansion ofthegeneral powerae:Choose a(positive) number cforwhich,onthe aonehand,cemayberegarded asknown, andontheother,0<c<2. Thenwemaywrite:=1+xwithIxI<1andsoobtain,astherequired expansion, ae=~.(1+x)?=c?[1+(0x~(~)x2+...J. (4~)1.(50)'=7.(45!)-i=~.(1_1)~' 2549 550 5 50 7 [(-!)1(-~) 1(--~) 1 ]=fi1-150+250i-3503+-... "-c~ _ isacOl1venient expansion ofv'2. Thediscovery ofthebinomial seriesbyNewton 34formsoneofthe landmarks inthedevelopment ofmathematical science. LaterAbel35 madethisseriesthesubjectofresearches whichrepresent aperhaps equally important landmark inthedevelopment ofthetheoryofseries(cf.below 170,1and247). §26.Thelogarithmic series. Asalready observed onpp.58and83,intheoretical investigations itisconvenient toemploy exclusively theso-called natural logarithms, thatistosay,thosewiththebasee.Inthesequel,logxshalltherefure alwaysstandforlog.x(x>0). Ify=logx,thenx=eYor y3_ylx-I=y+21+3!+... Bythetheorem forthereversion ofpowerseries(107),y=logxisthere- thus,foreveryIxI<1,wehavetheequation (1+x)frz.'(x)-a.f,,-(x)=o. Since(1+x)">0,thisshowsthatthequotient fa.(x) (1+x)" haseverywhere thedifferential coefficient 0,i.e.isidentically equaltooneandthe sameconstant. Forx=°thevalueisatoncecalculated and=+1;thusthe assertion fa.(x)=(1+x)'"isprovedafresh. 3.LettertoOldenburg, 13June1676.-Newtoll atthattimepossessed no proofoftheformula; thefirstproofwasfoundin1774byElIler. 3DJ.f.d.reineu.angew.Math.,Vo!.1,p.:Ul,1826. 212ChapterVI.Theexpansions oftheso-called elementary functions. foreexpansible inpowersof(x--I)forallvaluesofxsufficiently near to+1,oryoc=log(1+x)inpowersofx,foreverysufficiently smallIxI: y=clog(1+x)=x+b2x2+b3x3+.... Thecoefficients bnmayactually beevaluated bytheprocessindicated, provided theworking isskilfully setout36.Butitisadvisable toseekmore convenient methods: Forthispurpose, thedevelopments ofthepreceding sectionsuffice. ForIxI<1andarbitrary ex,thefunction fox=fox(X) thereexamined is =(1+x)"=e"log(!+x). Using,forthelefthandside,theexpression (b)oftheformerparagraph andfortherighthandside,theexponential series,weobtainthetwo powerserieseverywhere convergent: [X'x3 (_I)n-1n]1+x-2+-3--+...+--n-X+... (X+... =1+[log(1+x)](X+ .... Bytheidentity theorem forpowerseries97,thecoefficients ofcorre­ sponding powersofexmustherecoincide. Thus,inparticular 37,-and foreveryIxI<I - x'x· (--1)"-1120.(a)log(1+x)=x-"2+"3-+...-i-n-xn+... Thuswehaveobtained thedesiredexpansion, which,wealsoseeapos­ teriori,cannotholdforIxI>I.Ifwereplaceinthislogarithmic series, asitiscalled,xby-xandchangethesignsonbothsidesoftheequality, weobtain,-equallyforeveryIxI<1,- 1 X·x3 x" (b) log~=x+"2+"3+...+-;;+... Byaddition wededuce,-againforeveryIxI<1,- 1+x[x'x' X·k+1](c) log1_x~2x+"3+5'+...+2k+1+.... Thereareofcoursevariousotherwaysofobtaining theseexpansions j buttheyeitherdonotfollowsoimmediately fromthedefinition ofthe logasinversefunction oftheexponential function, ormakemoreextensive useofthedifferential andintegralcalculus 38. 38Berm.Schmidt,Jahresber. d.Deutsch. Math.Ver.,Vol.48,p.56.1938. 87Cf.thehistorical remarksin69,8. a8Wemayindicatethefollowmg twoways: 1.Weknowfromthereversion theoremthatwemaywnte log(1+x)=x+bzx·+b3x3+...; itfollowsfromTayloT'sseries99that b=1_(~Og (I_t~) =(_l)k-l kkldxkx=Qk' §27.Thecyclometncal functions. 213 Ourmudeufobtaining thelogarithmic senes alsuthetwo modesmentioned inthefootnote -donotenableustodetermine whether therepresentation remains validforx=+1orx= -1. Sincehowever120areduces, forx=+1,totheconvergent series (v.81c,3) thevalueofthisseries,byAbeZ'stheorem oflimits,is =Hmlog(1+x)=log2 . ..,-.-1-0 Ourrepresentation (a)thereremains validforx=+1;butforx-=-1 itcertainly nolongerholds,astheseriesisthendivergent. §27.Thecyc10metrical functions. Sincethetrigonometrical functions sinandtanareexpansible in powerseriesinwhichthefirstpowerofthevariable hasthecoeffi­ cient1,dIfferent from0,thisisalsotrueoftheirinverses, theso­ calledcyclometrical functions sin-1and!an-1•Wehavetherefore to write,foreverysufficiently smallIxI' y=sin-1x=x+b3x3+b~Xli+ . y=tan'-lx=x+b3'x3+ba'x~+ . wherewehaveleftouttheevenpowersatonce,sineeourfunctions are odd.HeretooitwouldbetedIOUS toseektoevaluate thecoeffiCients bandb'bythegeneral process ot107.Wcagainchoosemorecon­ venient methods: Theseriesfortan-1xistheinverse of ySy6y--+--+ ..•siny 315!(a) x=tany=--="4 , cosy 1_2'_+!'._+... 2!4' oroftheseriesobtained by103,4aftercarrying outtheprocess of division inthelastquotient. Ifhereallthesigns,innumerator and denominator, were+,thenweshouldbeconcerned withreversing thefunction BII-B- IIB~II-lx=.- =~-----.BII+e- 1IB~!J+l dlog(l+x) 1 ..2. =--=l-x+x"-+ ...+(-l)kxk+ •..=)'(_l)kxk. dx l+x "-;;;0 Integrating, itfollows atonce,byDU,theorem ;',-sincelog1=0,-that ..(l)k ..(1)k-l10g(l+x)=.2 -=---Xk+1=.s-xk•k=ok+l k=1 k Themethod inthetextissofarsimpler thatitproceeds entirely without theuseofthedifferential andintegral calculus. 8 (G51) 121. 122.214ChapterVI.Theexpansions oftheso-called elementary functions. Buttheinverseofthisfunction is,asweimmediately find, 111+x x8x·2ogI-x=x-+:I-+5-+.... Bythegeneralremarkattheendof§21,thereverseseriesoftheseries forx=tanyactually beforeusisobtained fromtheserieslastwritten downbyalternating thesigns 39again,i.e. tan-1x=-=x-;3-+:f-+ Iftherefore thispowerseries,whichobviously hastheradiusofconver­ gence1,issubstituted foryinthequotient ontherightof(a),andthis isthenrearranged, asiscertainly allowed, -weobtaintheterminating powerseriesx.Henceitssumforanarbitrary givenIxI<1isasolution oftany=x,andisprecisely theso-called principal valueofthefunction tan-1xherebydefined, i.e.thevaluewhichis=0forx=0andthen variescontinuously withx.Hencefor-1<x<-+1,itsatisfies the condition andisdefined, intheinteriorofthisinterval, withoutanyambiguity. ForIxI>1theexpansion obtained iscertainly nolongervalid; butAbel'stheorem oflimitsshewsthatitdoesstillholdforx-co±1. Fortheseriesremains convergent atbothendpointsoftheintervalof convergence andtan-1xiscontinuous atboththesepoints. Wehave therefore inparticular theseries,peculiarly remarkable forclearness and simplicity: givingatthesametimeafirstmeansofdetermining TTofsomepractical value.Thisbeautiful equation isusuallynamedafterLeibniz 40;itmay besaidtoreducethetreatment ofthenumber 7Ttopurearithmetic. It isasif,bythisexpansion, theveilwhichhungoverthatstrangenumber hadbeendrawnaside. 8.Adifferentmethodisthefollowing: Wehave dtan-1x 1 1 I--=--=----=---=I -.,,'+x4-1-•••,dxdtany1+tan"y1+x"dy--- thelatterforIxI<I.Astan-10=0,itfollowsby99,theorem 5,thatforIxI<I, xlx'tan-1x=x-3"+'5-+." . Amethodcorresponding tothatgivenfirstinthepreceding footnote issome­ whatmoretroublesome here,asthedifferential coeffiCients ofhigherorderoftan-1x -evenatthesinglepoint0-arenoteasytofinddirectly.-Theexpansion of tan-1xwasfoundin1671byJ.Gregory,butdidnotbecomeknowntill1712. 40Heprobably discovered itin1673fromgeometncal considerations and withoutreference totheinversetan-series. Exercises onChapLer VI. 215 Forthededuction ofaseriesforsin-1x,themethod whichwe havejustusedfortan-1xisnotavailable. Theprocess indicated inthelastfootnote, however, provides thedesired series:Wehave forIxl<1 dsin-':I: 1 1 1 o-! ~x- =(d~i;Y) =cosy=~l-x.=(1-x-), thepositive signbeinggiventotheradicalsincethederivedfunction ofsin-1xisconstantly positive intheinterval -1+1.From (sin-1x)'=1 _(~~)x2+(~~)x4-+ . itatoncefollows, however, by99,theorem 5,assin-10=0,that forIx1<1 .-1+1x31.3xi>+1.3.5x7+smx=x '2""3+2.4'"'5 2.4.0'"7 .••• 123. TInspowerseriesalsohasradms1,andonquitesimilarg-rounds to theaboveweconclude thatforIxI<1itssumistheprincipal value ofsin-1x,i.e.thatuniquely determmed solution yoftheequation siny=xwhichliesbetween -iand+i. Forx=+1,theequality isnotyetsecured. ByAbel'stheorem oflimit"itwillholdthereif,andonlyif,theseriesconverges there. Aswehaveamerechangeofsigninpassingfrom+xto-x,this onlyneedstestingforthepoint+1.Therewehaveaseriesofpo· sitivetelll1Sanditsuffices toshowthatitspartialsumsarebounded. Nowfor0<x<1,ifwedenotebysn(x)thepartialsumsof123, Sll(x)<sin-1x<sin-11=i. Andasthisholds(withfixedn)foreverypositivex<1,we alsohave Emsn(x)=sn(1):::;:i; x~l andasthisholdsforeveryn,wehaveprovedwhatwerequired. Thus 1t 1 1 1·311·3.5 1'2=1+'2' '3+2...·~+2....6·7"+.... 124. §§22to27havethusputusinpossession ofallthepowersenes whicharemostimportant forapplications. S4~=0 , i.e.Exercises onChapter VI. ofthefollowing functions 74.ShowthattheexpanSIOns inpowerseries buvetheformindicated ineachcase: 00 a)ersin:l:=l;~~xnwiths"=V2n.sinll-43r I,,=0'I. SU+l=(_I)k2'k,SU+9=(-1)k29k/1, 1 00 XU b)"2(tan-'.l;yJ.=~;(-l)k-1/;~.'iliwith 1:=1 216Chapter VI.Theexpansions 01theso-called elementary functions. 1 1+X <Xl X4k+9• 1 1 1c)-tan-1x·log-- =2;Ck'--- withck=I---+--+···+---;4 I-x k=O4k+2 3 5 4k+l 1 <Xl X9k+1 d)"2tan-1x·log(1+xH)=L;(-1)k-1nu·~l' k=l + . 1 1 1withnil=+2+...+n; 1 [ 119 <Xl11e)]"log1_x=2)nn--.!x",withthesamemeaning ofnilasind). ,.=2 75.Showthattheexpansions inpowerseriesofthefollowmg functions beginwiththetermsindicated: x xx2x9 a)---1=1-2--I2-24-...; log---I-x Xl :z:mx+-+,,·+- xm+1 b)(1-x)e2 m=1---+...,(m~1);m+l c)tan(sinx)-sin(tanx)=;0x7+72;6XO+...; 1 1(1)Z_ 1x112732447 4959, 5 • d)e+x--2+24 x-16x+5760x-2304x+'''' ~ 4128 e)-x--""""'l::-o-g-(·f-+-x) =2+'3x-9"x+1-3-5x3+•••. 76.Deduce, withreference to105,5,115and116,theexpansions in powerseriesofthefollowing functions a)logcosx; )Itanxcog----;-i x" e)l-cosx; xg)log--l-; 2sin2"x e'" i)er+1jsinxb)log---ix d)~'sinx' f)1 •cosx' 1k).•cosx-slnx 77.Showthat,fora=l=0,-2,-4,••• _l_.[~+~_x_+ 1·3.~+...J=\"l-x a2a+22·4a+4 =-!.r1+a+1x+(a+IH~+.?~x2+...]. aa+2 (a+2)(a+4) (2n+1)"78.Wehave2n..-::}-.e.Isthesequence monotone? Increasing or decreasing? What,inthisrespect, isthebehaviour ofthesequences (1~),,+a+n'O<a<H 79.Fromx"-+~itinvariably follows that (1+~"r-+e; Exercises onChapter VI. 217 andalso,ifx"and~arepositive, that n(Vx"-1)-+log;. 9 SO.If(x,,)isanarbitrary realsequence, forwhich~~-..0,andwewrite 'IJ (x")". 1-n=y",then,Ineverycase, Yn~e-:r •. SI.Provetheinequalities of114. S2.Express thesumsofthefollowing seriesbyclosedexpressions in ~ermsoftheelementary functions: (!lint:If((x)betherequired function, thenobviously whence(x)maybedetermined. Similarly inthefollowing examples.) x3x~x·xD b)---+----+- ....1·33·55·77·9 • S3.Obtainthesumsofthefollowing seriesasparticular yaluesofele­ mentary functions: 1 2 3 4, a)21+~1+41+5'+...=1; 1 1 1·31,35 b)2+2-:4+24,6+24.6,8+···=1; c)-21+-214'3_r;+-21~"-7_+-J~'3~5_:_7~.1l __+...=}{2;4·/)·8ID246,8.J0·12·14 2 d){-2\-~d+2.1~.36 58.710-2~.:~-:--~6:i~\4+-···= -V~~~~!. S4.Deduce fromtheexpansion inpartialfractions 117scq. thefollowing expressions forn: n=0:.tan~.[1__1_+_11_+_1_- +...J aa-Ia+-l2a-I 20:+1 • 'If=IX'sin:!:'[I+__1__1_-_1_ + -1+ - - + + ] IX IX-I 1X+1 21X-I 21Xt-1 ..., where IX*0,±I,±},±L.•.Substitute inparticular IX=:3,4,6. 218 ChapterVII.Infiniteproducts. Chapter VII. Infinite products. §28.Products withpositiveterms. Aninfiniteproduct UI•U2'U3••••• Un•••• is,by§11,II,tobetakenmerelyasrepresenting anewsymbolforthe sequence ofthepartialproducts Accordingly suchaninfiniteproduct shouldbecalledconvergent, with valueU, '" IIu"=U, n----"l ifthesequence ofthepartialproducts tendstothenumberUaslimit. Butthisisparticularly inconvenient, owingtothefactthatthenevery productwouldhavetobecalledconvergent forwhichasinglefactorwas =O.ForifUmwere0,thenthesequence ofpartialproducts alsowould tendtoU=0,sinceitstermswouldallbeequal'to 0forn~m.Simi­ larlyeveryproduct wouldbeconvergent -againwiththevalue0-for whichfromsomemonwards IunI<{)<1. Inordertoexclude thesetrivialcases,wedonotdescribe thebehaviour ofaninfiniteproduct bythatofthesequence ofitspartialproducts, butadoptthefollowing moresuitable definition, whichtakesinto account thepeculiar partplayedbythenumber 0inmUltiplication: 125. 0Definition. Theinfiniteproduct 00nun==ul.u2'Ua.•.• f1=1 willbecalledconvergent (inthestrictersense)iffromsomepoint onwards -sayforeveryn>m--nofactorvanishes, andifthe partialproducts, beginning immediately beyondthispoint Pn=~tm+l·um+2'" "un' (n>m) tend,asnincreases, toalimit,finiteanddifferent from0 Ifthisbe=U,,,,thenthenumber U=U1'U2··,"um'Um' obviously independent 01m,isregarded asthevalueOftheproduct!, 1Infinite products arefirstfoundinF.Vlcta(Opera, Leyden 1646,p.400) whogivestheproduct .!=1 /1 .1 /~+_1.-0::.1 /~+-.!_V-1+_~-0.~"V2V2 2V1fV2 2 2 2 V-2I §28.Products withpositive terms. 219 Wethenhavefirst,asforfiniteproducts, the oTheorem 1.Aconvergent infinite product hasthevalue0if. andonlyit.oneatitstactorsis=O. AsfurtherPn-l-+UrnwithPn-+Urn'andasUmis=f=O.we have(by41,11) andwehavethe oTheorem 2.Thesequence ofthefactorsinaconvergent infinite product alwaystends-+1. Onthisaccount, itwillbemoreconvenient todenotethefactors byun=1+an'sothattheproducts considered havetheform Ji(1+an)'"=1 Forthese,thecondition an-+0isthenanecessary condition torcon­ vergena. Thenumbers an-asthemostessentlal partsofthefactors­ willbecalledthelet'utHoftheproduct. Iftheyareall>0,thcn asinthccaseofinfinite scries.wespeakofproducts wlthpositive terms.WeWillfirstconcern ourselves withthese. Thequestion ofconvergence isentirely answered herebythe Theorem 3.AproductII(1+a..)withposltive termsanis convergent tt.andonlyit.theseries2)anconverges. Proof. Thepartial products P..=(1+at)'"(1+an)'since an:20,increase monotonely; hencetheFlrstmaincriterion (46)is avaliable andweonlyhavetoshowthatthepartialproducts Pnare bounded if,andonlyif,thepartialsumssn=a1+a!-1-'"-f--anare bounded. Nowby114It,1+a..<ea,.andsoforeachn ontheotherhand Pn=(1+a1),··(1+a..)=1+at+a~+...+an+ata~+...>sn' thelattcrbccause intheproduct, afterexpansion, wehave,besidesthe termsofsn'manyothers, butallnon-negative ones,occurring, Thusforeachn (cfEx.80) andinJ.Walhs(OperaT.Oxford1605,p.468) whoin1656gives theproduct :r224466 '2'=T'3'3'5'--;rf''7". Butinfinite products firstsecured afooting inmathematics through Eulel',who established anumber ofimportant expansions ininfinite product form.Thefirst criteria ofconvergence nreduetoCauchy. 220 Cbapter VII.Infinite products. Theformerinequality showsthatP..remains bounded whensndoes, thelatter,conversely, thats..remains bounded whenP..does,-which provesthestatement. 2 Examples. 1.Aswearealready acquainted withanumber ofexamples ofcon­ vergent seriesEa"withpositive terms,wemayobtain,bytheorem 3,asmany examples ofconvergent productsII(1+an).Wemaymention: 11(1+:0)isconvergent foret>I,divergent for C(~1.-Thclatter ismoreeasilyrecognised herethaninthecorresponding sene,',for (1+~)(1+~)...(I+~)=~.~.i....~+1=01+1~+CX)1 2 n12301 . 2.IT(1+x")isconvergent for0<x<1;similarly II(1+XO"). 3II""(I2)_IIex>(01-1)(01+2)_1 . - 01(01+1)=-01(01+1)--3". n=2 n=~ Withtheorem 3wemayatoncecouplethefollowing very similar Theorem 4.It,toreveryn,an20,thentheproductIl(1-an) alsoisconvergent it,andonlyit,.2anconverges. Proof. Ifandoesnottendto0,boththeseriesandthepro­ ductcertainly diverge. Butifa"-0,thenfromsomepointonwards, sayforeveryn>m,wehavean<~,or1-an>~.Weconsider thestriesandproduct fromthispointonwards only. Nowiftheproduct converges, thenthemonotone decreasmg sequence ofitspartialproducts Pn=(1-am+1).·.(1-an)tendsto apositive(>0)number Um•and,foreveryn>rn, (1-amu)···(l-an)2Um>o. Since,for0<a"<1,wealwayshave 1(1+a,,)~-I-a-; (asisatonceseenbymultiplying up),wecertainly have 1(1+am+1)(1+amH)...(1+an)<-er' m •Inthefirstpartoftheproofofthiselementary theorem, weusethe transcendental exponential function. V,ecanavoidthisasfollows:If,!Ja"=5 converges, choosemsothatforeveryn>m I am+1+am+.+...+a"<"2. As,obviously, forthesen's,wenowhave (1+a.+1)•••(1+an)<1+(am+1+ + an)+(am+1+...+an)2+...+(am+1++an)n<2, wecertainly have,foralln's, P"<2(1+a1)•••(1+am)=K, hence(P,,)isbounded. aInthiswehavetherefore, onaccount oftheorem 3,Itnewproofofthe divergence of.2~... 126.§'29.Products witharbitrary terms. Absolute convergence. 221 Accordingly theconvergence oftheproduct 11(1+an)'andhenceof theseries~an'resultsfromthatofII(1-an)'-If,conversely, Eanconverges, thensodoesE2amandconsequently byTheorem ~the productIl(1+2an)alsodoes.Hence,withasuitable choiceofK,the products (1+2am+!)...(1+2an)remain<K.Ifwenowusethefact that,for0~av~t, -asmayagainbeseenbymultiplying up-wcinfer 1(1-am+1)···(l- an)>](>0; andthepartialproducts onthelefthandside,astheyformamono­ tonedecreasing seqJence, therefore tendtoapositive limit:i.e.the product 11(1-an)isconvergent. RemnrlrsandExampies. 1.jj(1-;~)isconvergent forIX>I,divergent foret:;;;1. n-::,2 2.Ifan<1andIfEandiverges, then1I(I-an) i~notcomergent, with ourdefinition. Ashowever thepartmlproducts Pndecrease monotonely andremain >0,theyhavealimit,butonewhIch ISnecessarily =O.Wesaythattheproduct dIVerges toO.Theexceptional partplayedbythenumber 0thusinvolves usm someslightincongruity ofexpre~SlOn. Aproduct iscalleddivergent whosepartial products formadecidedly convergent sequence, namelyanullsequence, (Pn).The addltlon "inthestrictersense"totheword"convergent" mDef.125ISmtended toserveasareminder ofthisfact. Cl) 3.Thate.g.1I(1-~)diverges to0isagainveryeasilyseenfrom n=2 Pn=(1--~-)(1-~)...(1-:)=~.{_.~...~;:1=,~-+o. §29.Products witharbitrary terms.' Absolute convergence. Ifthetermsanofaproduct havearbitrary signs,thenthefollowing theorem -corresponding tothesecond principal criterionSIfor series-holds: oTheorem 5.Theinfiniteproductn(1+an)converges if,and •Atullandsystematic account ofthetheoryofconvergence ofinfinite products maybefoundinA.Pringsheim: LtberdieKonvergenz unendlicher Produkte, Math.Annnlen, Vot33,p.119-154, 1889. A. 222 Chapter VII.Infinite products. onlyit,given B>0,wecandetermine;; nosothatforeveryn>no andeveryk~1, [(1+an+l)(1+a"u)".(1+an+7c)-1]</3. Proof. a)Iftheproduct converges, thenfromsomepointon· wards,sayforeveryn>m,wehavean=l=-1,andthepartial products Pn=(1+aln+l)'" (1+an)' (n>m) tendtoalimit=l=O.Hencethereexists(v.41,3)apositivenumber f3 suchthat,foreveryn>m,IPn!~f3>O.Bythesecondprincipal criterion49wemaynow,given /3>0,determine nosothatfor everyn>noandeveryk~1, IPn+7c-Pn1</3.f3. Butthen,forthesamenandk,IP~~k-11=1(1+an+l)(1+anH)..•(1+an+7c)-11</3, whichisprecisely whatweasserted. b)Conversely, ifthe/3·condition ofthetheorem isfulfilled, first choose /3=~,anddetermine msothat,foreveryn>m, 1(1+am+1)·..(1+an)-11=IPn-11<~. Forthesen'swethenhave _1./IPI<;!2~"" n ~, showing that,foreveryn>m,wemusthave1+an=l=0;andfurther, thatitPntendstoalimitatall,thiscertainly cannotbeO.But wemaynow,given /3>0,choosethenumber nosothatforevery n>noandeveryk~1, /J'n33._ 11<..!...Pn 2 or AndthisshowsthatPnreallyhasauniquelimit.Thustheconver· genceoftheproduct isestablished. Asinthecaseofinfiniteseries,sosimilarly inthatofinfinite products, thosearethemostcasilydealtwithwhichconverge "abso· lutely". BythiswedonotmeanproductsIfunforwhich1I1unI alsoconverges, -suchadefinition wouldbevalueless, sincethen everyconvergent product wouldalsobeabsolutely convergent, -but wedefine,onthecontrary, asfollows: 127. 0Definition. TheproductII(1+an)issaidtobeabsolutely con- vergentiftheproduct1I(1+Ian\)converges. 6Or-v.SI,2ndform-ifinvariably [(1+an+1)(1+an+.)...(1+an+k,.)1.....1j or-v.SI,3rdform-ifinvariably [(1+aVn+1)...(1+aVn+kn)l-~1. §29.Products witharbitrary terms.Absolute convergence. 223 1;log(1+an) tI=m+ 1TheproductlI(l+an)converges it,andonlyitonlygainssignificance through thetheorem: Theconvergence ofII(l+IanI)involves thatofThisdefinition oTheorem 6. n(l+an)' Proof. Wehaveinvariably 1(1+an+1)(1+a"H)'" (1+anH)-11s:(1+Ian+11)(1+IanH\)...(1+IanHI)-1, asisatonceverified bymultiplying out.Iftherdore thenecessary andsufficient condition fortheconvergence ofTheorem 5issatisfied byII(l+IanI),itisipsofactosatisfted by11(1+an)'q.e.d. Inconsequence ofThcorem 3,wemaytherefore atoncestate oTheorem 7.AproductII(1+an)isabsolutely convergent it, andonlyif,;Eanconverges absolutely. Aswehaveanalrcady suffiClcntly developed theoryforthe determination oftheabsolute convergence ofaseries,Theorem 7solves theproblem ofconvergence inasatisfactory manner forabsolutely convergent products. Inallothercases,thefollowing theorem reduces theproblem dconvergence ofproducts completely tothecorrespond­ ingoncforseries: Theorem 8. theseries commencing withasuitable index6,converges. Andtheconvergence of theproduct isabsolute it,andonlyit,thatoftheseriesisso. Furthermore, ifListhesumoftheseries,then term,oftheseries sumL=logU""(1'1>m), by(42,2), ButlogPnis inquestion. AsUm=eL,ao [[(1+an)=(1+aJ...(1+a",).eL. n=1 Proof. a)Ifll(l+an)converges, thenan-0andhencefrom somepointonwards, sayforevery 1'1>m,wehaveIanI<1.Since, further, thepartialproducts Pn=(1+am+1)...(1+an)' tendtoalimitUm=1=0(hence positive), wehave logPn-logUt,,· thepartialsum,ending withthenth This,therefore, converges tothe wethushave II(l+an)=(1+a1)•••(1+a",)'eL• b)If,cunversely,theseriesisknown toconverge, andtohave thesumL,thenwehaveprecisely logp"_L,andconsequently (by42,1)P..=elOllJln_eL. Thiscompletes theproofofthefirstpartofthetheorem, sincee:L=1=o. 8Itsuffices tochoosemsothatforeveryn>mwehaveIa.1<1. 224 Chapter VIr.Infinite products. Todeduce, finally, thattheseriesandproduct are,inevery possible case,eitherbothorneitherabsolutely convergent, weusewith theorem 7and70,4,thefactthat(112,b),whena"~0 IIOg(12__a~21_1. a,. (Hereanytermsanwhich=0maybesimplyomitted fromcon­ sideration.) Although wehavethuscompletely reduced theproblem ofthe convergence ofinfinite products tothatofinfinite series,yetthe resultcannot entirely satisfyus,because ofthedifficulties usually involved inthepractical determination oftheconvergence ofaseries oftheformZlog(1+an)'Thewantherefeltmay,atleastpartially, besupplied bythefollowing 0Theorem 9.Theser£es(starting withasuitable in£t£alindex) 2,'log(1+an)andwith£ttheproductn(1-I-an)'iscertainly convergent, £f2,'anconverges andifZan'J£sabsolutely convergent 7. Proof. Wechoosemsothatforeveryn>m,wehaveIanI<~, andconsider II(1+an)and2'log(1+all)'starting withthe(m+l)th terms.Ifwewrite (n>m),n sn=2,;'a.., ..=m+lorlog(l+a,,)-a n_={) et:; ..' thenthenumbers {}nsodetermined certainly formabounded sequence, for8,asan~O,nn-->-l. Iftherefore XanandZjanl'J arecon­ vergent, 2'log(1+an)'andhencealsoII(1-I-a..),isconvergent. Thissimpletheorem leads ea~ilytothefollowing furthertheorem oTheorem 10.If2,'an'Jisabsolutely convergent, andIanIis<1 foreve1y n>m,thenthepart£alproducts n Pn=Jf(1+a..)andthepartialsums "=m+l aresorelatedthat i.e.therat£oofthetwosidesofthisrelation tendstoadefinitelimit, finiteand=F0,-whether ornoZanconverges. •2."a,,',ifconvergent atall,iscerlalnly absolutely convergent. Weadopt theabovewording sothatthetheorem mayremain trueforcomplex an's, forwhIchan·isnotnecessarily> 0(cf.§57). •For0<IxI<1wehaveinfact [Ix Xl ]log(1+x)=x+x·---+---+-...234 or AndthosetermswhicharepOSSibly =0maybeagainsimply neglected, as theyhavenoinfluence onthequestion underconsideration. 128.§29.Products witharbitrary terms.Absolute convergence. 225 Proof.Ifweadoptthenotation ofthepreceding proof,then,as log(1+an)=an+{}n·an~' wehaveforeveryn>m n (1--1-am+1)'"(1-1-an)=Ifeav+t~vav' =e~av.e~{}va.9, v=m+1 ifthesumsinthelasttwoexponents aretakenalsofrom,,=m-I-1 to,,~=n. Andas2,'{}nanll,the{}ntsbeingbounded, converges absolutely when2,'an~doesso,wecan,fromtheaboveequation, atonceinfer theresultstated.-Thistheorem alsoprovides thefollowing, often useful Supplementary theorem.It2'an1Iconverges absolutely, then Lan andII(1+an)converge anddivergetogether. Remarks andexamples. 1.Theconditions ofTheorem 9areonlysufficient; theproduct1l(1+an) mayconverge, without 2a"converging. Butinthatcase,byTheorem 10, 21a"19mustalsodiverge. '"( 1 '2.Ifweapplytheorem 10tothe(divergent) productIf1+-),thenit n=1 n followsthat ehn'"" th'I fh h.. ~1 1 1if"ndenotes thenparlIasum0t earmODlC serIes fin=+If+...+11. ehn Accordingly thelimitslim----;-=candlimr"n-logn]=logc=Cexist,the latterbecause cto,hence> O.Thenumber Cdefined bythesecondlimitis calledEuler'sorMascherom's constant. Itsnumerical valueisC=0·,)772156649..• (cf.Ex.86a,176,1and§64,n,4),Thelatterresultgivesusfurthervaluable InformatIOn astothedegreeofdivergence oftheharmonic senes,asItgIves Itn~logn. Further theestimates ofboundsabovemadefortheprootofTheorem 1:1show, 1evenmoreprecisely, ifwethereputav=-;,that eh..>ehn-l>norhn>"n-1>logn sothatEuler'sconstant cannotbenegative. "'((_I)n-l)3.If1+nisconvergent. Itsvaluer.lay,asithappens, be n=1 foundatoncebyforming thepartialproducts, andis=1. 4.jj(1+:)diverges forztO.However, theorem 10showsthat ..=1 '"(Z)x(1+~+...+!)JI1+-;""IJ 2 ..,or-whatisthesamethingby2,_""116, .'=1 i.e.(v.40,def.5)theratio n-z'H(1+~)=(z+l)(z+~)...(z+n) v=1 to n!nr 226 Chapter VII.Infinite products. has,forevery(fixed)x,whenn---+00,adetermmale (fmite)limitwhichisalso dlfferent from0ifxistaken'*-1,-2, 0.0(cf.below,219,4). '"(X9 ) 5.111--.-isabsolutely convergent forevery :1:. n=1 n "'(1)1 6II1--i=--. n=2 n2 §30.Connection between seriesandproducts. Conditional andunconditional convergence. Wehavemorethanonceobserved thataninfiniteseriesL:anis merelyanothersymbol forthesequence (sn)ofitspartialsums.Apart fromthefactthatwehavetotakeintoaccount theexceptional part playedbythevalue0inmultiplication, thecorresponding remark holds goodforinfinite products. Itfollows that,withthisreservation, every seriesmaybewritten asaproduct andeveryproduct asaseries. Asregards detail,thishastobedoneasfollows: '"129. 1.IfII(1+aJisgiven,thenthisproduct,ifwewrite n=1 n Jl(1+an)=Pn' ~=1 represents essentially thesequence (Pn)'Thissequence, ontheother hand,isrepresented bytheseries '"P1+(p~-P1)+(Ps-P2)+.. 0-P1+Z(1+a1)··.(1+an-I)all' ,,--::::~ Thisandthegivenproduct havethesamemeaning -iftheproduct converges inaccordance withourdefinition. Buttheseriesmayalso haveameaning without thisbeingthecasefortheproduct (eog.if thefactor(1+a6)is=0andallotherfactorsare=2). '"2.Ifconversely theseries2)anisgiven,thenitrepresents the n=1.. sequence forwhichsn=~'a~.Thisisalsowhatismeantbythe ,.'=1 prouuct 51.!-•.!-s...==51'if~n_==at'Il(1+ an), 5152 n=25n_1 n=2 al+a.+...+an_1 and and<Xl1l:-­ n-dn(n+l)or-provided ithasameaning atall.Andforthisobviously allthat werequireisthateachSn=f:O.Ingeneraltheconvergence oftheproduct impliestheconvergence oftheseries,andconversely. Inthecase,however, ofSn-+0,although wecalltheseriesconvergent withsum0,wesaythat theproductdivergestoO.'"1 Thuse.g.thesymbols l:2ii n=1 haveprecisely thesamemeaning. §30.ConnectIOn between seriesandproduclll. 227 Itis,however, onlyinrarecasesthatapassage suchasthis fromtheonesymbol totheotherWIllbeadvantageous foractual investigations. Theconnection between seriesandproducts which 1S theoretically conclusive was,moreover, established byTheorem 8alone, -orbyTheorem 7,ifweareconcerned withthemerequestion of absolute convergence. Inordertoshowthebearing ofthesetheorems ongeneral questions, wemayprove-asanalogue ofTheorem 88,1, and80,2,-thefollowing: oTheorem 11.Aninfiniteproductfl(l+a,,)isunconditionally 130 convergent -i.e.remains convergent, withvalueunaltered, however itsfactorsberearranged (v.27,3)-if,andonlyif,itconverges absolutelyD. Proof.Wesuppose givenaconvergent infiniteproductfl(l+a,,). Thetermsan'certainly finiteinnumber, forwhichIa"12~,were­ placebyO.Insodoing,weonlymakea"finitenumber ofalterations" andweensureIa"I<~foreveryn.Thenumber mintheproofof theorem 8maythenbetaken -~~O.\Vefirstprovethetheorem for thealteredproduct. Now,withthepresent valuesofa", fl(1+a,,)andZlog(1+a,,) areconvergent together, andtheirvaluesUandLstandintherelation U=eLtooneanother. Itfollows thatarearrangement ofthefactors oftheproduct leavesthisconvergent, withthesamevalueU,ifand onlyifthecorresponding rearrangement ofthetermsoftheseriesalso leavesthisconvergent, withthesamesum.Butthis,foraseries,is thecase1f,andonlyif,itconverges absolutely. Bytheorem 8thesame therefore holdsfortheproduct Nowif,beforetherearrangement, wehavemadeafinitenumber ofalterations, andthenaftertherearrangement makethemagainin theopposite sense,thiscanhavenoinfluence onthepresent question. Thetheorem istherefore trueforallproducts Additional remark. Usingthetheorem ofRiemann provedlater (187)wecanofcoursesay,moreprecisely: Iftheproduct isnot absolutely convergent andhasnofactor=0,thenwecanbysuitable rearrangement ofitsfactors, alwaysarrange thatthesequence ofits partialproducts hasprescribed lowerandupperlimits"andft'provided theyhavethesamesignasthevalueofthegivenproduct 10.Here "andftmayalsobe0or±00. 9Dm:.U.:Suiprodotti infimtl,AnnalidiMatem., (2)VoI.2,pp.28-38. 1868. 1UFornconvergent infinite product hascertainly onlyafinitenumber ofnegative factors; andtheirnumber isnotalteredbytherearrangement. 228 Chapter VII.Infiniteproducts. Exercises onChapter VII. 85.Provethatthefollowing products converge andhavethevaluesindi­ cated: a)Ifn3_~=~. b),,1=10(1+(2~)2")~2; n=2n3+13' 00( 2n+1 ) 4c)"II....1+(n2-:--=l)(,z-+1)2=3' 85a.By128,2thesequences 1 1x"~I+t+...+n--':1-lognandYlI=1+!+...+;;-logn havepositivetermsforn>I.Showthat(xnIy,,)isanestofintervals. Thevalue sodefined isEuler'sconstant. 86.Determine thebehavIour ofthefollowing products: 00((_1)11) 00((_l)n) a),~ 1+Vin-; b),~1+IOgn-; c)(1-~_)(I+_1_)(I-_!)(I--~)(I+~)(I-'~-9c)...<la <12 <15 v'7 v'4 v d)(I+1 )(11_)(1+ _1__)(1__1_)...,ex-I 2ex-l 3ex-l 4ex-1 1 1 forex=l=1,2'3'.•• 87.Showthat11cosx"converges if1:IxllI"converges. 88.Theproduct inEx.86dhas,forpositive integral valuesofex,the valueV2. [Hmt:Thepartialproduct withlastfactor(1-2kex1_1)is=,,_{!1(1--;;~)-1.] , TT 1T 1T 289.Prove,WIthreference toEx.87,thatcos4;'cos8'cos16... 71' (Werecognise Vieta'sproduct mentioned infootnote 1,p.218.) 90.Show,moregenerally, thatforeveryx x Xx x sinxcos2 .cos4 .cosEl•cos(6'• .= -x-. x x x x sinhxcosh2.cosh4.cosh8'coshla...x e'"+-x x-xinwhichlatterformula coshx=-2e--,sinhx=e_=;e-denotethehyperbolic coslOeandSlOeofx. 91.WIththehelpofEx.90,showthatthenumber defined bythenestof .I'E8' sin~h<>-'dficlhi" h'h Interva sInx.cIS=-~Ylowere orISeneast eacuteange.orWIC cos~=~.Similarly thenumber definedbyEx.8 dis=sin<>-h~Xl,if~isdefined YI or bycosh~=)". Xl way,showthatthenumbers defined inEx.8eand8f have92.Inasimilar thevalues: sinh2{}e)2{}XlwithExercises onChapter VIl. 229 sin2{} f)2{}"1withcos'{}=..:lYl 93.Wehave 1__a:.+x(x_~~l) at at.a.!_+...I-(_1)"X(x-a,)...Jx=an-=!2 al{1g•••an = (1-~~)(1-:J...(1-~J. WhatcanyOlldeduce fortheseriesandproduct ofwhichweherehavethe initialportions? 94.Withthehelpoftheorem 10of§29,showthat 1·35...12n-1)1 -- -~-'"--.2·4·G...2nVn 9:i.Similarly, showthat,for0<x<Y, ~xI-1)(x+?)(x+nt_..0 ,,(y+1)(y+2)(y+n). 96.Similarly, showthatifaandbarepositive, andA..andG..are respectively thearithmetic andgeometric means, ofthenquantities A..•G-;;-2a, (n=2,3,4,.•.)thena-1-b,a+2b,...,a+(n-l)b, 97.Whatcanbededuced, fromtheconvergence ofII(1+an)and 1I(1+bn),astothatof 1I(!+a..)(1+b,,) and (Cf.S3,3and4.) 9S.Given(U,.)monotone decreasing and->-1,is 1 Iul·--_·US·-·tl o••• 1t.,! u..I always convergent? (Ct.S2,theorem 5.) 99.Tocomplete §29,theorem 9,provethat1I(1+a..)certainly con­ vergesifthetwoseries ~:(a"-ta,,2)andIIan13 converge. -Howmaythisbegeneralized? -Ontheotherhand,show,by theexample oftheproduct whereweassumei<Cl::; ~,that1I(1+an)mayconverge evenwhen:!:a• •,nd:!:a..2bothdiverge. 230Chapter VIII.Closedandnumerical expressions forthesumsofseries. Chapter VIII. Closedandnumerical expressions forthesums ofseries. §31.Statement oftheproblem. InChapters IIIandIV,wewereconcerned mainly withour problem A,thequestion oftheconvergence ofseries,anditwasnot tillthelastfewchapters thatweconsidered alsothesumofthe series. Thislatterpointofviewweshallnowplaceinthefore· ground. Itisnecessary, however, inordertosupplement ourdeve· lopments ofpp.78-79and105,thatweshouldmakeitquiteclearonce morewhatisthesignificance ofthequestions whichariseInthis connection. If,forinstance, wehaveproved therelation122: :rr I I 14"=1 --3-+5"-'7+ -...J wcmayinterpret itintwoways.Ontheonehand,theequation indi­ catesthatthesumoftheseriesontherighthasthevalue ~,one quarter ofthevalueofanumber1whichwemeetwithinmanyother connections andtowhichapproximations arewell-known. Inthis sense,itmaybeclaimed thatwehavespecified thesumoftheseries written downabove. Butsuchastatement canonlyholdinavery relative sense;foritisnotpossible togiveacomplete specification ofthenumber:Il, otherwi'3e thanbyanestofl11tcrvals orsome equivalent symbol, andsuchasymbol isprecisely furnished bythe series;i.e.theexpression ontheright,intheaboveequation. We aretherefore equally justified inclaiming theexactopposite, namely thattheequation provides an(extremely simple) expression lorthe number :Ilinseriesform,-thatistosay,bymeansofaconver· gentsequence ofnumbers, -whichhappens indeedinourcaseto haveapeculiarly straightforward andconvenient formandmayalso (69,1)beimmediately expressed asanestofintervals 2, Thecircumstances areentirely altered whenwecometothe equation (cf.68,2b): 1 1 1r:2+2.3+3:4+...=1. 1Informer times,whenthesematters wereallinterpreted rathergeo· metrically, ~wasalways thought ofastheratiooftheareaofacircleto thatofthecircumscribed square. 11Namely: where 1 1 (_1)"-15"=1-3+5-+,,,+ 2n-l ' (n=2,S,...) §31.Statement oftheproblem. 231 Hereweareperfectly satisfied withthestatement thatthesum oftheseriesis=1,precisely because thenumber 1(andsimilarly everyrational number) canbefullyandliterally assigned. Insuch cases,wehaveaperfect righttoassertthatwehaveaclosed expressIOn forthesumoftheseries. Butinallothercases,where thesumoftheseriesisnotaratiollal number, oratanyratenot known tobeones,wecannot strictlyspeakofevaluating thesumof thesenesbymeans ofaclosedexpression. Onthecontrary, the senesoughtthentoberegarded asa(moreorlessimperfect) means ofrepresenting orapproximating toitssum.Byproceeding toexpress theseapproximations (usually intheformofdecimal fractions) and estimating theerrorsinvolved, weformwhatiscalledanumerical evaluation ofthesum. Lastly,asaboveinthecaseoftheseriesfor:'wemayhave ascertained merely thatthegivenserieshasforsumanumber related insomesimple (oratanyratespecifiable) manner toanumber which wemeetwithinotherconnections; ase.g.itfollows from122and 124that 1 1 1·31[1 1 ]l+--:foa+24°~+···= 21-3+5- +.... Inthatcase,weshould stillwelcome theinformation soobtained, smceitestablishes aconnection between resultswhereformerly we sawnone.Itisusual,insuchcases,stilltosay-though inan extended sense-thatwehaveevaluated thesumbymeansofa closedexpression; infact,thenumber concerned isthenregarded as "known" through thoseotherconnections, andwesimplyexpress the sumoftheseries"bymeansofaclosedexpression" involving this number. Herethestudent must,however, guardagainst self-delusion. Ifithasbeenascertained, forinstance (v.po211)thatthesumofthe senes 1+.L._1_+~~..~+I.:3~~...1_+ ." 2;,02·450·2·4·6 .503 hasthevalue{-V2,itisstillonlyinaveryrelative sense"deter­ minedintheformofaclosedexpression". Thenumbery2"isnot perseanybetterknownthanthesumofanyarbitrary convergent series. ItisonlybecauseV2occursinsomanyhundreds ofother connections andhas,forpractical purposes, beensooftenevaluated numerically, thatweareinthehabitofconsidering itsvalueasalmost asperfectly "known" asanyliterally specified rational numbero If 3Forinstance, ifwehavedetermined thesumofaseriestobeequaltoEulcr's constant, wedonotknowtothisdaywhether weareconfronted Witharationalnumber ornot. 232Chapter VIII.Closedandnumerical expressions Corthesumsofseries. instead oftheabovcsenes,wcconsider, forinstance, thefollowing binomial series: ~[1+-.!-..~~ __4_.~+_4~_ -.~-+...J 2 5 1000 5·101000' 5·1015 10008 6 _ anditssumhasbeenascertained tobeequaltoV100,weshall belessinclined toregard thesumasfullydetermined thereby; on thecontrary, weshallprefertoaccepttheseriesasamostuseful 5_ meansofevaluatingV100toadegree ofapproximation notsoeasily attainable byothermeans. Inotherwords,-withtheexception of thosefewcasesinwhichthesumofaseriescanbespecified asa definite rational number, -whenweconsider cquahties oftheform "s=.Ean",theemphasis willbelaidsometimes ontherighthand sideandsometimes ontheleft,according tothecircumstances ofthe case.Ifsmaybeconsidered asknown through otherconnections, weshallstill(though inanextended sens~)saythatthesumofthe serieshasbeenevaluated inthef01mofaclosede'tpression. Ifthis isnotthecase,weshallsaythattheseries ISameansofevaluatl11g thenumber s(ofwhichitprovides thedefinition). (Obviously both pointsofviewmaybetakenwithregardtothesameequality.) In theformer ofthetwocases,weshall,sotospeak,haveachieved our object,sincetheproblem B(v.p.105)alsoisthensolvedtooursatis­ faction. Inthelattercase,however, anewtasknowbegins, thatof actually expressing theapproximations, provided bytheseriesitself, toitssum,inaconvenient andsimpleform(e.g.indecimal fraction form,asthemostdesirable forourpurposes), andofestimating the errorsinvolved intheseapproximations. §32.Evaluation ofthesumofaseriesbymeans ofaclosedexpression. 1.Directevaluation. Itisobvious thatwemaywithout difficulty construct serieswithanyassigned sum.If5betheassigned sum, construct, byanyone ofthemanyprocesses atourdisposal, asequence (5,,)converging to5,andconsider theseries 50+(S1-50)+(S2-sJ+...+(sn-sn-1)+.... Sinceitsnthpartialsumisprecisely =sn'thisseriesisconvergent andhasthesums.Thissimpleprocedure affordsanincxhaUStlble meansofconstructing seriescapablc ofsummation intheformofa closedexpression; e.g.weneedonlyassumeoneofthenumerous null sequences (x,,)knowntous,andwrite 5"=S-xn'n=0,1,2,..., Examples oCseriesofsum1. gives111-+-+-+ ..·=11·22·33·4 §32.Evaluation ofthesumofasenesbymeansofaclosedexpression. 233 theterm0/aconvergent seriescanbespecified, for" " " "(- 1)")2(x,,)=n+1 3.(X")=(n~I)2 4.(x,,)=(n~lr 5.(x,,)==(21 ") (1) 00\11 6.(x,,)=y(;L+-l)"n-;;IVn(n+f)~'n·-:-· .-t--y-=n=+=--1t1. 7.Ifwemultiply thetermsofoneoftheseseriesbys,weohtalna convergent seriesofsums. Itisnotsuperfluous tobeabletoconstruct suchexamples, asweshaII seethatthepowertoprovide serieswithknownsumisanadvantage in thediscussion offurtherseries. Theconverse oftheprinciple justtreatedisexpressed bythe oTheorem. Givenaseriesian'whosetermsa"areexpressible 131, n=U rntheforman=x"-xn+l'wherex"is sequence ofknownlimit ~,thesumofthe wehave Proof.Wemaywnte s"=(xo-Xl)+(Xl-x~)+...+(X"-x"+1)=Xo-x,,+1' Sincex"--+~,thestatement follows. Examples. 1.Ifabeanyrealnumberof0,-I,-2,...,then(v.6S,2b): "', 1 1 [ 1 1 ] ~o«(+n)(a+n+l)=--;' asherea,,=a-!-n-a+n--t--f • 2.Similarly132. 00 1 1 n~(a+n) (a+n+l)(a+n+2)=2a(a+l) ashere or1[11] an=-2-(a+n)Ca+nTf)-(a+n+ 1)(a+n+2). 3.Generally, ifpdenotes anypositive integer, ~ 1 =.t... ~.J... . ~o(a+n)(a+n+1)...(a+n+p)Pa(a+I)..,(a+p-1) 4.Putting a=t,wethusobtain,forinstance, from2.: 1 1 1 1 1.4.7+47--:TO+'i-=-"10--13+...=24' 5.Putting a=1in3.weobtain111 1·2...(p+1)+2·3...(p+2)+...=p·pl ao1p+l ,,~(p+n+l) =-p-' P+l 234Chapter VIII.Closedandnumerical expressions forthesum!lofseries. Thefollowing isasomewhat moregeneral theorem. 133.°Theorem. Ifthetermanofagivenseries.4anisexpressible intheformXn-xn+q'wherexnisthetermofaconvergent sequence ofknownlimit ~,andqdenotesafixedinteger>0,then., .2an=Xo+Xl+...+Xq-1-q~. n=O Proof. Wehave,forn>q, sn=(xo-xg)+(Xl-xq+l)+ + (xq_1-X2q-1)+(xq-x2g) ++(X"-xn+q) =(xo+Xl+...+xq_l)-(xn+1+xnH+...+xn+q)· SinceX.--+~(by41,9),thestatement atoncefollows. Examples. 1.1;( )1 =~(~+~1+...+11),,,=0a+n(a+n+q) qaa+ a+q- sinceherewehave 1 ( 11)a.=qCl:+n-Cl:+n+q • Inparticular, writing Cl:=1, )1 1 1(11 1 ) n:;'O(2n+1)(2n+2q+1)=2q+"3+...+"\fq-=-f. 2Fora=1andq=2wehaveaccordingly: 1 1 1 3 r:a+2.4 +3-5+'" ="4; andforCl=hq=3: 3.Somewhat moregenerally, ifkIaswellasqIdenotes afixedinteger> 0: QC 1 n~(a+n)(a+n+q).••(IX+n+kq)= 1'1-1 1 =kq.~ (IX+v)(IX+V+q)...(a+v+k-iq) 4.ThusforCl=~,q=2,k=2wefind 1 1 1 13 I:S--:]+3-~7.IT+ 5·9~13+...=420' Theartifices hereemployed maybeextended toobtain,finally, thefollowing considerably furtherreaching 134.°Theorem. Iftheterms01aseries.4anareexpressible, for everyn,intheform an=ClXn+1+C2XnH+...+CkXn+I<(kconstant, ~2) where(xn)denotesaconvergent sequence ofknownlimit ~,andthe coefficients Clsatisfythecondition Cl+c2+...+c"=0, §32.Evaluation ofthesumofaseriesbymeansofaclosedexpression. 235 then;2'a"isconvergent andhaslorsum' S';an=ClXl+(Cl+C~)X2+...+(Cl+C2+···+C,,-I)X h-1n=O+(C2+2C3+...+k-=-i Ck).;• Theproofisatonceobtained bywriting theexpressions for apa2,•••,am'onebelowtheothersothattermsinvolving x"occupy thesamevertical. Carrying outtheaddition incolumns, -whichof courseisallowed evenwithout reference tothemainrearrangement theorem--wefind,form>k,takingintoaccount thecondltion ful· filledbythecoefficients c", mk-l k-t .2Jan=.2J(Cl+C2+...+c,,)x"+.2J(CHI+...+Ck)Xm+.t+t' n=O ).=1 ).=1 whichisagainthesumofalinitenumber ofterms. Lettingm--+00, weatonceobtaintherequired relation. Examples. n2 Putting Xn=n2+I'k=2, c1=-I, c2=+I, weobt'\in 3 5 7 2,.t-I I-+-++ ...+------+ ...=-2·55·1010·17 (n2+1)(n+12+1) 2' 00 11 00(1 1) 13 2.n~(3n+1)(3n+10)=27n~-i+n+1--f+n+4.=84. Theseexamples mayofcourseeasilybemultiplied toanyextentdesired. 2Application totheelementary functions. Theabovefewtheo· remshave,speaking generally, madeusfamiliar withalltypesofseries whichmay,without requiring anymorerefined artifices, besummed intheformofaclosedexpression. Byfarthemostfrequent series,inallapplications, arethoseob­ tainedbysubstituting particular valuesforxinseriesexpansions of elementary functions andinseriesderived fromthesebyeveryspecies oftransformation orcombination, orotherknownprocesses ofdeduction. Examples, obtained inthismanner, ofsummation byclosedexpres· sionsareinnumerable. Wemustcontent ourselves withreferring the readertotheparticularly ampleselection ofexamples attheendof thischapter, intheworking outofwhichthestudent willrapidlybe­ comefamiliar withaUthemainartifices lIsedinthisconnection. The devt'lopments inthisandthefollowing section willaffordfurther guidance inthispartofthesubject Letusmerely observe quite generally, forthemoment, thatitisoftenpossible todealwithagiven seriesbysplitting itupintotwoormoreparts,eachofwhichagain represents aconvergent series;orelsebyadding toorsubtracting from.J:an'termbyterm,asecondseriesofknownsum.Inparticular, ifatlisarational function ofnitsexpansion inpartiaZtractions will frequently beaconsiderable help. 236ChapterVIlI.Closednndnumerical expressions forthesumsofseries. s.Application ofAbel'stheorem oflimits. Afurthermeansof evaluating thesumofaseries,-oneofgreattheoretical importance, differing fromthatjustindicated intheprinciple itinvolves, though inmostcasesintimately connected withitinvirtueof101,-con­ sistsinapplying Abet'stheorem oflimits. Givenaconvergent se­ riesZan'thepowerseriesf(x)=Zanx"converges atleastfor - 1<x~+1,andhence,by101, - ~a..=hmf(x). x~1-0 Ifwesuppose thatthefunctionf(x)whichthepowerseriesrepresents issofarknown, thatthelatterlimitcanbeevaluated, thesummation oftheseriesisachieved. Thedevelopments ofChapter VIoffera widebasisforthismodeofprocedure, andinfactAbel'stheorem has alreadybeenusedtheremorethanonceinthesensenowexplamed. Weshallgivehereonlyafewrelatively obvious examples, with areference totheexercises attheendofthischapter. 13:>. Examples. 'Vearealreadyacquainted withtheseries: 00(_1)" 00 x"+11..J;..--=Hm.J;(_I)n--1=Hmlog(1+x)=log2. X=On+1x~1-0n=O n+x~1-0 00(-1)" 00 X9n+1:Jr2.)'.-- =lim)'(-1)"--=limtan-1x=-. n~02n+l x~1-0n"';;;"0 2n+l X-~1-0 4 Wehavethefurtherexample 00(_1)". ( x'x? )3.2--=hmx--+--+.., . n=O3n+1x-)-1-0 4 7 Theseriesinsidethebrackethasforderivedseries 1_x'+x._+...=_1_ 1-1-;c' andtherefore represents thefunction (v.§19,Del.12) z fdx1(x+1)91 2x-I 1r 1+x·=(flogx9-x+1+VStan-I~-3 +6{3" o Accordingly, thesnmotthegivenseriesis=..!.log2+-.!!-.3 3{3 4.Similarly wefind(v.§19,Del.12) 1;(-1)"=1-..!.+..!.- .~+_.".=..!.v'2[:Jr+ 100'(3+2V2)] n=O4n+1-5 9 13 8 ,., • Forfurtherseriesconstrncted onthesamelines,theformulae ofcoursebecome moreandmorecomplicated. 4.Application ofthemainrearrangement theorem. Equallygreat theoretical andpractical significance attaches, inourpresent problem, totheapplication ofthemainrearrangement theorem. Thisapplication weproceed atoncetoillustrate byoneofthemostimportant cases; additional examples willagainbefurnished bytheexercises. In115and117,weobtained twoentirely distinct expansions ofthefunction xcotx,bothvalidatleastforeverysufficiently smallIx/. §32.Evaluation ofthesumofaseriesbymeansotaclosedexpression. 237 If,inthefirstofthese,wereplacexbynx,weobtain,certainly for everysufficiently smallIxI' 00 22nB 002x2 1+~(-lr-(2n)~'(nx)~n =1-k~k2-X2' Eachtermoftheseriesontherightmayobviously beexpanded in powersofx: (k=1,2,...fixed) Thesearetheseries Z{k)ofthemainrearrangement theorem; sincetheseries C{k)ofthattheorem inourcaseonlydifferinsign fromtheseries Z{k)themselves, theconditions ofthattheorem areall fulfilled, andwemaysumincolumns. ThecoefficIent ofx2ponthe rightthenbecomes <Xl1 :~-2.2~­ k=1k2p(pfixed) (pfixed)136.andsince,by97,ithastocoincide withthatontheleft,weobtain theimportant result(oncemoredenoting theindexofsummation byn) 001 B(2]I;)2p ~_=(_l)p-l 2p • tl~1n2P 2(21))! Thisgivesusthesumoftheseries 1 1 11+22P+32P+"'+ n2p+··· (pfixed) intheformofaclosedexpression, sincethenumber :nandthe(ra­ tional)Bernoulli's numbers mayberegarded asknown'. Inparticular, 001,1l;1l 001,1l;4 ao1]1;6 ~-=-, I.=90' 2-=-.••='ln2tl tl""in tl=ln61145 •Quiteincidentally, formula136showsthatBernoulll's numbers B2"are ofalternating signsandthat(_I)n-1BI•ispositive; further, thattheyincrease Withextreme rapidity asnincreases; forsineethevalueof.i;_1_liesbetween I k=1k2n and2,whatever bethevalueofn,wenecessarily have 2(2n)! n-1 2(2n)! 2(231)2n>(-1)Bin>(2.31)""' whence itfollowsthatIB1~:11--+00.Finally, astheabovetransformation holdsforIxI<1,italsofollows thattheseriesIUSconverges absolutely at leastforIxI<.31.ButforIzI>nitcertainly cannotconverge absolutely, forthencotxwouldbecontinuous forx=:n,by9S,2,whichweknowisnot thecaseithustheseriesIllShasexactly theradius 11.Itfollows fromthis that116lahastheradius-;-,116btheradius :tr. 238ChapterVIII.Closedandnumerical expressions forthesumsofseries. Itisnotsuperfluous totrytorealiseallthatwasneededtoobtain eventhefirstoftheseelegantformulae 5.Thiswillbeseentoinvolve muchofourinvestigations uptothispoint. Theaboveprovides uswiththesumofeveryharmonic serieswith anevenintegralexponent; weknownothingyetofthesumofaharmonic serieswithoddexponent(>1);thatistosay,wehavenotsucceeded asyetinfindinganyobvious relations thatmightresultinconnecting suchasum(e.g.El3)withanynumbers occurring elsewhere. (There isofcoursenoobstacle toourevaluating thesumofanyharmonic series numerically, toanydegreeofapproximation 6;v.§3!i).Ontheother hand,ourresultsreadilyyieldthefollowing furtherformulae: Wchave <Xl1 001 <Xl1'\'---)'--- +.L;'-­ n~ln2P-'~1(2v-l)2P .=1(2v)2P' Thelatterseriesisprecisely thesamethingas-i-i;-~.Subtract· 2Pn=1n-P ingthisfrombothsides,weobtain i-(211)2P=(1--2-~P)i-~P,,=1n- n=1n or 137. For1 1 221J_l" l+-+-+ ...=(-I)P-l B., .:rr:~Pa2"52p :.l(:.lp)!~l' • P=1,2,3,..•,thesumsarcinparticular n:2n:' n:6 or 13S.S'96'!l60' 1 <Xl1Ifweagainsubtract thesameseries-2-2J-.,-,weobtain 2Pn=1n-P ~(-1)n-l(2) <Xl,1~--n'P-=1 --2'21'~2P n=1 n=ln 1 1 1 22p-1_l1-22p+a21'-42p+-...=(_1)p-l (2p)!B2p':rr:21'. ~JamesandJohnBernoulh didtheirutmosttosumtheseries 1 1 11+4+9+16+.... Theformerofthetwodidnotlivetoseethesolution oftheproblem, which wasfoundbyEulerin1736.JohnBemoulll, towhomitbecame knownsoon after,wroteinthisconnection (Werke, Vol.4,p.22):Atqueitasatisfactum est ardenti desidcrio Fratris mei,quiagnoscens summae huiuspervestigationem dzflzczhorem quamquzsputavent. ingenue fassusestomnemsuamindustriam fuisse elusam. "UtinamFratersupcrstes esset'Asecondproof,ofaquitedifferent kind.willbefoundin156,athirdin189,andafourthin210. <Xl1 6T.J.Stzeltjes (Tables desvaleurs dessommes Sk=.2-h'Actamathe­ n=1n matica, Vol.10,p.299,1887)evaluated thesumsofthcseseries,uptotheex­ ponent70,to32placesofdecimals. or§32.Evaluation ofthesumofaseriesbymeansotaclosedexpression. 239 Inparticular, forp"-~1,2,3,...theSUIUSare 1~7431 6 12:n,720:n, -30240n..... Hereagain,however, weknownothing ofthecorresponding series withoddexponents. ---Thelasttworesultsmightofcoursealsohave beenobtained bystarting withtheexpansions inpartialfractions of thefunctions tanor~,andreasoning asaboveforthatofthefunc-sm tioncot.Wemaydeduce furtherresultsbytreating theexpansion inpartialfractions, giveninlIS,ofthefunction _1_,i.e.eos ;rt 1 3 5 ac,(-1)"(2v+1) ---n-x =1".-x"-3"-x"+5"-x"-+...=L,-(2-~+i)"-x' .4cos__ ,-=0.2 The"thtermIShereexpressible bythepowerseries 00 x9le (1)vy--~ -k~(2..+1)~k+1; afterrearranging, thecoefficient ofx2pthusbecomes: "',(-1)"_ 1 1 ,,~(2v+1)2P+1~1-32p+1+-52P+1-+.... Letusdenotethesesumsprovisionally by02P+1;then ;rt=01+03x~+00x4+...;rtX4cos-2 -co1-sz=~.[01+a3(~~r+°0(~y+...J. Ontheotherhand,thispowerseriesmaybeobtained bydirectdivision anditscoefficients -justlikeBernoulli's numbers in10~,[)--.. bysimplerecurring formulae. Weusuallywrite ThisgivesEo=1,and,foreveryn~1,recurring formulae 7which maybewrittenasfollows(aftermultiplication by(2n)I): E2n+(22n)E2n-2+(2,t)E2n-4+···+Eo=O, 139. 7Thenumbers determined bytheseformulae (whicharemoreover ,.atlonal integralnumbers) areusuallyreferred toasEuler's nU1nbers, Thenumbers Evup tov=:10havebeencalculated byW.Scherk,Mathem. Abh.,Berlin1!l:!5. 140.240Chapter VIII.Closedandnumerical expressions forthesumsofseries. orintheshortersymbolical form(cf.106): (R+l)k+(E-l)k=O, nowholding foreveryk~1. Wededuce without difficulty: E~=E8=E.-,=...=0 and Eo=1,E'J=-I, E4=5,E6=-61. Es=1385,...• Intermsofthesenumbers, whichweareperfectly justified IDcon sidering asknown, wehave,finally, E2P4 22p (-l)p(2Pfj=~t1'Jp+l' n~P' i.e. 1__1__+_1__+,..=(-1)1' R2p .7l2p+1 321'+1 52J,+1 - 221'+2(2p)! • Inparticular, forp=0,1,2,3,...,thisgivesthevalues 13 32n,5 ~ 1536n, forthesumsofthecorresponding series. §33.Transformation ofseries. Inthepreceding section(§32),webecame acquamted withthemost important typesofserieswhichcanbesummed bymeansofaclosed expression -eitherinthestricterorinthewIdersenseoftheterm. Intheevaluations lastmade,whicharereallyofaprofound nature, themainrearrangement theorem playedanessential part;indeed, in virtueofthistheorem, theoriginal serieswaschanged, sotospeak, intoacompletely different serieswhichthenyielded furtherinforma. tion.Weweretherefore principally concerned withaspecialtrans­ formation ofseriesS•Suchtransformations arefI:equently ofthegreatest use,andindeed evenmoresointhenumerical calculations which formthesubjectofthefollowing twosections, thaninthedetermina. tionofclosedexpressions forthesumsofseries. Tothesetrans· formations wewillnowturnourattention, andwestartatoncewith amoregeneral conception ofthetransformation deduced fromthe mainrearrangement theorem andrepeatedly applied toadvantage alreadyinthepreceding section. SSuchtransformations werefirstindicated byJ.Stirling (Methodus diffe­ rentialis, London 1730);theyarebased,inhiscase,onsimilar linestothe above,excepting thathefailstoverifythefulfilment oftheconditions under whichtheprocesses arevalid. §33Transformation ofsenes. 241 <Xl Givenaconvergent series.2z(k),leteachofitstermsbe k=O expressed, inanymanner, (e.g.by§32,p.232)asthesumofan infiniteseries: !z(O)=ao(0)+a1(0)+a2(0)+...-I-an(0)+... (A) ~(1)..ao:l ):.a<1):-.a2~1)~f-:".~a:(1).~..: Z(k)=ao(lc)+al(k)+a 2(1c)+ + an('C)+-. .. . Weshallassume furtherthatthevertical columns inthisarraythem­ selvesconstitute convergent series, anddenote theirsumsby s(O),sw,...,s(n),••..Underwhatconditions maytheseries.fsin) ,,=0 formedbythesenumbers beexpected toconverge, with )1z(k)=~S(n)? ~ ~ k=O ,,=0 Ifthisequality isjustified, wehavecertainly effected atrans­ formation ofthegivenseries. Themainrearrangement theorem im­ mediately givesthe °Theorem. Ifthehorizontal rowsofthearray(A)allconstitute 141. absolutely convergent seriesand-denoting bye(k)thesum,1'1a"(k)I,of 11=0 theabsolute values01thetermsinonerow-,iltheseriesze(lc)is convergent, theseries2's(n)alsoconverges and=:Ez(k). Itisthistheorem thatwehaveapplied inthepreceding para­ graph. Thequestion ariseswhether itsrequirements arenotun­ necessarily stringent, whether thetransformation isnotallowed under verymuchwiderconditions. A.Inthisdirection, anextremely far-reaching theorem wasproved byA.Markofr. Heassumes firstonlythatthescriesconstltutcd bythe vertical columns ofthearray(A)converge, aswellastheoriginal series andtheseriesconstituted bythehorizontal rowsofthearray.The numbers sln)havethusdeterminate values. Since1;Z(k)and.fao(k) k=O k=O converge, sodoes.J;(zlk)-aolk»;andalso,similarly, foranyfixedm, A=O thcseries ~((Ic> (k) (k) (1')) £..JZ-ao-at-•••-a"'-l k=O(mfixed). 9Mcmoire surlatransformation deseries(Mem.del'Acad.Imp.de St.Petersburg, (7)Vol.37.1891).Cf.anotebytheauthor, "Einige Bemer. kungen zurJ(ummcrschen undMarkojjschen Reihentransformation", Sitzungs­ berichte derBerl.Math.Ges.,Vol.19.pp.4-17,1919. 242ChapterVIII.Closedandnumerical expressions forthesumsofseries. Thetermsofthisseriesare,however, precisely theremainders, each withtheinitial 10indexm,oftheseriesconstitutcd by theindividual rows ofthearray.If,forbrcvity,wedenotethesercmainders byr:,~l,sothat r~)=1:a~h) (/~andmfixed), n=m thesenes (mfixed) 142.isconvergent. Thefurtherassumption isthenmadethat Rm-+0whenm-:>-oc). Itmaybeshownthatunderthesehypotheses}; S(II)converges and=EZ(h). Thetheorem obtained willthusbeasfollows: oMarkoff's transfor1nation ofseries. Letaconvergent series 00 EZ(h)begivenwitheachofitstermsitselfexpressed asaconvergent series: h~U (A) z(l<)=ao(h)+a1(h)+...+an(h)+... (/~=0,1,2,...). (mfixed). also.00 Lettheindividual columnsEan(h)ofthearray(A)soformedrepresent h=O convergent serieswithsumS(II),n=0,1,2,••• Jsothattheremainders r(h)=1:a(k) (m>0) III 11 1l=11l oftheseriesinthehorizontal rmvsalsoconstitute aconvergent series ir(h)=R k=U III III Inorderthatthesumsbyverticalcolumns shouldformaconvergent serie~ ES(II),itisnecessary andsufficient thatlimRm=Rshouldexist;andin orderthattherelation llll ..:tS(D)=:tz(k) n=O k=O shouldholdaswell,itisnecessary andsufficient thatthislimitRshouldbeO. Theproof isalmosttrivial,forwehave (a) s(U)+s(1)+...+S(II)=Ro-Rn+1, whencethefirststatement isimmediate. Sinceitfollowsthat 00 ES(II)=Ro-R, 11=0 andsinceRoissimply;;r~k)=J:z(k),thesecondstatement nowfollows h=O h=O 10Hereweofcoursetakem--0togivethewholeseries,i.e.Z(k)itself. §33.Transformation ofseries. 243 B.Thesuperiority oflIIarlwff's transformation overTheorem 141 consists, ofcourse, intheabsence ofanymention ofabsolute conver­ gence,onlyconvergence pureandsimplebeingrequired throughout. Its applications arenumerous andfruitful: thosebearingonnumerical evalua­ tionswillbeconsidered in§35,andweshallonlyindicate inthisplaceone oftheprettiest ofitsapplications, whichconsists inobtaining atrans­ formation givenbyRuler 11-ofcourse,inhiscase,without anycon­ siderations ofconvergence. Itisadvantageous heretousethenotation ofthecalculus offinite differences, andthiswewillaccordingly firstelucidate inbrief.Given anysequence (xo,Xl'X2,••.),thenumbers arecalledthefirstdifferences of(xn)andaredenoted by Ltxo,LtXl'•..,LtXkl Thedifferences ofthefirstorderof(Ltx,,),i.e.thenumhers Ltx"-LtX"q, kC~U,1,2,...,arccalledtheseconddifferences of(x,,),denotedby Lt2'~o,Lt2xl!...,Lt2XI". Ingeneral, wewritefor11::;:1 Lt"+!XI.=Ltnx"-LtnXk+l (1~=0,],2,...) andthisformula mayalsohetakentocomprise thecase 11=Uifwein­ terpretLt°:>.:"asheingthenumber X~itself.Itisconvenient toimagine the numbers XkandLtnXkarranged inrowssoastoformthefollowing tri­ angular array,inwhicheachdifference occupies theplaceinitsownrow immediately belowthespace,intherowabove,between thetwoterms whosedifference itis: (Lt)Xo, Xl!x2, X3, XJ,••••• Ltxo•Lt.\,!,Ltx2,Ltx3,••••• Lt2Xo,Lt2Xl'Lt2.'l:2••••• Lt3Xo•Lt3Xl!••.• Lt4Xo• andsimilarlyThedifference Lt"X"maybeexpressed intermsofthegivennumbers X"directly. Infact Lt2Xk=LtXk-LtXk+l=(Xk-.'l:k+1) -(x'''+l-X,,+2) =Xk-2Xk+l+X"+2 11Institutiones calculidiffcrentiahs, 1755,p.281. 244ChapterVIII.Closedandnumerical expressions forthesumsofseries. 143.theformula Ltnxk~Xk-(~)Xk+l+(~)Xk+2 -+...+(-1)'1(:)XI'HI forfixedh,isthusestablished inthecasesn=1,2,3.Byinduction, its validityforeverynfollows. For,supposing 143provedforaparticular positiveintegern,wehaveforn+1: Ltn+1Xk=Ltnxk-Ltnxk+l =Xk-G)Xk+l-/-C)Xk+2-++(_1)'1(:)Xk+f1 -(~)Xk+I+(7)Xk+2-+ +(-1)"(n:1)xk+f1 +(_1)'1+1 (:)Xk+nt-h whencebyaddition, sinceC)+C':I)=(n~I),wehavetheformula 143forn-+-1insteadofn.Thisprovesallthatisrequired. Makinguseoftheabovesimplefactsandnotation, wemaynowstate thefollowing theorem: 144. 0Euler's transformation ofseries. Givenanarbitrary con- vergentseries12 00 E(-l)kak ~ao-a1+a2-+..., k~O weinvariably have: i.e.theseriesontherightalsoconverges andhasthesamesumasthegiven series13. 12Theseriesneednotbeanalternating series,i.e.thenumbers anneednot allbepositive. Therearehowever small,thoughbynomeansessential, ad\an­ tagesinwritingtheseriesinalternating formasabove,wheneffecting thetrans­ formation. 13Thisgeneraltransformation isduetoRuler(Inst.calc.diff.,pp.281seq., 1755).Theparticular transformation givenbelowinexample 2istobefound alreadyinalettertoLeibnizdated2.8.1704,from.!.Bernoulli, whoattnbuted the discovery toN.Fatzius. (Cf.alsoJ.Hermann, lettertoLeibnizof21.1.1705.) Anearlyinvestigation ofamoresearchmg kind,usingremainder terms,wasunder­ takenbyI.V.Poncelet, Journ.f.d.reineu.angew.Math.,Vo!.13,pp.1seq.,1835. Theproofthatthetransformation isalwaysvalid,provided onlytheseries1:(-l)kak isassumed alsoconvergent, wasfirstgivenbyL.D.Ames(Annals ofMath.,(2) Vol.3,p.185.1901).Cf.alsoE.Jacobsthal (Mathem. Zeitschr., Vol.6,p.100. 1920)andthenotebearingonthatbytheauthor(Ibid.p.118). §:l3.Transformation ofseries. Proof. Inthearray(A)ofp.211,wesubstitute foran'I..): (b) a<k)-(-l)~[1L1na--1_Anf-lCl]n- 2n k2n+1LI'.'245 By131,ifwenowsumforeveryn,keepinghfixed(i.e.furmthesum ufthe I~Lhhorizontal row),weobtain Foren .::;(1..)=EanCk)= ( n=O(hfixed). (71)rlj-(nl)a"I1+ -...limL1n.rz,,-lim_0.•_-'-''---_ '"l----?OO 2'-1.11->00 2!t ISequaltozeroby44,R,bccause a,,,-a"u,a'e\-2'•••certainly form anullsequence. Accordingly (b)givesanexpressIOn fortheindividual termsofthegivenseriesE(-1)~a,.ininfiniteseries. Forming thesum ofthenthculumn, wcohtaintheseries (nfixed); thegenerictermofthisseries,asL1n+la"-L1nOk-L1nak+l'canbewritten inthcform sothattheseriesunderconsideratiun mayagainbesummed directly,by 131.Weobtain ;;Cl'k)=-1.[L1na-lim(-1)'"L1"a] k.oOn2"+1 0k-+oo le(Ilfixed). Since,however, thenumbers a,.formanullsequence, sodothefirstdiffer­ encesandthe11thdifferences generally, foranyfixed 1/.Thevertical columns arethuss(:cntoconstitute convergent seriesufsums L1nas'n)=-2"+1°' ThevalidityofEuler'stransformation willaccordingly heestablished when wehaveshownthatRm_.,..O.Nowthehorizontal remainders areseen tohavethevalues Am r(k)=(-I)k ~_.f1Tc ni, 2"1• 9 (G51) 246Chapter VIII.Closedandnumerical expressions forthesumsofserieIJ. following precisely the~amelineofargument aswasusedabove fortheentirehorizontal rows.Thus R=~...~,(-1)'<,dma (fixedm). m2m~~' k Ifwewriteforbrevity (-l)k(ak·-ak+l+ad~-+...)="k' thisseriesforR",maybethought ofasobtained byterm-by-term addition fromthc(m+1)series: "0'(7),,1'G)r 2,...,(:)""" Hcnce R_~O+cnrl+(~)r2+···+G;:)rm. m- 2m ' therefore, asrmistheternlofa!lullsequence, soisR""by44,8. Thisprovestheva!Jchty ofEuler's transformation withfullgenerality. Examples. 1.Take 1 I 1 s=1-~+3-4+-"" Thetriangular array(,1)takestheform 1,1 2'1 3'1 4'1 5' 1 1 1 1n' 2·;}'34' 4·5' 1 2 1·2 2 i.·2-~q' ~:f·4' 3--4-5'..• 1·2·3 1·2·3 1·2:r:-4'2·3·4·5' Thegeneral expression ofthentlldifference isfoundtobe nl £I"a"=Ch+1)(h+2r~-.(h+n+1), sothatinparticular 1 A"au=;j-+-1. Thisiseasilyverified byinduC'tion. Accordingly wehave 111It 1ts=log2=1-:r+3'-"i+ -...=1.21+2.2~+8.2"+4.2~+...• Thesignificance ofthiStransformation e.g.forpurposes ofnumerical calcu­ lation(§34)isatonceapparent. 2.Withequalfacility, wemaydeduce :Jr: 1lIt [ 1 1.21.2.3 ]-;:=1--+---+_ ...=-1+-+-+--+ ....4 857 ::l83·5:i.5.7 Inwhatcasesthistransformation isparticularly advantageous forpur· posesofnumerical calculation Willbeseeninthefollowing section. §34.Numerical evaluations. 24-7 .-15.c.0K1lmmer's transformation ofseries. Another veryobvious transformation consists simplyinsubtracting fromagiven ~eriesone whosesumiscapable ofrepresentation bymean5ofaknownclosed expression andwhichatthesametimehastermsassimilar incon­ struction aspossible tothoseofthegiven5cnes. Bythismeans, subtractmg forin~tance flOms=2,'d2theknown sene~(v.6S,2b) co11-\'-----,--­-n-;;;'In(n+1)' wededuce thetransfonnatlon co1 co1 s==2,;'n2=1-+2)n"-\n+1)'n=l n=l Theadvantage ofthistransformation fornumencal purpose!;. ISat onceclear. Simplcandobvious a~thistransformation is,ityetformswhat isreallythekernelofRmnmer's transformation ofseries 14;theonly ditTerence beingthataparticular emphasIs isnowlaidonasuitable choiceofthesene~tobesubtracted. Thischoice ISregulated as follows: Let2,'all=sbethegivenseries(ofcourse, byhypothesi5, convergent). Let2'cn=Cbeaconvergent seriesofknownsumC. Letussuppose thatthetennsotthetwoseriesareasymptotically proportional, say I·an --LIm'-,=r-1-O. tI-)-lX)C71 Inthatcase er> (70( C)s=~a,,=yC+ It-y~ (t", n=O n=U Rn andthenewseriesoccuring onthenghtmayberegarded asatrans­ formatIon ofthegivensenes. Theadvantage ofthistrdn5fonnatlon liesmamlyinthefactthatthenewserieshastermslessinabsolute valuethanthoseofthegivenseries,asinfact(1-r::)-+O.Con­ sequently itsfieldofapplication belongs forthemostparttothedo mainofnumerical calculations andexamples illustratmg itwillbe foundinthefollowing paragraph. §34.Numerical evaluations. 1.General considerations. Asrepeatedly explained already, it isonlyonveryrareoccasions thataclosedexpression, properly so­ called,existsforthesumofaseries. Inthegeneral case,thereal 11Klimmer. E.E.:Journ.f.d.reineu.angew.Math.,Vo!.16,p.206.1837· Cf.alsoLee/ertandCatalan, lVll'mOlres couronnes etdesavantsctnmgers del'Ac. Delglque, Vo\,33,1811G-lii, andthenotebytheauthormentioned infootnote U. 248Chapter VTlI.Closcdandnumerical expressions forthesumsofseries. number towhichagivenconvergent series,orthesequence ofnum­ bersforwhichitstands, converges, is,sotospeak,firstdefined (given, determined, ...)bytheseriesitself,intheonlysenseinwhichanumber canbegiven,according tothediscussion ofChapters IandIIu.In thissense,wemayboldlyaffirmthattheconvergent senesisthe number towhichitspartialsumsconverge. Butformostpractical purposes wegainverylittlebythisassertion. Inpractice, weusually require toknowsomething moreprecise aboutthemagnitude ofthe number andtocompare different numbers amongthemselves, etc.For thispurpose, werequire tobeabletoreduce allnumbers, defined byanykindoflimiting process, tooneandthesametypicalform. Theformofadecimal fraction isthatmostfamiliar tousto-day,and theexpres5ion, inthisform,ofnumbers represented byseriesaccor­ dinglyinterests usfirstandforemost 16.Thestudentshould, however, getitquiteclearinhisownmindthatbyobtaining suchanexpres­ sionwehavemerely, atbottom, substituted forthedefil1ltlOn ofa number byagivenlimiting process, arepresentation bymeans of another limIting process. Theadvantages ofthelatter,namelyofthe decimal form,aremainlythatnumbers sorepresented areeasllycom­ paredwithoneanother andthattheerrorinvolved interminating an infinitedecimal atanygivenplaceiseasilyevaluated. Opposed to thisthereare,however, considerable disadvantages: thecomplete ob­ scurityofthemodeofsuccession ofthedIgitsinbyfarthegreater number ofcasesandtheconsequent labourinvolved intheirsucces­ siveevaluation. Thtseadvantages anddisadvantages maybeconvel1lently illustrated bythetwofollowing examples: (:Jr) 1 1 1 ."4=1 -"3+-"5--i+--=0'785398 ..• (log2==)1--~+-~-~+--=0'6U3147 .•• Bytheseries,distinct lawsofformation aregiven;buttheyaffordus nomeansofrecognizing whichofthetwonumbers isthelargerof thetwo,forinstance, orwhatisitsexcessoverthesmaller number. Thedecimal fractions, ontheotherhand,exhibitnosuchlaws,but giveusadirectsenseoftherelative andabsolute magnitudes of bothnumbers . ..Indeed aninfinite series-ourprevious considerations giveample conflrmation ofthefact-isoncofthemostusefulmodl'sofsodefimng a number, oneofthemostsigmficant bothfortheoretical andpractical purposes. 1dAndonly IIIspecial casestheexprc~..ioninordmary fractional form. Thcreasonisalwaysthatofconvenience ofcompari!>ol1j whIch, ofttorli, isthelarger, wecannotsayatonce,whereas th('answer tothesamequestion lor0647and0·641requires nocalculation whatever. §34.Numerical evaluatIons. 249 Weshalltherefore henceforth reserve thetermnumerical eva­ luationtortheexpression ofanumber indecimal form. Asnoinfinite decimal fraction canbespecified intoto,itwill benecess,Hy tobreakitoffafteradefinite number ofdigits.We havestillafewwordstosayastothe~ignificance ofthisprocess ofbreaking oftdecimal fractious. Ifitbedesired, forinstance, to indicate thenumber ebyatwo-digit decimal fraction, wemaywith equaljmtification write2'71and ~'72,-theformer, because thetwo firstdecimals areactually 7and1,-thelatter,because itappears toinvolve alessererror.Weshalltherefore makethefollowing con­ vention: whenthenspecified digitsafterthedeCImal pointarethe actualfirstndigitsofthecomplete infinite decimal whichexpresses agivennumber, weshallinsertafewdotsafterthenthdigit,writing forinstance e=2'71...;when,however, thenumber isindicated by theneare5t possible decimal fractlOl1 ofndIgits,weinsertnodots afterthenthdigit,butwrite 17e.g.eR;j2,72,inthelattercasethe11thdigit written downisthusthenthdigItoftheactualinfinitefraction raisedor notbyunityaccording asthesuccceding partoftheinfinite fraction re­ presents moreorlessthanonehalfofaunitinthenthdecimal place. Inpointoffact,eitherspecification hastheeffectofassigning an interval oflengthl/lOncontaining therequired number. Intheone case,thelefthandendpointisindicated, intheother,thecentreofthe interval. Themargin, fortheactualvalue,isthesameinbothcases. Ontheotherhand,theerrorattaching totheindicated value,relatively tothetruevalueofthenumber considered, isintheformercaseonly knowntobe2::()and~]/lOn,inthelattertohavemodulus<1/1010. Wemaytherefore describe thefirstindication astheoretically the clearer, andthesecond aspractic.llly themoreuseful. Thediffi­ cultyofactualdeterminatwn ofthedigitsisalsoinallessential par­ ticulars thesameinbothcases.Forineither, itmaybecome ne­ cessary, whenaspecially unfavourable caseisconsidered, todiminish theerrorofcalculatIOn toveryappreciably lessthan1!10nbefore thenthdigitcanbeproperly determined. Ifweare,forinstance, concerned withanumber ex=5'2799!J999326..., -todetermine whether a=5'27...or5'28... (retail~ing twodecimals), wehave todiminish theerrortolessthanaunitinthe8thdecimal place.Onthe otherhand,ifweareconcerned withanumber {J=2'3850000026 ..., thechoicebetween {JR;j2'38and2'39wouldbeinfluenced byan uncertainty ofone11l1ltinthe8thdecimal place18. 17Ine=2'71...,thesignofequality maybejustified asrepresenting alimIting relation. 18Th£'probability ofsuchcasesoccurring isofcourseextremely small. Bymentioning them,wchavemerely WIshedtodrawattention tothesigni­ ficanceofthesefacts.InEx.131,however, aparticularly crudecaseisindicated. 250Chapter VIII.Closedandnumerical expressions forthesumsofseries. 2.Evaluation oferrorsandremainders. Whengivenaconver· gentseries2:an=s,weshallofcourseassume thattheinchvidual termsoftheseriesare"known", i.e.thattheirexpressions indeCimal formcan'easilybeobtained toanynumber ofdigits. Byaddition. everypartialsumSnmayaccordingly alsobeevaluated. Thequestion 19 remains: whatisthemagnitude oftheerrorattaching toagivensn? Heretheworderrordesig-nates the(po'iiuve ornegative) number which hastobeaddedtosntoobtaintherequired values.Sincethiserror iss--sn'i.e.isequaltotheremainder oftheseries,startmg im­ mediate:lyafter thenthterm,wewilldenoteitbyrn'andtheprocc5S ofdetermining thiserrorwillalsobedesignated bythetermevaluatwn ofremainders. Inpractical problem", evaluations ofremamders almostinvariably reduce tooneofthetwofonowing types: A.Remainders ofabsolutely convergent series.Ifs=.:sancon­ vergesabsolutely, determine aserieslEan'ofpositive terms,capable ofsummation inaconvenient closedexpressIOn, andwithtermsnot lessthantheabsolute valuesofthecorresponding termsofthegiven series(though alsoexceeding thesebyaslittleaspossible). Obviollsly 11'1<la1+la1+,··<a'Ll-I--a'+,)-L ...=r'n=n+1 ,,+~ =n,n•,- n andthenumber rn',whichisassumed known, thusprovides ameans ofestimating themagnitude oftheremainder l',i.e.Il'I<r"andn f&-_n thisallthemorecloselythelessan'exceedsIani. Aparticularly frequent caseisthatinwhich, forsomefixedrn, andeveryk2.1: inthatcase,otcourse,laml-I,I~ [a",l·a" withU<a<1; 11'",1<Ia",h-:a' andinparticular, if0<a<~: 11'",I:::::IamI· Theabsolute valueoftheremaincler isIIIthisC;Jsenotgreaterthan thatofthetermlastcalculated 20. B.Remainders ofalternating series. Givenasenesoftheform s=2:(-l)nan'andsupposing thatthe(posItIve) numbers anform amonotone (decreasing) nullsequence, wehave(cf.82,Theorem 5): 0<(-1)n+1rn=(all+1-a"H)+(an+3-anH)+... =an+1-(an+'J-a,,+:l)-...<an+1' 19Orinmorepractical form:Uptowhatorderofdecimal does5"com. cideWiththerequired values? 20Informing theseestimates, itshouldbenoticed thattheygivenoin. dications astothesignoftheremainder r",onlyastoitsabsolute value. §34.Numerical evaluations. 251 Hencewemayassertthattheerrorrnhasthesamesignasthefirst neglected term,buthasasmaller absolute value. Whenneither ofthesetwomodesofprocedure isapplicable, the evaluation ofremainders isusually moretroublesome, anditbecomes necessary toadoptspecial artifices ineachparticular case.Weshall, then,designate theseriesconsidered asrapidly orslowlyconvergent, according asrndoesordoesnotfallwithmthedesired limitoferror formoderate values 21ofn. Afewfurtherfundamental remarks maybeelucidated bythe 3.Evaluation ofthenumber e.Wefound 1 1 1 1 e=1+Ti+-21+3T+...+n!+.... Already, onp.194,wehavementioned thatthe(positive) remainder r" waslessthanthenthpartofthetermimmediately before, sothat 1s<e<s+--.u nn!n Ineffecting thenumerical calculations, weh,wcnowtotakeintoac­ countthefollowing fact:Whenweexpress theindividual termsofthe seriesindecimal form,wehaveevenatthatpointtobreakoffthe decimals atsomeparticular digit,andwetherefore incuracertain error.Unlessnremains comparatively small,theseerrorsmayaccu­ mulate to~uchanextentthatthewholecalculation isindanger of becoming illusory. Themodeofprocedure isthenasfollows: Sup­ posingthatweareretaining 9digits,weWrIte 22 ao+at-/-a2~~~=2'500000000 a,l =016666666T a4=0-O'.41666667- a:; =0'..8333333+ an ~0'..1388889- a7~,O·...198413- aH~O'...~4802- all ~0·.....275()- a10=0·... 276- all--().... . 25+ a12=0·... 2+ [1'12<0'... ..0+] Herethesmall+and-signsareintended toindicate whether the errorintheterminquestion ISpositive ornegative. Ineithercase itisinabsolute valuelessthanonehalfofaunitinthelastdecimal place.Byaddition, weobtainthenumber 2'718281830. 91Amoreprecise definitIOn ofrapidconvergence willbegivenin§37. lIJanisdeduced froman-1bysimpledivision byn. 252Chapter VIII.ClosedandnumerIcal expressions torthesum"ofseries. ButS19itselfmaypossibly (namely ifallpositive errorsarenearly0 andallnegative onesnearly ~ofaunitinthelastdecimal place) fallshortofthenumber required byasmuchas~ofaunitinthe lastdecimal place;oritmay,ontheotherhand,beasmuchas1of aunitinexcess, sincethereare7negative and3positive errors. Taking alsointoaccount theremainder, wecanonlydeduce with certainty, sincesn<e=sn+rn'that 2'718281H2G <e<2'718281832. Ourc.llculation thussecures onlythefirstseventruedecimals, while theapproximate value 23isobtained witheightdigits:eR::j2·7182SUI3. Inpractice itwillgenerally suffice toprocE'ed afewtlecimal places furlher(2or3atmost)withtheevaluatIOn ofthetermsthanitisdesired to proceed forthesum.Thenumber noftermstakenmtoaceo,:ut WJllbechosen solarg-ethattheremamtler Y"contributes atmostoneunitinthelastdecimal placeconSidered. Theerrormtheindividual termswillthen,ingeneral, have noappreciable effect. Buttoobtainperfect security fortheresulting dig-its, itisnecessary toproceed asdescribed above. Forwcmayretainalarge number ofdigitsbeyond thedesired number mcalculating theIIldividual terms, -yetasanerrorattaches toeachofthedecimals broken offandtheseerrors accumulate, theymay,inparticularly unfavourable cases(cf.theexample 011 p.249),influence someofthemuchearlier digits 4.Evaluation ofthenumber:r.Thechiefmean')placedat ourdisposal, uptotheprc!:>ent, fortheevaluation ofthenumber :n, aretheseriesexpansions ofthefunctions tan-1andsin-1;ofthese, theformerhasthepreference, owmgtoitssimplemodeofformation. Fromthisseries,wededuced theexpansion 1 1 1-;r:n=1- 3+5 -+"', whichfornumerical purposes ispractically valueless. Infact,by p.250,wecansaynomoreoninspection abouttheremainder rnin thisexpansion, thanthatithasthesign(-l)n+1andisinabsolute value<-213'Inordertosecure6decimah, weshouldthereforen+ beobliged totaken>106,butanevaluation ofamilliontermsis, forpractIcal purpose'i, quiteimpossible. Therapidity oftheconver­ gencemaybeincreased verymaterially byEuler's transformation 144,2.Inthenextparagraph, weshalldiscuss theutilityofsuch transformations forpurposes ofnumerical calculation. Ourpresent objectistodeduce moreconvenient seriesexpressions for:ndirectly fromthetan-1seriesitself. Theseriesexpansion fortan-11_=!!-isalready ofappreciableV36 use:thisgives !!...=_1_fl-_!-+-.!_---_1-+ -...J6 V-:-~ 335·3"7.33• •3Cf.p.249. §84.Numerical evaluations. 253 Thelollowingmodeofprocedure, however, provides considerably more convenient series 24. Thenumber _}1 1 1 1 1,,=tan5=-Ij--3.53+5.56-7.5'+-... iseasilycalculated fromtheseriesitself(seebelow). Forthisvalue ofet,tanet=~,andso tan2et= __2~anex 5I-tanJa I~ and Consequently 4etexceeds wehave120tan4a=m' ~-4-byonlyasmallamount. 4"-~=P,Wnting HencefJcanveryea'illybeevaluated fromtheseries _._}1__1 1 1 fJ-t,m239-239-~-239"+-.... Thetwonumbers (;(,andfJgIveus n=4(4,a-fJ) =l(j.[~-:J.~a+a.1f)iJ-+···J -4L~!)-3.21 ;19:1+_...].146. Ifitbede~lred toobtainthefirstseve"truedeCllilals 0/n,wcmay endeavour toattainthISendbytaking, say,9decimals foreachoftheterms andfortherem:llnder~6 - a,canty enoLlgh margm, fortheerrorsIncurred onthenumbers exandfJhaveultllllately tobelllultlplied by16and4respec­ tively Denoting thefirstseriesbya,- aJ+ab-+.."thesecond by a,'-aa'+a.'-+"',andthecorresponding partial sumsbys~andso',the calculation proceeds asfollows: 0,~0200000000 a,--o0000(i4000 (/"0-...••..fi7'-03=0002(WG(i07­ (/,-,000000IS2l)­ (/"-0·........2- aII-a,+a.--020001i4057- a,-I-a,+all Hence, astheerrorschanges'gnsinasubtractIOn, s"=0·197395559+++- and0<rll<10111 Accordingly 3158328930 <11)IX<3,158328970,o01l2liliS-!US--- ".f.M(/chin (inW..lanes: Synopsis, London 170li). 25Theresultalonecanshowwhether thISsuffices. Infactwedonotknow aprioriwhether wearenotinthepresence ofoneoftheparticularly unfavourable casesdeSCribed onp.24!!. 9- (051) 254Chapter VIII.Closedandnumerical expressions forthesumsofseries. foraftermultiplying by16wehavetosubtract1;=8unlt~ofthe9thdecI. malplace,oradd'\,fl.=24ofthe~eunit,>,toobtainbounds oneitherside for16s1l'Since - 0<l(j"11<2·10-°, wehavefinallytoadd2unit,totheboundabove,toobtainthecorrespond­ ingbounds of16IX.Further a/=0004184J00+ a."=0· 024+ aI'-~:-c=OOI)4184076± hence - 0016736307<-4P<-0'016736302. Combining thetwofl''iults, weget 3·141592629 <n<3'141592668.0<,./<10-1', Thisbriefcalculation thusreallygive,>usthesevenfirsttruedecimals ofn: n=3'141[,926 ., (Thesameprocedure wouldonlyhavesecured SIXdecimal, fortheapproxI­ matevalue;cf.calculation of/I,wherecirenmstances, IIIthisrespect, were theexactreverse) Theserieshereutilized forthecalculation ofnareamongthemo~t convenient; bytheirIIle:tl1S, averymuchgreater number ofdecimals may alsobesecured's withrelatIvely smalltrollblC" andwearetherefore fully justified inregardmg nhenceforth asoneofthe"known" numbers. 147. 5.Calculation oflogarithms. Thestarting pointforthecal- culation oflogarithms reSIdes inthesenes (lxl<1). Thisseriesconverges Withconsiderable rapidity forx=~,andat oncegives log2=2 [~-+-3~3'3+5~:P+...J. Denoting byao'at'...,thetermsoftheseriesinsidethesquare bracket, wehave I an=-(2n+I)-3",+1 and or ••Thenumber 7Thasbeenevaluated to810placesofdecimals (Mathematical Gazette, Feb.1948,p.37). §34Numerical evaluations. 255 Ourcalculations thenproceed a~follows, Ifweagaintake9decimals foreachofthetermsan' ao=0333333333+ al~=0012345679+ a~~000082304;)+ aa=0,065321+ a.=()'00;,645+ ar•=~0· 5J3+ all-0'..'"()48+ a,=0',.00;,- [",<:0' 001J______ _0.--__-o:146573[,89 Whence itfollows, taking- intoaccount theremainder andthesmall+and -sig-ns: log-2=0'6931471 ...orlog2R::0'6931472 withsevendecimals secured'7. Oncelog2ISevaluated, thecalculation ofthelogarithms ofall othernumbers involves verylittlefurthertrouble. Infact,our"cues . f JgIVes,orx~=2pT1' [1 1 J ]log(p+1)=log])+2'!.1J+1+a(2p+1):\+5(2'/)+I)"+...;148. therefore iflogpISknown(p=2,3,...),we(JbtallJthevalueof log(p+1),bytheabovefonnul.1. Moreover, since-21 1=+,71 I•••,1'+ .) theexpression involves aseriesconverging i'cryrapidly. Infact (cf.above,casep=1) 0<r< 1 . 1 <__an_ n(2n+:3)(2P+1rn+a1-1__41'(1'+1)' (2Pt-1)2 sothattheremainder ISalready verysmallforquitemoderate values ~lfn.TherapltlIty ofconvergence ofcourse il1crcase~ whl'npis givensomewhat largervalues, 1.e.assoonasthefirstfewloganthms h,lvebeensuccessfully determined. ItISusefultoobserve thatby :17,1,onlylogarithms ofprimenumbers 2,3,5,7,11,13,...needbe evaluated; thoseofallothernumbers followbymerecombination. Nowsupposing thatwehaveeffected thecalculations forthe logarithms ofthefirstfourprimenumbers, 2,3,5,7,thelabourin. volvedincalculating theloganthms offurtherprunes issmall.Thus, forinst.lIlcc, takingp=10,wehave log11=10"2+Iocr5+2['1+1_-1_-_!-+ ] b b 213.21315.2P... with , Oan<'n<fC40' 27Theseries1- ~+k-~+...for109"2isofcourseinappropriate for theevaluatlOll ofthisnumber; evenitsEulcr'stransformation effected in144,1 islessconvenient thantheseriesutilized above. 256Chapter VIII.Closedandnumerical expressions forthesumsofseries Thusalready oforn=3, 1 1 1 1 rn<7-:-21"711-=-40 <-20R:2~Yf~-i<10".2''-'--7<-101ii ensuring adegree ofapproximation sufficient evenforthemostrefined scientific needs. Itwouldaccordingly appear desirable topossess somewhat morecon­ venient methods ofcalculation forlog2,log3,109"5,andalso,atanyrate, log7.Diverse artifices maybeapplied forthepurpose, allofwhichconsist malllly infinding rational number~!._,asnearaspo~sible to1,whosem numerators anddenominators areproducts ofpowers ofthesefirstfourprimes. Ifqoftheseprimeshavebeenutlli7ed, qfractions willbeneeded todeduce thelo~anthms ofthoseqprimes fromthoseofthefractions. Foractually effecting thesecalculations, itisconvenient tofollowthemethod indicated . 102581byAdams28:Evaluate theloganthms of-9'24'SObymeans, notoftheseries 120,cjustemployed, butoftheoriginal series120,aandb,whIchheregive log~~=__la"(1-~) =-.!...+_1_+_1,+...9 ~10102·10" 3.103 25 log24 81 log80= Owingtotheoccurrence, inthedenommator, ofpowers of10,thecalculatior. herebecomes extremely simple \\<'iththeaidoftheselogarithms, wethen obtain,asmaybeverified immediately: 10 25 81log2=710<'- -2log-+3log,--...9 24 RO ]0 25 81log3=1]log--9--3log24+5log80 10 25 81 log5~1610g"if -4log24+71dgffo' Ifweproceed further toevaluate, aswemaywitheql:alfacilitY,3' log~2~=log(1+_8 __)=~_.!..~~+.!..~ _+...125 101)0 1032106310° • wealsoobtain 10 25 81 121ilog7=1910CT-- -410g-+810g---+Iog-",9 24 80 125' 28Proc.oftheRoyalSociety, Yo1.27,p.88,1878. 2.Thefacility withwhichthiscalculation iseffected maybeseenby thefollowing, whichin5simplelinesprovides log~;~with10decimals secured: +0'0080000000001 -0 032000000 +0: 170667- 1; -0 001024 +0· 007-]26 log125=0'0079681696 ... §34.Numerical evaluations. 257 Wehavethus,fortheactualcalculation ofnaturallogarithms, amethod which isconvenient andeasilyapplicable inpractice. Intofurtherdetailsofthecom­ putation oflogarithmic tableswecannotenterinthisplace. Havingobtained log2andlog5,wehavealsothevalueoflog10; andhence,in1. M=log10=0'4342!)4Mll90.., the"modulus" ofBriggs' system oflogarithms tothebase10,or factorbywhichthenatural logarithm ofanumber mustbemultI­ pliedtogivetheBriggian logarithm 30• 6.Calculation ofroots.Oncelogarithms havebeenma.,tered nogreatpractical importance attaches totheproblem ofobtaining simplemethods ofcalculation fortherootsofnatural numbers. We shalltherefore bequitebriefIIIthefollowing eXpLll1atlOns. Thera­ pidityofconvergence ofthebinomial series 'P-.! increases asIxIdiminishes. Nowthecalculation ofapowerVq=qP I canalwaysbereduced tothatofapoweroftheform(1+x)p-,",ilh somesmallvalueofIxI. Afewexamples mayservetoillustrate theabove.allp.211,wegave149. . . 7 ( 1 )-tfor,/2theseriesexpansIOn of-5I -.-:..)0 ..)2=2[1+Ll_+~. __~_+~_3.S._1+...J. I)2502·450"2·4·(j50" Since(-I)"(-})iscon~tantly positive andformsamonotone decreasing se· quence, theremainder r"maybeestimated bymeansoftheinequality 0<rn<an'([jl0+I)~"+...)=:;j, ~howing that,evenforsmallvaluesof",acon~iderable deg-ree ofapproxi. mation isattained Jl.Themethod isevenmoreeffective ifwcwnte ( -~- (~rgiJ(11) \'=70+IJl:lUO'-141(119)-~ or..)2=1001-2000-6 I 30'Vemayremark inpassing thatwehavecertainly foundample jUSll' licatlOn, bythistime,forwhatseemed atfirsttheratherarbitrary designation ofthelogarithms withtheremarkable baseeasthe"natural" logarithms. 31How ~implythecalculation proceeds isshewnbythefollowing details: au-I-al=1010 01hence-indeedWithout anyerrorI a.=O· 15 0 5.=1-010152544;,375 a3=0·25.;0f 0<'.<17·10-I" a4=0·, 4.175.0 V2 a.=0· 7875 2=1-4142135623 , bywhichthefirst10deCimals arethusalready secUlcd. 258Chapter VIII.Closedandnumerical expressIOns forthesumsofseries. orothersimIlar expressions, obtained bytakinganyroughapproximation Q tof2(~~inthefirstcase,1·41inthesecond). andputting f2=aV~· Since a~iscllO~en tobeverynear2,thequantity underthe{"isoftheform 1+x>withsmallIxI.-Sinlllarly, Ifwearealready awarethatvS=1'732.", wehaveonlytowrite vs=1'732~~2)i=1732[1-30~~~00rl toobtain,withthegreatest ease,anexpansion of";T3to50ormoreplaces01 decimals. Wemay,without further explanation, indicate theexamples: --10( 1)t-18( 1 )-~ ~1l=3I -fo-6'VI3O~51-325 3-.5 ( 3 )~s..10(29)1V2="41+I25' V:3=71+1006. 150. 7.Calculation oftrigonometrical functions. Theseriesexpansions ofsinxandcosxconverge withevengreater rapidIty thantheex­ ponential series,sinceonlytheevenoronlytheoddpowers occur inthem,andthesehave,moreover, alternatmg signs.Accordmgly, no specialartifices arerequired; foranglesofno['xcessive magIlltude, theseriesfurnishallthatcanpossibly bedesired. Todetermine, forinstance, sm10,wchavefirsttoexpre~slOin circular measure. Wehave10=1~0=0'017453292 ...•i.e.ccr· tainly<5~'Denoting thisquantity bya, 3 ; sin1()=ex-~-i+~!-+...~ao-al+a!-+..., andtheerror1'"mayatoncebeestImated (p.2:)0,il)by °()"+1 a2n +3 <-11'n<(2n+3)1' whichlastexpression isalready lessthan{-·10--1~forn=2. Circumstances aresimilarinthecaseofcos10;thISquantity may also,however, sincesin21°<25100'beobtamed eaSIlyfromtherelation cos10=(1-sin210)l bymeansofthebinomial series:-tanxandcotxarethenobtained bydivision, orfromtheirexpansions 116and115,whoseconver· genceisstillquitesufficiently rapidwhenIxIissmall. Theselatterseriesalsoleadtousefulexpansions forthelog­ arithms ofsinxandcosx,-whichforpractical purposes areof -§34.Numerical evaluattons. 259 greaterimportance thanthevaluesofsinxandcosxthemselves. We have32(cf.§HJ,Def.l:l) x logsinx=logx+logSi;x=logx+f[cotx-~]dx o '" k22k.B'k=logx-t-"(---1)- -x~k A--::l 211.(2iI)! andsimIlarly from116 r f-n 2"(22k_1)BH 2-logcosx=tanxdx=.20'(_l)k-l --------- Xk. A=I 211.(211)" o logtanxandlogcotxmaybeobtallled fromthesebysunpieaddition. Asregards theconvergence oftheseseries,\\ecanonly ~tateinthe firstimtance thattheycertainly doconverge forallsufficiently small valuesofIxI·However, theremarks ofp.2:37,footnote 4,showfurther thattheseriesinunhastheradiusn,tlutin152theradiusi. Further detaIlsinthecomputation oftrigonometncal tableswill notbeentered intohere,astheydonotconcern thetheoryofin­ finiteseries. 8.Moreaccurate evaluation ofremainders. Inthecasespre­ viouslyconsiclel ed,thesumofagivenscneswasinvariably deduced byevaluating suitable partialslimsandestimating theerrorinvolved inthecorresponding remainder. Itisobvious thatthi"method isim­ practIcable unlesstheconvergence ofthe~eriesisrelatively rapid.If itbedesired toevaluate, withsomedegree ofapproximation, for instance [hisdirectmethod isprettyhopeless 33.Evenifwearcverycautious IIIthemarginweallow,wccanonlydeduce, asanupperestimate oftheremainder 1 1'n=(n+l)2+(n+2)2+..., 3'Thefunction inthesquarebracket hastobeunderstood tostandfor 1theserieslUiafterdiviSion byxandsubtractIon oftheforemost term Thefunction istherefore defined andcontinuous alsoforx=O. 2 33Aswehappen toknowthatthesumis~,itsevaluation indirectly bymeansofthevalueof1risofcoursequitesimple. Butforthemoment we areassuming thatweknowaslittleaboutthis~umase.g.aboutthesum of2,'~ii'n131. 152. 260Chapter VIII.Closedandnumerical expressIons torthesumsofserie!l. theinequality I 1 1 "n<n(n-I-I)+(n-I-I)(n-I-2)+...=n; according tothis,itwouldbecome necessary tocalculate amillion terms,inordertosecure 6placesofdecimals. Thisofcourseisout ofthequestion. Thi"stateofthingsmayfrequently beimproved tosomeextent, ifitispossible tosupplement theUppf'restimate oftheremainder rn byalowerestimate, i.e.todeduceaninequality forr"ofopposite sense totheaboveinourcase.Inourexample, thesamepnnciple asthat already usedgives 1 1 1"">(n--~f)-(n-l-2)+(n-l-2)(n-l-3)+...=n-l-1; wcarethusabletoassertthatoursumssatisfies theconditions 1 1 1 1 1 1 1--I-if'+...+n-.--I-n-I-1<s<1--I-2"--I-..•-1-n"--I-n' foreveryn.Tosecure 6decimals, wemayaccordingly needonly 1000terms. ThisisstIlltoolargeanumber forpractical purposes. Butinspecial examples thismethod ofupperandlowerestimates 01therema£nder (cf.Ex.1:31)mayleadtoasatisfactory result. Thesecasesare,however, sorare,thattheydonotcomeinto account forpractical purposes. Greater importance attaches tomethods fortransformation ofslowlyconvergent intorapidly convergent series, because theyadmitofafarwiderrangeofapplications. Tothese methods weproceed togiveourattention. §35.Applications ofthetransformation ofseries tonumerical evaluations. Incasesofslowconvergence, onenaturally attempts tochange thegivenseriesintooncwithamorerapidconvergence, bymeans ofsomesuitable modification. Weproceed toexamine inthislight thetransformations discussed in§33,soastoseehowfartheywill beofusetoushere. A.I{mnmel"s transformation. Forthistransformation itisim­ medIately obvious whether andtowhatextentanincrease inthera­ pidityoftheconvergence canbeobtained byit.Infact,usingthe notation of145,wehave ian=rC+i'(1-r~n)an; n=O n=O n as(1-?'~:)-+(),thetermsofthenewseries(fromsomeinclexon· wardsjarelessthanthoseofthegivenseries.Themethod willac· §35.Applications ofthetransformation ofsenestonumencal evaluations. 261 cordingly beallthemoreeffective the are,fromthefirst;orIDotherwords, aretothoseofXan' Examples.smaller thefactors (1_t'en) all thenearerthetermsof2'en liS:! 1.Wefoundonp.247that.2~=1+2J-2-(1 1)-'Thetermsofthen nn+ newseriesare a~Yl1lptotically equaltothoseofthe~eries rJ:n(1\+l~(n+2)=-}n-E(n(n1 +1)-en+1)1(n+2»)=-{-; thushereC=~andr=1,andso4 {aarbitrary -1'0,-1,." p,integer 2:;1Thelatterseries,evenformoderate valuesofP,shows convergence. 2.Consider thesomewhat moregeneral series co co 1\'a=,-,-c------o-:;--c-----=c--------~ n~O11-n':;:O(n+Cl)2(n+Cl+l)"· ..tn+a+p-1)S> Herewetake Cn=(n+y)an-(n+1+y)an~1, n=0,1,2,..., andwctrytodetermine y(independent ofn)sothat Cnisasnearanas posslble34•HerewchaveC=yaoandasimplecalculation givesr=__1_.2p-1 Henceweobtain ~1=1'11"-__"(1-en+y)all(It+1+y)an+1) "-Jan2p-1f-......, (2p-l)a an'n=O n=O n TheexpressIOn inthelargebracket is 1_(n-+-y)(n-+-a-+-P)2-(n+1+y)(n-+-a)B (2p-l)(n-+-a+p» ' Since,bysimplificatIon, thetermsInnBanunBmustdisappear ofthemselves. thisgives (2a+3P-2 - 2y)pn+(2P-1-y)(a+P)o+a2(1-+-y) (2P-1Y(n+a+p)" 3Ifwenowchoose ysothatthetermsinnalsodisappear, i.e.takey=Cl+2P-I' thentheexpression inthclargebracket abovenowbecomes pS 1 2(2P-1)' (n+a+p)B' ".Thechoiceofanumber enoftheform Xn---InClwill,by131,always pl'ovemostconvel:ient, asinthatcaseCatallYratemaybespecified at onceandthechoicestillbesoarrangeu thattheen'sareneartothean's. 262Chapter VIIl.Closedandnumerical expressions torthel>umsofseri!:s. andaccordingly ThetransformatIOn thushastheeffectofintroducing anadditIOnal quadlatlc factorinthedenominator. -Particular ca~('s: a)ex=1. WriteOD 1 n~'en+1)'(n+2)9• , •(tl+P)'J 3 1 '2P'2p-l p' 0011 12"?X:-.----:·ji-+~f(2P-1)n~o(n+1)'...(n+W(n+P+1)11 forbrevity OD 1 00 1 A~k"(k+l)"~(k+,p--j)" ==n~~(n+1)'...(tITp)2 =Sp; therebultthentakestheform 3p P' 5p=2(2-p-'=1)~ P."2"-:-:'.·p';'-+2(2P_1),5p+l' Thisformula enables useasilytoobtainveryrapidly convergent seriesfor \--.1 5=51=,,;.Jn"' Thisformula similarly leadstorapidly convergent seriesfor Forfurther examples, seeExercises 127seq.1b)Similarly, forex='2': i 1n=O'(2n+1),---CC20-1-1+~3CC-).0-,.-.-.(=2-n-+--=2-p--~1 )-=-. 3p-l 1 2p' 00-, 1 =2(2P-=I)"T'-=32•••(2P-1)'+2"P-=-in~o(2n+1)'...(2n+2p+1)9' 1 27(211+1)' B.Ettler's transformation. senes Buteventhe thetransformation ofn~(~rgives evidently hasalessrapidconvergence,1<r(3)n 2n~'4'whichEuler's transformation 14/1neednotbyanymeansIl1volve an increase intherapidity ofconvergence35oftheseriestowhichitis applied. 134. For1I1stance a.Theexplicit definition ofwhatwemeanbymoreorlessrapidcon. vergence Willbegivenin§37:2'an'ISsaidtobemoreorlessrapidlycon· vergentthan2'an,according as Ir~I=I~':_+.!~!-,.a..~+.~ -t...:..-.:1--0or--+00. "na.+l+an+I1+··· ~35.ApplicatIons ofthetransformation ofseriestonumerical evaluations. 263 "serieswiththesamerapidity ofconver-"inthecaseofalternating serie~,theeffectneednotbeanincreased rapidity ofconvergence; indeed thefollowing threeexamples show thatallconceivable casesmayactually occurhere: '" 1 11.~'(-It2ngivesamorerapidlyconvergent series,'2 n=O 2.i'(-1t-b n=O...1 gence,T..1':F' n=O ...("1 l'ell . 1 ~(3)"3..2)-1)-4"" "essrapIyconvcrgent sencs'2 n"':-'__'oH. n=O Wcshallnowshow,however, thatsuchanincreasl: intherapi­ dityoftheconvergence doesresult,intheca'>cofthosealternating series2,'(-1)"an,an>0,whoseterms,thoughnotshowing rapidity ofconvergence, stilltcndtozeroinaparticular regularmanner, which weproceed todescribc. Thesearetheonlytypesofalternating series ofanypractical importance. Thehypothesis required willbethatnotonlythenumbers a" formamonotone decreasing sequence, i.e.havepositive firstdiffe­ rences LIan'butthatthesameistrue of alldifferences ofeveryorder. A(positive) sequence ao'ai'a2,•••issaidtohavep-foldmonotony36 ifitsfirst,second,...,pthdIfferences areallpositive, anditissaid tobefullymonotone ifallthedifferences LlI,an'(k,n=0,1,2,...) arepositive. Withthesedesignations, thetheorem referred toIS: Theorem 1.If1;(-1)nan2Sanalternating seriesforwhich15:i n=O the(Positive) numbers ao'ai'...formafullymonotone nullse a 1quence, while,fromthefirst,.nII?:a>2(foreveryn)37,thenthe a" tra~lsf()rmed seriesE21~+J.LJ"aoconverges morerapidly thanthegiven senes. aTheproof isverysimple. As,,+1:-;a,wehavea,,:::;;au•an. Cl" Further, fortheremainder rnofthegivenserieswehave (__1)n+lrn=anH-an+2+-...=LJan+!+.1ann+.1an+5+..., hence,since(.1av)isitselfamonotone nullsequence, II'>1(A A+A+ )_1>1 . 11+1rn:--=2.uan+l+.uanI2 .ua"+3 •••-2an+l=2ao a • 38Cf.Memoir ofE.Iacobstlzal refcrred toin144. 37Thisassumption isthepreciseformulation oftheexpression usedabove, thatthegIvensenesshouldnotconverge particularly rapidly. TheseriesWIllIn fact,astheexample showsmoredistinctly, converge lessrapidly than1:(~)n. Cf.furthertheworkbyF.V.Poncelet quotedinfootnote 13. 264Chapter VIII.Closedandnumerical expressions forthesumsofseries. Ontheotherhand,asAnan-AnilaO=Anal20,thenumerators ofthetransformed seriesalsoformamonotone nullsequence, and inparticular areall~ao'Consequently theremainders rn'ofthe transformed series,-which,moreover, isaseriesofposItive terms, satisfy ,if"+1ao ~ao(+1+1+) aorn=~~2n+T+"'22"+2 1 -2~4"...=2"+1' Consequently, wehave I!!.~\<~(L)" "1£=:a2a ' whichproves ourstatcmcnt completely. Further, weseethatthe largerais,thegreater willbctheincrease 111therapidity ofcon- vergence, i.e.themorerapidly will~{-+O.Inparticular, wemay f'" transform intoserieswhichconverge withpractically thesamerapid- ityas27(~)",allalternating senesforwhichtheratiooftwocon­ secutive termstendsto1inabsolute value;suchserieshaveusually aslowconvergence. Examples. Thetwomoststriking examples ofEuler's transformation, (_1)" (-I)"thatof'"---and'\'--- wereanticipated in144.Forfurther appli-~n+l ~2n+l' callons itisessential toknowwhichnullsequences arejullymonotone. We mayprove, inthisconnection, byrepeated application ofthefirstmean valuetheorem ofthedifferential calculus (~19,Theorem 8),thefollOWIng theorem: Theorem 2.A(poslllve) sequence ao'ai''"JSjullymonotone decreasmg ~j ajunctionf(x)eXIsts,dejmed jarx::::::0,andpossessmg dljjerenllal coejjlclents oj allordersjarx>0,jarwll1chf(n)=a"whIlethekthdeYlved functIOn hasthe constant sIgn(-1)k,(k=0,1,2,...). Accordingly thenumbers en,(O<a<I);1 (n+p)a'(p>0,a>0);1 log(n+P)'p>1; forinstance, formfullymonotone decreasmg sequences; andfrom the~emany furlhersequences ofthISkindmaybededuced, bymeansofthe Theorem 3.Ijthenumbers ao'a"...andbo'b"..•constItute jullymono­ tonedecreasing sequences, thesame ~strueojtheproducts aobo,a,bl,a2b2t••• Proof. Thefollowing formula holds,andiseasilyverified hyinduction relatively totheindexk: .kb~(k).k-" '~b LJann=.:;:o V£Jan+v·£j 11- Itshowsthat,asrequired, allthedifferences of(a"b,,)arepositive, ifthose of(a,,)and(b,,)areso. Thefollowing maybe!Iketched asaparticular numerical example: Theseries co 1 1 1 .J)(_1)"a"=log10-IoU"rr+log12-+... ft=O e ~35.Applications otthetranstormation ofseriestonumerical evaluations. 265 hn~extraordinarily slowconvergence; infnct,itconverges withpractically a'lsmall arapidity asAbel'sseriesI11n(logn)". YetbymeansofEuley'stransformation, itssummaybecalculated withrelative ease.Ifweuseonlythefirstseven term'!(to----~1.inclusive), wecandeduce thefirstseventermsofthetrans-log{) formed series.Ifweuselogarithms tosevenplacesofdecimals, wefind, with6decimals secured, thevalue0221840...forthesumoftheseries"s. C.Marlwff's transformation. Asthechoiceofthearray(A),p.241,fromwhichMarkol!,s transfurmation wasdeduced, islargely arbitrary, itisnotsurpnslng thatwcshould beunabletoformulate general theorems astothe effectofthetransformation ontherapidity oftheconvergence. We shalltherefore havetobecontent Withlayingdownsomewhat widcr directing linesforitseffective use,andwithillustratll1g thisbyafew examples: Denoting asbeforeby.:EZ(k)thegivenseries(assumcd convergent), wechoosethetcrmsoftheatllcolumn inourarray(A)tobea'i ncaraspossible tothoseofthegivenseries,andatthesametime topossess asums(O)whichwecanindicate byaconvenient closed expression; thisisanalogous tothecondition ofKummer's trans­ formation. Theseries ~(z(k)-ao(k»)nowcertainly converges more rapidlythan2'z(I');proceed withthisnewseriesinthesameway,for thechoiceofthenextcolumn inourarray,andsoon.Theeffect ofthetransformation willbesimilartothatofanindefinitely repeated Kmnmer's transformation, -thepossibility ofwhichwasalreadyindi· catcdintheexamples 153,2a(cf.Ex.130). Asanexample, wemaytaketheseriesikt.,whichispractically useless156. k=l 2 forthedirectevaluation ofitssum~.Herewethinkoftheothrowand column asconsisting entirely ofnoughts, whichwedonotwritedownThe choiceoftheseries2k(/-tI) fortheI~rstcolllmn, whichwasalready used onp.247,thenappears obvious enough. Thisgives ~(Z(k)_a(k)=~__1_ ~ I-~k2(k+1). Assecondcolumn, weshallthen,asin1iS:I,1,choosetheseries y' 1 .:...Jk(k+l)(k+2), andsoon.Thekthrowofthearraythustakestheform 1 O! 11 2! k"k(k-+-I)+liCk+Ink+-2)+k(k+-I)(k+-2)(k+-3)+...(kfixed). Thefurther calculations are,however, simplified bybreaking' offthisseriesat the(k-1)lhtermandadding' askilltermthemissing' remainder rk'after 3.Thisexample istakenfromtheworkofA.A.Markoff: "Dlfferenzen­ rechnung", LeipZig, p.184,1896. 266Chapter VIII.Closedandnumencal expressions torthesumsofseries. whichtheseriesisregarded asconsisting entirely ofnoughts. Theklbro\'f nowhastheform: 1 O! I! (k-2)1 kg 11(k+1)+k(k+1)(k+2)+...+k(k+f):-~~(2~k-=-f)+ rk' Subtracting thetermsoftherighthandsidefromtheleftznsuccessIOn, weeasilyfind (k-I)! rk=k-2(k+1)...(2/~1). Inourcase,theprocess ofsplitting uptheseries'\'~~-intoanarrayofthe--'k2 11 +223 11+-­3·4·5form 1= 1 2J 1 32(A)ofp.241thusgives: 1 01 23 01 3421 +:f°-:-4.5 101 11 (k-2)! (k-l)! k"=k~lk+1)+k(k+l)(k+2)+...--J-k(11+1)~2 k-1)+11'(11+1)..-.(2~n ~inceallthetermsofthiSarrayare>0,themainrearrangement •;If2 theorem90itselfshowsthatwemaysumincolumns andmustobtam-Ifas ultimate result. Nowinthentbcolumn wehavetheseries r..+(n-l)! [(n+l) .1.(2-1~+i)+(n+25 ..\2n+-2)+· ..J,(nfixed). By132,3foret=n+1andp-~n,theseriesinthesquare brackets has thesum 1 n(1I+1~2n' Hencethenthcolumn hasforsum s.nJ=(n_l)1 [., 1 +__IJn"(n+1)...(2n-I)n(n+1)...2n =3(n-l)! ~__=3~n-l~. n(n+l)...(2n) (2nl) Therefore wehave 001 00(n_l)12I-..=3.2:(2), •k=lk~ n=1 n• Thisformula issignifieant notonlyfornumerical purposes, inviewof theappreciable increase intherapidity oftheconvergence, butalmostmore sobecause itprovides anewmeansofobtaining theclosedexpression forthe 1sumoftheberies.2k2'whi<,hweonlysucceeded indetermining indirectly byusingtheexpansion inpartial fractions aswellastheseriesexpansiO;:l of thefunction cot.Infactwecaneasilyestablish directly (cf.Ex.123),that 123implies theexpansion, forIxI.:;:1: (sin-1x)2=..!-i'(n_l)2 (2X)211. 2..=1(2n)1 E1crcises onChapter VHf. 267 1 PUtlillg X=-i'wcatoncedednce 3. \'~=3\'(n-~)~=32'(-6~)9 =.7t(.)_.9. k-;:l'~ 2n':;;:l(2n)I Afurther applIcation offundamental importance ofMarhof/'s transforma­ tionwehavealready comeacross (v.144)inEuler'stransformation, which wasindeeddeduced fromMarhof/'s. Forfurther applications ofMarkof/,s transformation wemustrefertothe a('count~ of1'I'farkotf himself (v.p.\:'G:i,footnote 38)andofE.Fabry(Theorie des series;\termes constants, ParisUnO).Theirsucccss depend~ forthemost partonspecial artifices, buttheyaresonwtimes surprisingly effective. Nu­ merous examples willbefound,completely worked out,inthewritings re­ ferredto. Exercises onChapter VIII. T.Direct formation ofthesequence ofpartial sums. lOO.X2x24x'8x· x a)l+x+r+-Xii+I+x.+r+x.+···=l-=X for x x" x'x·j1~-; b)I-x' j-T=-x.+T--xH+1"=-;'0+...=__.~ x-IforIxI<I. forIxI>1. 103.~ an101.~-- "---.-- is,foranpositive. znvartably convcr- n=1(1+aJ)(I+ag)..•(I+an) !!entWhendoestheseriesstillcontinue toconverge forarbitrary an.and whatisit~sum? ex;, 2311'102.a).2)hm-1-=-; n=l 7'124 (hint:tan-l~1- tan--1~I=tan-12.). 7'1- 7'1+ n' ex;,l:Itb).2)tan-1---- =-. n=l n2+n+l 4 ~ 71 1 1 a)n::I(.c+IH2x+I).'-:Tllx+l)=X-' ifx+O,-1,-2'-S.··· b)1+:l: x(x+l) Iify>x>O.yy(y-tlj+y(y+I)(y+2)+..=y-x' c)1+a.,.aCa-t...!2+'!Ca+.J)Ja.J:. 2)1-'"=.b.-1__ b b(b+1)b(b+l)(b+2) b-a-1' itb>a-j-I>I. 104. -hI+hI-b..!.+(hI-l1i~-b)•.!.+...=.!.,ifbof0, 1hIh2 ht•hg Ha b everyh">°and2,;'k:isdivergent. a9Cf.anotcbyI.Scliurandtheauthor: "OherdieHerleitung derGlei­ chung.2):.=~9..,Archiv derMathematik ulldPhysik, Ser.3,Vol.27, p174.1918. 26BChapter VIII.Closedandnumerical expressions torthesumsofseries. 10~.a)""1 :J<4.2)2ntan2~=7i; n=O ""1x1b),-,~tan~= - -cotx. ,,'7:12n 2nx ' (hint:coty-tany= 2cot2y)• 106In~ g(n) letPI'pz•..,f}~• n~o(a+PI+n)(cc+pg+n)...(a+Ph+n), denotefixedgivennatural numbers, alldiHcrent, anda=l=0,-1,-2,".any realnumber, whileg(x)denotes anintegral rational function (polynomial) of degree<k-2.Weassume theexpansion inpartialfractions: g(x-a) Cl Ck (X+PI)~X+P~j= X+-1:\+"'+ x+P~' Thegivenseriesthenhasthesum _;,CO'[2.+_1_+...+__1__J. ,~ aa+1 a+p_-1 107.a)1.2~6~i-3.f\-.9+5:-G-~~TI-+...=6~(.7C-16~); 1 1 1 5 1 • b)12-4.5-i3T67+5.6.8.9 +"'=S6-tilug2j 1 1 1 1 c)1.2.4.;' +3.r0:7 +5.6.8.9+...=36; 1" 3" 5" d)14+4-3 4+-4+ 54+4-+...=0; 1 I 1 :1r e)1CF+4)-n~4+4)+5--=-(54-+4)-+...=16 1 1 I 1 f)1.(4.14+1)-2~4--:-24+1)+3:"[T:.:.-fl)-+...=log2 -2'; 1 1 1 n g)1.2.34+5-""6.7.8+···="4log2-24; 1 1 1 n./-1 h)r:-2~+4.5.6+7-:-8~+...=12V3 -4log3. 11.Determination ofclosedexpressions bymeansoftheexpansions ofelementary functions. 1 1 1 1 ,.,-,10S.a)1-53'-7,3"+11.3'+13:-i3"0--+ +...=logV7 , 1 11·311,3·51b)_.-+_._+-_ ..-+...=log2j2 22·442·4·66 1 1 1 1 1 ."c)1+3-s-'f+g+TI--++'" ="4v'2; I I I I I nnzcl)-+--+--+_._-+- -+"'=-'C01-xx-yx+yx-2y x+2y yy givesfory=7andx=1,2,3: ] I I 1 1 !- !~:1r 1+"2--3+"4-"5-"6+ 0 + + -+- -+ 0...=,/7' 109. 110. 111.IfExercises onChapter VJII. 1 1 1 1 a)1:-2-=-3+ ~4-:S+5.6-7+...=log2-~; 1 1 1 1 b)1.2-3-3 45+~6-7-+"'=2"(1-log2): c)__~__1+__1_+...=}_(n-3); 2·3·44·5·6 67·8 4 001 1 001 n9-8 d)2-,'40n-!~i=2"'2-'(4n9-=-1;2=~,n=t n=t 001 32-3n3 .2(41Z"-lV'=--64- ;n=l 1 1 1 1 1 nc)1--+---+-----+-··· =--.5 7 111317 2J3 1·21·2·:3123·4 n a)1-:-3+13.5+T.i3:5-:'i+"·=2; 11·2 1·2 3 2,n b)2:-11+34:-,';;+4.5:-67+"'=--- _-1;3v!i:l 1 1 2 1·23 n91c)-+---+- .-+...=---;2·343·4·,5·6 4·,5·6·7·8 182 001 d),,------ =210g2- 1i n--::'ln(4n9-1) 00 1 3e)'\'-._.---=--210g2' n~lnI4n9-1)9 2 ' f)~,1 - ~9_~(I2'9 -.J2n.n2-12 2og) .n=l QC(nI)9wewrite2)-(--)-, =Tp,then n~Op+n. _:It" _,.939 _59197 T"-3-3ITa-4--I6'T.-54:1t-216.269 conP112.Ifwc\\riteLJ-/=gpe,(P=1,2,.,.),thenthenumbersrpare n=ln integers obtainable bythesymbolical formula gP+l=(1+g)P.Wehaveg,=I g"=2,ga=5,.... 113.1 1 1 1- ---+----------+-... :I:x+y:I:+2y :I:+3y maybesummed intheformofaclosedexpression bymeansofelementary functions when:c/Yisarational number. Special casesare: 1-2.-+2.---!-+-...=2.-(..!!--+log2)4 7 10 3l/3 I ]-_2.-+_~_2.-+_...=2.-(~-log2)2 5 8 11 3l/3 I I-J+_11_+-...=~(n+210g({2+1»). 5 9 13 4l/2 270Chapter VIII.Closedandnumerical exprcsslOns torthesumsofseries. 114.Writing1;1~"-;.-=L(x)I(IxI<1)Iwehave,if(x,,)denotes n=l X Flbonacci's sequence 6,7, 1;~-==1+,1_-I-}--I-~+...={5[L(?_--:V§\-L(7--~_\(5)J. !=1x2k3821 2 - ) 2 ""1 OD(_l)k-1k 5-Andifwewrite2)2-=5,and~------- =5',wehave-,=V1)• !=1xJk-1 '.-1 X2k 5 Ill.Exercises onEttler's transformation. lU').Wehave(forwhatvalucsofx?) 00..(-1)" ""~1 k! a)n~o--;;+-n-=k~2,·-+1' X(x+1)...(.l:-I-k) OD,(-1)" 1 [ 1 ( x) 1·2(X)"Jb).2-~a-l-nx"=a(l-1-x) l+a+1l+x+(a+l)(a+2) f:i-x+...; 00, (_1)" 1OD, 1 c)n~o(n+1)(n+2f.-~-(1LT- P+1)=p-'!~O-2"+-1(P,:.l~+-1)' 116.Ifweput -x. ~lx"_'"(-I)"b x"e ..::..JannI -..:..J "nI'n=O n=O wehavebn=Ll"ao'Inparticular, therefore, a)e-x[1+x-I-Cl-I-2~+_(a+2LC~_±- 4).l:~+...J a+1 2 I(a+1)(a-I-2)3! 1x" 1 x4 = 1+a-I-121-=-fj+(a+l)(a+3)2".21'" ""(_1)"-1 CIOx"(11)b)eX."------x"=2)h -,h,,=I-t2-+"'+n--'n~n·nI n=1"n' 117.Quitespecial casesare: (n)1(n)1(n) (_1)"-1(n) 1 1a)1-22-1-33--I-"'-I---n--n=h,,=I-1--2--I-"'-I--:;;-i 1(n)1(n) 1(n)2·4,..(2n)b)l-a 1-1-52--I-,..+(-I)"2n-l-l n=-3~2n+I5' 118.IfLl"ao=b",thenLl"bo=an'\Vhataccordingly aretheinverse equations tothoseofthepreceding exercise? 119.If(an)beanullsequence with(p+I)-folddecreasing monotony (p~I), thesumsofthe"eries];(-1)"an"atlsfies themequalitlcs n0 aoLlaoLtP-1aoaoLlaoLlP-1aoLlPao --+---+...+<5<-+- +...+---+--.22J2P 2292P 2~' Usethistoprovetheequality lim[~-~-+-~- +...J=~. X-H-O 2l+x l-1-x24 Exercises onChapter VHf. 271 120.IfSkand5"denotethepaltial sumsofboththeseriesin144, wehave ("+1) (n-Il) (n+l)150+ 2 5,-+...-+1l+ 15" Sn~ ---2"+' ----. Usethisrelation toprovethevalidity ofEuler's transformation. 121.Thefollowing- relations hold,ifthesummatIon oneithersideis takentostartwiththeIl1dex0andthedIfference-symbols Lloperate onthe coefficIents ontheleft,aT"a~k' a~k+lrespectively: a)L(-I)'·akxk=(1-y)LLlnao·yn wIth(l-+x)(l-y)=I; b)L(-I)'·a27,x2k=(I-y')LAnao·y2n with(l+x')(I--y')=l; c)2J(-1)"a2kHX2k+'={I_y22,,'Ana".y2n T1with(1-+x")(1--y")=1. 122.Thuse.g. tan-1x=~[1+2_ ..~-+~(~'__)9-+...J1-+x'3 1-+x'3 5 1-+x' . 1112 PUlling- x=2-'.~-'-iIIf' seriesfor.1<.asforinstance:3 I' "d I I 79'...,tI1Sprovl L'SpecuIaryconvenient -4.1<=tan--'-21-+tan-1.1.=~_[1-+~_(2)-+...J+3_[1+2_(_!.)-+...]310 310 10 310 ' :n: 1 1 1 3-4-=2tan-1-3--+tan-1'(=1)tan-1-7+2tan-1 79,andothers. 12:1.Thepreceding seriesfortan-1XmayalsobePlltIIItheform sin-1y 22·4 71~~=Y+-3 y3+3-5y"+..·· IIencededuce theexpansion 2(Slll-'y)2=5;(~-=-!)!" (2y)2". nd (~It)! IV.Othertransfornations ofseries. 001124.\Vriting- ~'--Sp.wehave nc211P a)59-+5a+54+...=1 ; 1c)5J-+56+5.+...=-4;3b)S2+S.+56+...=-4-; 1 1d)S2+2-S4+-3Sa+...=log-2; 1 1 ~n_~-J1 t)SJ-"254+S5a-+...=log--:r-~-; 1 1 1g)2-59+-3'Sa+-.{54+...=1 - C ; 1 1h)-2-52-;~Sa+-...=log2+C-1 • whereCdenotes Elller's constant, defined in12S,2andEx.~51l. 272Chapter VIII.Closedandnumerical expressions forthesumsofseries. 12~.'Viththesamemeaning for5pasinthepreceding exercise, writina (11).nle" ----=bkandhm---=A. k+1 2k nn+-~ wehave beSe+b3SJ+...=1-logL (Theexistence ofthelimitAresults fromtheconvergence oftheseries. We haveA={2---;'.) 126. "'; 1rIll 1 1 ] a)n-?:ox(x+l)...(x+nf=e. x-ll:~+i+2Ix+2-+'''; ~1a"a, a3-"a'b),,- =e-xe+"'-e-+..., n"';;;'on!1+x"a"" 00 1 b)2:-(--1-)3-(-2-)3 =ID-ne,»=0n+ n+ 12S.Withreference to§35A,establish therelation between 00 1 ~ 1Y - and~ --;---:---,----, n~l(n+0:)3(n+0:+1)3.,,(n+0:+P-1)3 n=l(n+o:-1)3...(n+0:+P)I and,bygivingspecialvaluesto0:andp,provethefollowing- transformations: <XI1 9 254 00 1 a),,~:n3~8+2~.31-ifn~en+If3(n+2)3(n+3)3 913334' ~, 1 =8+26.33+-5-n)j1n3(n+1)3(n+2)3(n+3)3(n+4)": )~1 S323.34~ 1 bn~ln3(n+if~63U--ssn'7:1--;a-en+-1)3-(n-t2)3C.i+3)" Evaluate thesumofthefirstseriesto6placesofdecimals. 129.Provesimilarly thetransformations: b)y}_=.2._i~~(n+1>-+24n+5 , "-':::1n66n=l 12n6en+1)6 • ~ (_1)"-1 c)".{:1-;;g(n+lr:-~:cn+p-=lV 5p+2 1P(p+l)8 "", (-1)"-1 =4(P+1)(PI)"---4--n~1 -n9en+Jy9...(n+P+1)9 Evaluate thesumsoftheseriesa)andb)to6placesofdecimals. Exercises onChapter VIII. 273 130.a)Denotmg byTvthesumoftheseriesc)intheprecedmg eXl'reise, weobtam relations between TIan:!T'lch'T,andT.k+1•Whatarethese relatIons? IstheproceS3 k---+00allowed Inthem?WhatISthetransformation thusobtained? Isitpossible todeduce Itdirectly asaMarlwff transform.ltlOn? b)\Vehave "",(_1)"-\ 1 1 "",(-1)"-' 3 1 log2=nJf:t--n-- =2+2n~l-n-(11t=1-Y="4-"4T2• Givetheformnowtakenbythetram.formations ofthesenesforlog2 whichwereindicated ina). :ITc)Carryoutthesameproces,; withtheseries122for"4' '\;' 1131.Thesumoftheseries~1 1 (1)2'wherenstartsfromnognoggnog.,n thefirstinteger satisfying logan>1,evaluated to8decimal places, isexactly Rc:1'00000000. -Howmaywedetermine whether theactualdecimal expan­ sionbegmswith0·...orwIthI·...?-Thesolution ofthISproblem requires aknowledge ofthenumerical valueofe'''=e(eeJtoonedecimal placeatleast: thisISe'"=3814279·1...Itsuffices, however, toknowthatclll-[elll]=0'1.... (Cf.remarks onp.249.) 132.Arrange inorderofmagnitude allnaturalnumbers oftheform p~. (PtqpOSitive integers>2)anddenote thenthofthenumbers soarranged byp",sothat (P.,P2,•••)=(4t8,9,16,25, 27,32,...). v.'ethenhave ""1"......---- 1n7:JP"-1 - . (Cf.6S,5.) PartIll. Development ofthetheory. Chapter IX. Seriesofpositive terms. §36.Detailed studyofthetwocomparison tests. Intheprecedmg chapters wecontented ourselves WIthsettmg forththefundamental factsofthetheoryofinfiniteseries.Hcnceforth weshallaim!:lomewhat further, andcndcavour topenetrate deeper intothetheoryandproceed togivemore cxten~ivc applications. For thispurpose wefirstresume theconslCleratlOns statedfromaquite elementary standpolIlt inChapters 1IIandIV.'vVebeginbyexall1ll1­ ingingreater detailthetwocomparison testsofthefirstandsecond kinds(72and73),wluchwerededuced immediately fromthetlrst maincriterion (70),fortheconvergence ordivergence ofseriC'sof pOSitive terms.These,andallrelatedcriteria, willinthesequelbe expressed moreconcisely byusingthenotation2:cnand2'dntode­ noteanyseriesofpositive termsknownaprioritobeconvergent anddivergent respectively, whereas2:a"shalldenoteasenes­ also,inthepresent chapter, ofPositive termsonly-whosecon· vergence ordivergence isbelllgexammed. Thecriterion72canthen bewritteninthesimpleform 157.(I) ~. Thisindicates that,ifthetermsofthesenesunderconsideration satisfythefirstinequality fromandafteracertainn,thentheseries willconverge; if,ontheotherhand,theysatisfythesecond inequality, tramandafteracertainn,thenitmustdiverge. Thecriterion73becomes inthesameabbreviated notation I:is.(1I) ~. Beforeproceeding wemaymakeafewremarks inthisconnexioIl. ButletusfirstinsistOHeemoreononepoint:Neither thesenorany 274 §36.Detailed studyofthetwocomparison tests. 275 oftheanalogou!:> criteria tobee~tabh~hed belowwillnecessarily solve thequestion ofconvergence ordivergence ofanyparticular given series.Theyrepre,ent sufficient conditions onlyandmaytherefore verywellfailinspecial cases.Their succe~3 willdepend onthe choiceofthecomparison series~cnand~dn(seebelow). Thefol­ lowing pageswillaccordingly beelevated toestablishing tests,as numerous andasefficacious aspossible, soastoincrease thepro­ babJlity ofactually solving theproblem ingivenspecial cases. Remarks onthefirstcomparison test(1:)7). 1.Sinceforeverypositive numbergtheseries ~gc"and~gd" necessarily converge anddiverge respectively with2,'cnand~d",the firstofourcriteriamayalsobeexpressed intheform: ~1I<g«-+(0) 8, ~">g(>0) ".. or,evenmoreforcibly, IIItheform --'-a8, lim~:>0 lun·1I<+00e,l 2.Accordingly wemust alway.~ have:HiD. -dlun..!!=+00, C..I·cn0IIn·.=_d" or,otherwise expressed: hll1~" =c=+-00isanecessary condition forthedivergence c"of:;'an' lim11"=0_d"" "necessary"" "convergence" 2,'an' 3.Here,asinallthatfollows, itisnotnecessary thatactual U1l1quelImItSshouldexist.Thismaybeinferred, totakethequestion quitegenerally, fromthefactthattheconvergence ordivergence of aseriesofpositive termsremains unaltered whenthesenesissub· jectedtoanarbitrary rearrangement (v.SS).Thelattercaninevery casebesochosen thattheabovelimitsdonotexist.Forinstance 2'c"canbetakentobe1+~+~-\-"~+,.."and2'alltobetheseries ~+1+?i+i+li'2+16+"" obtained fromtheformerbyinterchanging thetermsineachsuccessive pair;theratioa..certainly tendstonounique limit;infact,ithas Cn distinct upperandlowerlimits2and~. Similarly, let~dnbechosen tobetheseries1+~+~+i+.."andlet2'anbetheseries 1+~+~+g+~+~+~+-/i.+~-+.." 276 l:hapter IX.Seriesofpos!t1ve terms. 160.deduced fromtheformer byrearrangement, (inthisseneseverytwo aodddenominators arefollowed byoneevenone).Here--"hasthe d" twodistinct upperandlowerlimits ~and~. Inas!lnilarmannerwe mayconvince ourselves byexamples intheothercasesthatanactual uniquelimitneednotexist.11,however, suchanuniquelimitdoes exist,itnecessarily satisfies theconditions indicated forhrriandlim, sinceitisthenequaltoboth. 4.Inparticular: Nocondition oftheform:n-+0isnecessary n fortheconvergence of2'an-unlessallthetermsofthedivergent seriesXdnremain greater thanafixedpositive t5.For,evenifwe onlyhaveIlmd..=0,bychoosing ki<k'J<...<k~<..., sothat 1d<-- 1,~ 2~ andwntmgak=dk'a=0or=thecorresponding termcofany,. ,.n " convergent senescnforeveryothern,weevidently obtainacunvergent seriesan'butitisequally evidentthat=ndoesnot-+O. n Remarks onthesecond comparison test(I:'iS). 1.Thevalidityofthecomparison testIImaynowbeestablished moreconcisely asfollows: Inthecasemarked((?),wehave,fromandafteradefiniten, an"'-an+l'(an). dd' I1"-~---, 1.e.-ISamonotone escen lI1gsequence, wloseumt en.e..+1 en risdefinedand:2O.Inparticular lim:n=r<+00,and,by1:'i9,1, n .:Ea..isconvergent. Inthecasemarked(5)), (~,:)ismonotone ascend· ingfromandafteraparticular n,andaccordingly alsotendstoa definitelimit>0,orto+00,Ineithercasethecondition lim~~> 0 of159,1isfulfilled andthisshowsthat.:Ea"isdivergent. 2.Thecomparison testIIthusappears asanalmostimmediate corollary tothecomparison test1.Iftheconvergence ordivergence ofaseries.:Eancanbeinferred bycomparison witha(definitely chosen) series,2'cnorXd"inaccordance with158,thenthismay alsobeinferred bymeansof157(orl:'i9,1),butnotconversely, i.e.ifIisdeCIsive, Ilneednotbeso. Examples ofthishavealready occurred inthepairsofseriesof 159,3.Forthefirstpairwehaveliman=2,whilean+1alter- en an §36.Detailed studyofthetwocomparison tests. 27i nately==2and-~{,i.e.itissometimes greater,sometimes lessthanthe corresponding ratio ~.+"sincethisconstantly =-21•Thesecond c" pairofseriesrepresents anequally simplecase. 3.Thisrelation between thetwotypesofcomparison testsbe­ comesparticularly interesting whenwecometodealwiththetwotests towhichwewereledin§13asimmediate applications ofthefirst andsecond comparison tests.These\\eretherootandratiotests, inferred fromIandIIbytheuseofthegeometric seriesascam parison series,andtheymaybestatedthus:- Ourremark 2.showsthattheratiotestmayverywell fail whenthe roottestapplies (theseriesZangiventhereareobvious examples of this).Ontheotherhandourremark 1.showsthattheroottestmllst necessarily work,iftheratiotestdoesso.Thisrelation between the twocomparison testsisexpressed inmoresignificant formbythe following theorem, whichmayberegarded asanextension of43,3. Theorem. 11xl'x2'•••arearbitrary positive terms,wealways161. have1 x -!'j----!'j- -x+1lim~<limyx.:-s;:limyX<1Im~-._x..-_ 11- 11- Xn Proof. Theinnerinequality isobvious2,andthetwoouterin· equalities arcsocloselysimilarthatwemaybecontent withproving oneofthem.Letuschoosetherighthandinequality andput _n_ hm)lxn=p.. 1-·-X"~1 ,Im--=p..Xn sothatthestatement reduces to"It<It'''.Nowifft'=+00,there isnothing toprove.But,ifp.'<+00,wemay,given8>0,assign anintegerp,suchthat,foreveryv;;:::p,wealwayshave X"+I<'-I8~-ft-2' 1Thistheorem isofthesamecharacter as43,3.Infact,writing ~~~ .Y,,YJ'YsI•••fortheratios-1'-,--,•••,weareconcerned Withacom­x,xgn _ parison oftheupperandlowerlimitsofYnandofy..'=VY,Yg•..Yn' gForthisreason, itisusualtowritemoreshortly: x+_n_-x+lim---'!.-!<lim;-jx<!im---'!.-!-x..=-y11=x,.' implying thatinthecentre,either!imor!immaybeconsidered indifferently. Suchanabbreviated notation willfrequently beusedbyusinthesequel. 10 (G51) 278 Chapter IX.Seriesofpositive terms. Thisinequality maybesup}.lOsed written downforeveryv=!J, P+1,...,n-1,andwethenmultiply alltheseinequahlles together, deducing, forn>p, Letus,forbrevity, denotetheconstant number xp'(ft'+i)-P byA; then,forn>p,wealwayshave YXn<YA.(//+i). ButVl-1,andhence(ft'+~)y:;:f-+ft'+i.Wecantherefore sochooseno>pthat,foreveryn>no'wehave(ft'+i)V~.f<ft'+8. Wcthenhaveafortiori, foreveryn>no' VX~<ft'+8 andhencealsoft:::;::ft'+E,or,asasserted, since Eisarbitrary, It<,It'. (Cf.p.68,footnote 10.)Moreover wecanshowbysImpleexamples thatthesignofequality neednotholdinanyofthethreeinequalIties of161,whichisnowcompletely established. 4.Theprecedmg theorem showsinparticular thatiflim_~".+1, Xn n-exists,lim\lxnmustalsoexistandhavethesame\aluc.lIencein particular: Iftheratiotestworksintheformgivenin76,2,thenso willtheroottest,necessarily, (butnottheconversel). Tosumup:­ Theratiotestistheoretically lesspowerful thantheroottest.(Never­ thelessitmayfrequently bepreferred, asbeingeasierofapplicatIOn.) 5.Inthisplacewehavealsotorefertotheremarks 7;S,1 and76,3. §37.Thelogarithmic scales. Wehavealready observed thatsuchcriteria asthosejustdis­ cussedonlyprovide sufficient conditions andmayaccordingly faili.1 particular cases.Theirefficiency willdepend onthenatureofthe chosen comparison series ~cnand ~:dn;ingeneral termswemay saythata@·testwillpresent abetterprospect ofsuccess thegreater themagnitude ofthecn's,a~-test,onthecontrary, thesmaller the magnitude ofthedn·s.Inordertoexpress thesecircumstance" more precisely, weproceed firsttodefinetheconcept oftherapidity of convergence: Aconvergent serieswillbesaidtoconverge moreor lessrapidlyaccording asitspartialsumsapproach moreorlessra­ pidlytothesumoftheseries;andadivergent serieswillbesaidto diverge moreorlessrapidlyinproportion totherapidity with,vhich itspartialsumsincrease. Moreprecisely: §37.Thelogarithmic scales. 279 Definition 1.Giventwoconvergent series~cn=sand'Ecn'=s '162. ofpositive terms,whosepartialsumsaredenoted bys"andsn',the corresponding remainders bys -sn=rn's'-s,.'=rn',wesaythat thesecondconverges moreorlessrapidly (orbetterorlesswell) thanthefirst,according as .r/lun--=0 ""0'.1',/hm-=+00. "" Ifthelimitofthisratiocxistsandhasafinitepositivcvalue,or ifItbeknownmerelythatitslowerlimit>0anditsupperlimit <+00,thentheconvergence ofthetwoserieswillbesaidtobe atthesamekind.Inanyothercaseacomparison oftherapidity of convergence ofthetwoseriesisimpracticable 3. Definition 2.It2'dnand2,'d,,'aretwodivergent seriesofposi. tiveterms,whosepartialsumsaredenoted bysnandsn'respectIVely, thesecondissaidtodivergemoreorlessrapidly (ormoreorless marl~edly) thanthefirstaccording as s '!im-'!..=+00s.0'.5,.'hm---=O.s. Iftheupperandlowerlimitsofthisratioarefiniteandpositive, thenthedivcrgence ofbothserieswillbesaidtobeotthesamekind. Inanyothercaseweshallnotcompare thetwoseriesinrespectof rapidity ofdivcrgcnce 4. Thetwofollowing theorems showthattherapidIty ofthecon· '<ergence ordivergcnce oftwoscriesmayfrequently berecognised frnmthetermsthemselves (without refercnce topartialsumsorre­ mainders): Theorem 1. rapidly than.:Ecn'e 'It-?--+0(+(0),then.:Ecn'converges more(less) . , , •Inthecaselim~-=0(>0)andlim""<+00(=+(0),wemig-htalso-"n I'n speakoftheseries:ze..'as"noless"("nomore") rapidly convergent thanthe series:ze.ithishowcver pre~cnts noparticular advantages Inthecaseofthe lowerlimitbeing0andtheupperlimit+00,therapidity oftheconver­ genceofthetwoseriesistotallyincommensurable. Asimilarremark holds fordivergence. (Thestudent shouldillustrate byexamples thefactthatall thecasesmentioned canreallyoccur.) -Thesedefinitions maybedirectly transferred tothecaseofseries 01arbitrary terms,replacing "..and"..'by theirabsolutc valucs. 4Theproperties referred tointhesedefinitions areobviou~ly transltlVc, i.e.ifafirstg-lvenseriesconverges morerapidly thanasecond, andthis againmorerapidlythanathIrd,thefirstserieswillalsoconverge morera· pidlythanthethird. 280 Chapter IX.Seriesofpositive terms. areeven163.Proof. Inthefirstcase,givene,wechoosenosothatforevery n>nowehavecn'<ecn'Wethenalsohave Consequently thisratiotendstoO.Thesecondcasereduces tothe firstbyinterchanging thetwoseries(cf.thetheorem of40,4,Rem.4). Thisprovesallthatwasrequired. Theorem 2.11~,{-..0(+(0),then2'd,,'diverges less(more) n rapidly than2:dn• d'Proof. By44,4itfollowsimmediately from -~-..0that d" Thisprovesthestatement. Simple examples. 1.Thescrics aresuchthateachconverges morerapidly thanthepreceding. Infactwe havee.g.forn>3: ]-;-~=3"=(3~a~)-=-3~=-=----~~<~.(~)"-3 nl1l"n!1·2·3.4...n2 4 ' whichtendstoO.Similarly log"n-.0(by38,4); theothercasesn simpler. 2.Theseries .L;n,.L;1,12)--nlogn'y'~_1~_ ~nlognlog.n'.•, aresuchthateachdiverges le!>srapidly thanthepreceding. Besides theabovesimpleexamples, themostimportant casesof serieswithrapidity ofconvergence forming agraduated scaleare afforded bytheserieswhichwecameacrossin§14.Aswesawin thatparagraph, theseries 2)~,.2__1_,.L;-- 1 ,.."1: 1 n" n(logn)" nlogn(log,n)" nlogn...logl'_ln.(logpn)" converge forCl:>1anddiverge forIX::;1.Ourtheorems 1and2now showmoreprecisely thatwhenpisfixedeachoftheseserieswill converge ordiverge lessandlessrapidlyastheexponent IXapproaches unity(remaining> 1inthefirstcaseand<1inthesecond). Simi· larlyeachoftheseserieswillconverge ordiverge lessandlessra· §37.Thelogarithmic scales. 281 pidly,aspincreases, whatcver positivc~ valuemaybegiventotheex­ ponenta(>1inthefirstcase,<1inthesecond). Thesecondaloneofthesestatements perhaps requires someJusti· fication. DIvidethegeneric termofthe(p+l)thserieswiththeex· ponentclbythecorresponding termofthepthseriestakenwiththe exponent IX.Weobtam (Iogpnt --'------' logpn.(logp+1n)"" Inthecaseofdivergent senes, IXanda'areposItive and<1;the ratiotherefore tendsto0,qe.d.Inthecaseofconvergence, i.e." andIX'both>1,theratiotendsto+00;in£.lct,-byreasoning analogous tothatof3S,4,-wehavetheauxiliary theorem that thenumbers (lo!!p1-1n)'"{log(Iogpn)}'" l1og--nl-= -(lol!n)/I -p p formanullsequence, fJ=IX-1denotmg anypositive exponent and panyposItive integer. ThlsprovesallthatwasreqUIred. Thegradation IIItherapidity oftheconvergence anddivergence oftheseseriesenables ustodeduce complete scalesofconvergence anddivergence testsbyintroducing theseseriesascomparison series inthetestsIandII(p.274).WefirstImmediately obtainthefol· lowingformofthecriteria: (1)an<} I------------ with an>nlogn ...logp_ln(101'1'n)"{">1 1X<1~ ~ 164. (11) Thesecriteriawillbereferred tobrieflyasthelogarithmic tests ofthefirstandsecondkinds-alsointhecasep=O.Theireffi· ciencymaybeincreased bythechoiceofp,and,forfixedp,bythe choiceof()("inaccordance withourprevious remarks 6 • &For Cl:= -fJ<0,eachseriesofcoursediverges morerapidlythanthe (logn)p.preceding onewiththeexponent replaced by1ithuse.g.'"----With4.Jn ' fJ>0,diverges morerapidly thanZ.!..n eTheconvergence anddivergence ofseriesoftheabovetypewasknown toN.H.Abelin1827,butwasnotpublished byhim(CEuvres 11,p.200). A.deMorgan (Thedifferential andintegral calculus, London 1842)wasthefirst 282 Chapter IX.Seriesofpositiveterms. ~6:S.Forpractical purposes itI~advantageous togiveotherformsofthesecriteria. Suchtransformations aregivenbelowwithafewremarks appended, butwithout completely carrying outthenecessary calculations. Transformation ofthelogarithmic testsofthe19tkind. 1.Whenaandbarepositive, thetwoinequalities a;<;:bandloga~logb areequivalent; afteraslightalteration theinequalities 164,Iaccordmgly become: (I') inlogan-I-logn-I-log2n1-···-I-lOgvn{;S;-~<0 logvn ~0 2.Denoting foramoment byAntheexpression ontheleftof(I'),wehave, (I") lunAn<0 ;I), anuifatestofpractically thesameeffect.Thepartsrelatmg toconvergence areindeed completely equivalent in(I')and(I");thatrelatmg todivergence isnotqUite sopowerful in(I")asin(1'),smceitISreqUired in(I")thatAllshouldremain, fromsomevalueofnonwards, notmerely =0butgreaterthanafixedpositive number 7. 3.Ifweusethesomewhat moreexplicit notation An=A::'>,andconsider bothA~)andA/tI),weobVIOusly have A(p+-1)=1+)()gpn.A'P) n logp+lnn• Ad' b384logpn logpnd' +I"Insmce,y " fOg"~1n=log(iog pn)tensa~nmcreases toex;,tliSslmpe transformation leadstothefollowing result:IfforapartIcular poneofthelimits ofAn=A~)isdIfferent fromzero,itisneeessanly fX)forthefollowing p,infact +OC)or-fX)aecorumg asthepreceuing pwaspositive ornegative. Morepre­ cisely,ifwedenoteby1-'"andx"theupperandthelowerlimitsofAn=A~'),for everyp,thenifwehave,foranyparticular p, x";?-1-'"<0,wehavexp+1=I-'v+1= -00, If,however, Xv<0,1-'">0,wehaveX,,+1=-00,1-'1'+1=+00. Thescalesofreference (I)thusleadtothesolution ofthequestion ofcamer­ genceordivergence If,andonlyif,foraparticular p,thevalues XvamIJ1p havethesamesign.Ifthesignisnegative, theseriesconverges; ifpositive, it tousetheseseriesfortheconstruction ofcriteria. Essentially, thesecriteriaare consequences of164,IandII;numerous transformations ofthemweresubse­ quently published asspecialcriteria, e.g.byj.Bertrand (].demath.puresetappl., (1)Vol.7,p.35.1842),O.Bonnet(Ibid.,(1)Vol.8,p.78.1843),U.Dini(Giornale dimatematiche, Vol.6,p.166.1868). 7Itwouldclearly, however, bewrongtowritethelast;I)-testintheform HmAn~0,sincethelowerlimitmayverywellbe0without asingletermbeing positive. §37.Thelogarithmic scales. 283 diverges; ifthetwonumbers haveopposite signsforsomevalueofP.thenfor allhigherp'swehave hmA~)= -00, li~A;~)=+00 andthescaletherefore isnotdecisive. Similarly itfailswhenbothnumbers are zeroforeveryp. Transformation ofthelogarithmic testsofthe2"dkind. 1.Thefollowmg Lemmas areeaqllyproved: Lemma 1.Foreveryintegerp::;;U,foreveryrealatandeverysufficiently large n.mlequalityoftheform (Iogp(11-I»)" at ~n log;11-=1 -nI()g~ logpn-n" holds,where (~n)isbounded". Themdexnishereassumed tostartwithavaluefrom andafterwluchallthedenominators aredefinedandposItive. WeImmedlatelv mferthat,foreverymtegerp~0,foreveryrealatandevery suffi~lently large11, ~=_l.lo_g (,,-::1)...Iogp_~l(n..:-n.(~~g"(~~J2)" nlogn log}'_1n logpn166. where('I,,)isagamcertainly bounded o. Lemma 2.Let::::anand::::all'betwoseries0/positive terms.'I1theseries167. whosentbtermIS n=(:Z!'5 I~Il'_1)rallan+l ISabsolutely convergent, thetwogivenseYlesareeitherbothconvergent orboth dIVergent. Infact,wehave r~>-1foreveryv;takmg, then,anypositive in­ tegerm.writing downtherelations a,.t-l a.,..'-a-.--,-=1+r.. ~a.+l forv=m,111+1•...,n-1,andmultiplying themtogether, weatoncede­ ducethattheratioat.'Iallforn>mliesbetween twofixedpositive numbers. - 8Anequality oftheaboveformofcourseholdslmderanyClrCU1IIstances. Infactwecanconsider thenumbers {}"asdefined precisely bytheequation: {}_n"[1_ cc _(~g-I'(n-1»)"J. n- nlogn...logp11 logp11 Theemphasis lIesonthestatement that(0,,)ISbounded. -Theproofisob­ tainedmductlve!y, wIththehelpofthetworemarks thatif({}n')and({}n")are defIned, foreverysufficiently large 11,by (1-xn)"=1-ccXn-{),.'x,,"andlog(1-_~)=__1__'!.n",nYnnYIl11" theyarenecessarily bounded, provIded (x,,)isanulls('quence andthenum­ bersYIIareinabsolute value2:1,say. oTheinterpretation inthecasep=0isimmediately obvious. 284 Chaptcr IX.Sericsofpositive terms. Theconditions oftheLemma arefulfilled, inparticular, whcnthcratios ~,,_±! atl and~~~1liebetween fixedpositive bounds andtheseries.J;'Ian+1 _~~~11 an an an converges. 2.Inaccordance withtheabovewemayexpress thelogarithmic testof thesecond kinde,g.inthefollowing form10: ~.Itlogn ••.log'"It 169.•(1,,,+1;;;;;}1 1 1 I168.---a:-~--n- Itlogn-...-nloglt•••logp_1It {a'>l ewitha':Sl ~, or,afterasimpletransformation, [a.,+l_l+-!..+ ...+ I ].nlo"I' II0••••ngI.n(I,,, n Itlogn...ogl'n {;;;;;-fJ<O e ~U ~, or,finally,denotmg theexpression onthelefthandsideforbrevity byHn, andslightly restricting thescopeofthe~·test(cf.163,2), IimBn<0 e, IimBn>0 Remarks analogous tothoseof16:'),2holdhere. 3.Thedevelopments of165,3alsoremain valid,withquiteunessential alteratlO,ns. For,Ifweusethemoreexplicit notation Bn=B~),wchaveob· viously And,aslogp+ln-..+00,wemayreasonwiththisrelation inprecisely the samemanner aswithitsanalogue in16:'),3.Itisunnecessary todevelop this indetail. 4.Stillmoregenerally, wemayatonceprovethataseriesoftheform 1 .J;e(a1)n.nao(logn)a,(log.n)'"•..(Iogqn)"q converges il,andonlyif.thefirstoftheexponents IX,1X0'IX"".,IXqwhich differsfrom1is>1.Thevaluesofthesubsequent exponents havenofurther influence. -Whenthecomparison seriesisputintothisform,Raabe's t('st (§38)andCauchy's ratiotestappearnaturally astheot.andthe(_I)th term~ ofthelogarithmic scale. §38.Specialcomparison testsofthesecondkind. Thelogarithmic testsdeduced inthepreceding articleareun· doubtedly ofgreatertheoretical thanpractical interest. Theyaffordin· deedamoreprofound insightintothesy!>tematic theoryofthecon· vergence ofseriesofpositive terms,butareoflittleuseinactually testingtheconvergence ofsuchseriesasoccurinapplications ofthe 10Herewemakethenthtermoftheinvestigated series ~'ancorrehpond tothe(n-l)thtermofthecomparison series,which,byS2,theorem 4,Is allowable. §38.Specialcomparison testsofthesecondkind. 285 theory. (ForthIsreasonwehaveonlysketched theconsIderations relating tothem.)Forpractical purposes thefirsttwoorthreeterms, atmost,ofthelogarithmic scales maybeturnedtoaccount; from theseweproceed todeduce byspecialization anumber ofsimpler tests,whichwerediscovered atvanous times,ratherbychance, and eachproved initsownway,butwhichmaynowbearranged in closerconnexion withoneanother. Forp=0thelogarithmic scaleprovides acnterion alreadyestab­ lishedby].L.Raabel1.Wededuce itfrom169,firstintheform [a"+l_1+.!-Jn{< -P<0 ann:2 0 or,aswemaynowwritemoreadvantageously, [""+'_1] It{<-a<-1 t".. :2-1 The \T~ryelementary natureandgreatpractical utilityofthisCrl­ terionmakesitworthwhiletogiveadirectproofofitsvalidIty: the e-condition meansthat,foreverysufficiently largen,170. ornan+l.<(n-1)an-pan wherep="-1>O.Hence (n-1)an-nan+!:2pan>0 andtherefore nan+1isthetermofamonotone descending sequence, forasufficiently largen.Sinceitisconstantly positive, ittendsto alimit.,,:2 O.Theseries2:cwithc=(n-1)a-11a+1there· 1_ n n n n foreconverges, by131. Since an<1cn'theconvergence of :Eanimmediately follows. Similarly, iftheS)-condition isfulfilled, wehave or(n-1)an-n an+1<O. Accordingly nan+tisthetermofamonotone increasing sequence andtherefore remains greater thanafixedpositive number /'.As an+1>~,/'>0,thedivergence followsimmediately. Iftheexpression ontheleftin170tends,whenn-++00,to alimitt,itfollows fromthereasoning already repeatedly applied (v.76,2)that1<-1involves theconvergence of:Ean'and1>-1, itsdivergence, while1= -1leadstonoimmediate conclusion. 11Zeitschr. f.Phys.u.Math.vonBaumgllrten u.F.ttinghausen, Vol.10, p.63,1832.Cf.Duhamel, J.M.C.:]ollrn.demllth.pllrcsetllpp!.,(1)Vo!.4, p.214,1839. 10- (0:il) 286 Chapter IX.Seriesofpositive terms. Examples. 1.In§25weexnmined thebinomial seriesandwereunable todecide therewhether thesenesconverged ornotattheendpoints oftheinterval of convergence, thatis,whether forgivenreala'stbeseries 5;(:),,=0andi(-1)"(:),,=0 ,,"(a+l)IanI=IamI·If1- - • '-=mf-l ..were,orwcrenot,convergent. Wearenowabletodecldethisquestion. Forthesecondserieswehave a"+._a-n_(n+I)-(a+l)---a::--n+f-- n+i---. Sincethisratioispositive fromacertain stageon,itfollows thattheterms thenmaintain onesig-n;thiswemayassumetobethesign+,sincechanging the signsofallthetermsdoesnot,ofcourse, affecttheargument. Further, ac­ cording tothis. (a"+l_I)n=_(a+1)._n__-(a+I). an n+1 fromwhichweatoncedcduce, byRaabe's test,thatthesecond ofour serie~ converges fora>0,anddiverges forCl<O.Fora=0,theseriesreduces toitsinitialterm1. FortheItrstserieswehave an+l=_I+a+~ an n+1 and,sincethisvaluebecomes negative fromsomestageon,thetermsofthe se~have analternating signfromthatstageon.Ifnowwesupp&fea+1<0, we eforehave \ fa:~-!12:1 whetlt weit-that ultimately thetermsa"arenon-decrcasing. Theseries musttherefore-.diverge. Ifhowever we~upposea+1>0,wehaveultimately, sayforeveryn>m, , (a) Ia.+~I=1 -~1<1an n+1 ' andthetermsultimately decrease Inabsolute value. ByLelbmz's criterion for serieswithalternately positive andnegative terllls,oursenesmusttherefore converge, provided wecanshowthat(:)_0.Ifwewrite'downtherela- tlOns(a)form,m+I,...,n-1andlllultiply themalltogether, we'deduce foreveryn>m Since,however, theproductII(1-a~1),br126,2, 3,diverges to0,anmust also_0.andtherefOre.2(:)mustconverge. Summing up,wetherefore havethefollowing resultsrelating tothebinomial series: Theserles1;(ex)x"converges tl.andonly7./.ettherIxI<1.orx=-1,,=0'1 §3tl.Specialcomparison testsofthesecondkind. 287 andIX>012,orx=+IandIX>-I.ThesumoftheseriesisthenbyAbel'stheore:n oflimitsahvays(I+x)".IfIXISrwintegerandisnon-negative, thentheseriesisfinite andhenceconverges foreachx.InallothercasestheseriesisdIVergent. (Anappreciable addition tothistheorem ISprovided by247.) 2.Thefollowmg cntenon doesnotdifferessent13lly fromthatofRaabcj itISduetoO.Schlomllch: 1an1I{:c:-ex<-1 noga~';-1 1 Inf,ICt,mtheedse(CD),wehave,by114,lln+1~:e-/I>1 _ 1an 11' fromwhichthedivergence follows byHaabe's test.Incase(<?)wehave, ulttmately, a ~1I+1<e-n<1-!:" an= = n' ifa>a'>1.By170,thiSinvolves convergence. If,intheloganthmic scale,wechoosep=1,weobtainacn· tenonoft11Csecondkllldwhich,omitting thelimiting case (X=1,. wemaywrite (l1l+1=1-..!.-~ withfa">et>1 <?171. (I" nnlogn1a"<a<1 '3)• Adirectproofofthevahdityofthiscriterion canbegivenas follows AsintheproofofRaabe's testwefirstputthecriterion in thefollowing form: {2fiawithfi>0<? [-I+(n-1)logn]a n-[nlogn]a n+1<-fi'a: with~'>O '3). Ifnowthe<?.condItion isfulfilled, since,aswemayimmediately venfyby114, (x, • ~(n-1)log(n -1)> -1+(n-1)logn, wehaveafortiori (n-l)log(n -1).a"-nlogn·an+1::2fian' Accordingly nlogn·an+1isthetermofamonotone descending se· quenceandaccordingly tendstoalimity2O.By131,theseries who5enthtermis cn=(n-1)log(n-1)·an-nlogn.an+l mustconverge. Asans~,cn'thesameistrueofXa", If,ontheotherhand,the'3).condition isfulfilled, wehave (n-1)log(n -1).a"-nlogn·an+1 <[-fi'+1-(n-1)log(1+n~I)Jan· Forn-+00,however, theexpression insquarebrackets --fi' 19Fora=0Iseeabov.. 288Chapter IX.Seriesofpositive terms. 172.(by112,b),andistherefore negative foreverysufficiently large 11 Henceforthosen'stheexpression nlogn·an+1increases monotonel)' andconsequently remains greater thanacertainpositive numberr Asa+1~-1"-,'V>0,itfollowsthat~amustdiverge.n-n.n { n Hereagainwemayobserve, asrepeatedly inprevious instances, that,ifantendstoalimitt,thenl>1involves convergence, and l<1involves divergence, while,froml=1,nothing canbedirectly inferred. Eventhis,thefirstproperly logarithmIc criterion ofthescale,will rarelybeactually applied inpractice. Infact,theserieswhichare amenable tothistest,andnotalready toasimpler one(Raabe's test, ortheratiotest),occurexceedingly seldom; andastheirconvergence isDOmorerapidthanthatof.L;__1__,(a>1),theseseriesare n(logn)Q uselessfornumerical calculation. Itenables us,however, todeduce easilyoneortwoothercri· teria.Wewillaboveallmention Gauss's TestIS:11theratioan+1canbeexpressed tntheforman wherel>1,and({}n)isboundedH•then 2~anconverges whena>1 anddivergeswhena<1. Theproofisimmediate: whena:Z1,Raabe's testitselfproves thevalidityoftheassertion. Fora=1,wewrite a.+1=1--!..1_(On.logn). an nnlogn nl-1' andasnowthefactorinbrackets tendstozerosince(l-1)>0, theseriescertainly diverges, by171. Gaussexpressed thiscriterion insomewhat morespecialformas follows:"11theratioan+1canbeexpressed intheform aft an+1nk+blnk-l+ +bk ( '-a;;-=nk+b/nk1++bk'kaninteger 0=:;1) then~anwillconvergewhenbI-b/< -1anddivergewhenbI-b/ ~-1."-Theproofisobvious fromthepreceding. 11Werke, Vol.3,p.140.-Thiscriterion wasestablished byGauss in1812. 14Cf.footnote 8,p.283. §38.Specialcomparison testsofthesecondkind. 289 Examples. 1.Gaussestablished thistestinordertodetermine theconvergence of thesocalledhypergeometric series 1+a'.0x+~,:_+_!2.fI (.0+_!2x2+~~+1)_~a+2).fljP:,-_I)J.8+2~x'+... 1.1' 1·21'(,,+1) 1·2·3 1'(,,+1)(1'+2) =i;a(a+1)~..(a+71,-1).,8(p-±=-l)-.:._~~+ n-=_I~xn n=O 1·2•..71,"er+1)..·(r+n-1) wherea,.0,l'areanyrealnumbers H'dIfferent from0,-1,--2•.... Here an+1(a+71,)(,8+71,) --a,:-=(1+71,)(r+n)x whichshowsinthefirstinstance thattheseriesconverges (absolutely) for !xI<1,anddiverges forIxI>1.Accordingly itonlyremains toexamine thevaluesx=1andx= -1.Thisisanalogous tothecaseofthebinomial ~enes,towhich,ofcourse, thepresent onereduces whenwechoosefJ="(=1) andreplace a:andxby-aand-x. Forx=1,wehave an+ln2+(a+p)n+afJ ---;,;-=n2-=+-er+1)n+". Thisshowsthatforeverysufficiently large 71"thetermsoftheserieshave oneandthesamesign,whichmaybeassumed positive. Gauss's testnowshows thattheseriesconverges forex+.0-"-1< -1,i.e.fora+.0<YJbutdi­ vergesfora+.0>r Forx=-1,theserieshas,fromsomestageon,alternately positive and negative terms,sincean+1 _ -1,i.e.isultimately negative. Therelation 18 an an+1=_71,2+(a:+p)n+(X~=_[1+a+,8-l'-1+~nJ' an n2+(,,+1)71,+1' n n2 withwordforwordthesamereasoning aswasemployed in170 J1forthe binomIal series,nowshowsthatthehypergeometric serieswill diverge when a+.0-">1 converge whena+.0-"<1. Wehaveonlytoverifyfurther thatitalsodiverges whena+.0-"=1, asthisdoesnotfollowfromprecisely thesamereasoning asbefore.Iffor everyn>p> 1wehave an+l=_(I+~~) withl~nIS~ forevery 71" all n then,assuming pchosensolargethatp2>~, IanI>IapI(1-;)(1-(p:1)2)...(1-(71,!1)2). Sinceontherighthandsidewehavetheproduct ofthefirst(71,-P)factorsof aconvergent infinite product ofpositive factors, itfollows thatianI,forall thesevaluesofn,remains greater thanacertain positive number. Theseries cantherefore onlydiverge. 15Forthesevalues, theserieswouldterminate orbecome meaningless. Forn=0,thegeneral termoftheseriesshouldbeequated to1. 1ftAsbefore, (~..)denotes abounded sequence ofnumbers. 290 Chapter IX.Seriesofpositive terms. 2.Raabc'se·testfailsifthenllmbers a"intheexpression ~•..±-'.=l-!:.". a,l n thoughconstantly> 1,havethevalue1forlowerlimit.Inthatcase,writing an=1+{l",thecondition isanecessary condition fortheconvergence otEa,..InC,ct,If/I13nv.ereboundcd. weshouldhave an+t1{)n-------=1- - -- all nnil andIa"wouldbedivergent byGauss's test". §39.Theorems ofAbel,DiniandPrin(Jsheim andtheir application toafreshdeduction ofthelogarithmic scale ofcomparison tests. Ourprevious manner ofdeducing thelogarithmic testsinvests these,themostgeneral criteriayetobtained, withsomething ofafor· tuitouscharacter. Infacteverything turnedontheuse,ascomparison series,ofAbet'sseries,whichwereobtained themselves onlyaschance applications ofCauchy's condensation test. Thi~character offortuitous· nessdisappears tosomeextentifweapproach thesubject froma different direction, involving agreater degreeofinevitableness. Our starting pointforthisisthefollowmg • 173. Theorem ofAbelandDini18:It2.'dnisanarbitrary divergent n=1 series01positiveterms,andD"=d1+d~+...+d"denotes itspartial sums,theseries J;a==idn{converges whena>1 ,,=1" ,,=1D:diverges whena<1. Proof. Inthecasea=1, !"-±..I-!-...+",,__tk~~n+t+· ..+dn+7<=l __~. D,,+t Dn+k---- D"+k Dn+k AsD~-+00byhypothesis, wecantherefore choosek=k",for eachn,sothat Dn1.+ + + 1 P--<'2'1.e.a"+1 a,,+~...an+k">2; n+k" 17Caken.E.:!\ouv.Annales deMath.,(3)Vol.5,p.535. 18N.H.Abela.f.d.reineu.angew. Math,Vol.3,p.81.1828)only provedthedivergence of.J)Dd";U.Dini(Sulleserieaterminipositivi, An.n-l naHUniv.Toscana Vol.9.1867)established thetheorem intheabovecom- pleteform.Itwasnottill1881thatwritings ofAbelwerediscovered (CEuvres 11, p.197)whichalsocontain thepartrelative toconvergence ofthetheorem givenabove. §39.Theorems ofAbel,DiniandPringsheim. 291 bySI,2,theseriesIanmustaccordingly diverge when (X=1,and afortioriwhen (X<1- Theproofofitsconvergence inthecase (X>1isslIghtly more troublesome. Wemayatthesametimeprovethefollowing extension, duetoPrtngsheim1lJ• Theorem ofPrillf}sllei'ln: Theseries 174. Vdn_vD"--Dn-l ~-D--D0---.L.J --()-, n=~,,'n-ln=2D".Dn_1 wherednandD"havethesamemeaning asbefore,converges for everye>o. 1Proof. CholJse anaturalnumberpsuchthatp<e.Itthen suffices toprovetheconvergence oftheaboveseneswhentheex­ 1ponenteisreplaced byf=p'Since,further, theseries no(11))'---- n7JD,~_1 D~ converges, by131,sinceDn-1<D,,-+00,andsinceitstermsare allpositive, itwouldalsosufficetoestablish theinequality D"-D!'-_l<~(_1__1_)or1_!?n-=-!.::::;::-!(1__l}~-l) D.D'=rDrDr Dra-r DT '"n-l n-l n n thatistosay,toprovethat foreveryxsuchthat0<x<1.Butthisisobvious atonce,from (1-x1J)=(1-x)(1+x+...+xP-1). Therefore thetheorem isestablished. Additions andExamples. 1.Inthetheorem ofAbel-Dmi, wemayofcoursereplace thequantities Dnbyanyotherquantities D,,'asymptotically equaltothem,orforwhichtbe .Do'l'b fd . . b fratiO-lesetween twoIxeposItive nunters,oreveryn(atleastfromDn somestageon).By70.4theconvergence ordivergence oftheseriesZa. c~\nnotbeaffected bythischange. 2.Bytbetheorem ofAbel-Dml, ~d'='\'dn -fI--DJ! diverges wllhZd".Wemayenquire whatistherelation astomagnitude between thppartialsumsofthetwoseries.Herewebavethefollowing elegant leMath.Annalen, Vol.35,D.329.1890.17ii. 292 ChapterIX.Seriesofpositive terms. dn11D-0,wehaveJ1 "(11d.. dn-+~+ ...+-~logDn·.1)1.lJ:J 1.)71- ThenewparltalsumsthusmCl'ease essenttally ltkethelogarithms ojtheoldones. Proof. Itx..=~"-O. wehave,by112,b, " dn x,. Dn-1.1=---D-,.-- log---- log---- I -X" Dn-l Theundefmed number Dowehereassume =I,alsoreplacing theaboveratio by1forallindices 71forwhichx"=O.Bythetheorem oflimits44,4,since logD.._+00,wethenhave d1dg d"--+-~+...+--___D.....1_1?g Dn =-_1_-r~+ d2_+...+~~]_1. D.. D..logDnD1D. D..logD1+log__e+...+log ~D1 D"- l Thisprovesthetheorem. Further, itisatonceclearthatinthestatement ofthistheorem, the numbers D"mayonbothstdesbereplaced byothersD,,'asymptotically equal tothem. 3.Theseremarks nowenableustoelaborate inthesimplest manner the considerations indicated atthebeginning ofthISsection: 00 a)Theseries2Jdn,withdn=1,i.e.Dn=n,mustbeconsidered asthe n=1 simplest ofalldivergent series,forthenaturalnumbers D"=nformtheproto­ typeofdivergence to+00.Thetheorem ofAbel-Dtnt thenshowsatoncethat theharmonic series1;~{converges n=1nadiverges andthetheorem in2.showsfurtherthatinforIX>1 forIX<1, thelattercasewehaveforIX=1, 1111+"2+3"+...+n~logn theorem ofAbel-Dini, theseries2)~ 71 mayby1.and2.,(cf.12S,2). b)Nowchoosing forIdn,inthe newlyrecognised tobedivergent bya),andreplacing, aswe DnbyDn'=logn,weconclude that 1;1 { converges when n=an(logn)adiverges when Thetheorem in2.showsfurtherthat111-212+-31 3+"·+-I--~loglogn=logg1l.og og nogn- 90v.Cesaro,E.:Nouv.Annales deMath.,(3)Vol.9,p.353.1890. 91Th,scondition iscertainly satisfiedifthenumbers dnremainbounded, ­ benceinalltheserieswhichwilloccurinthesequel. IX>I, IX<1,~31).Theorems ofAbel,DiniandPrmgsheim. c)Byrepetition ofthisextremely simplemethod ofinference, afresh,andquiteindependently ofourprevious results' Starting fromasuttably largeindex(e"+1),theseries 22 .2 1 {converges when nlogn...logp_tn(logpn)" dIVerges when293 weobtain whatever valuet5giventotheposttwe integerp.Theparttalsumsofthesertes forIX=1sa/lsfyIheasymploltc relatIOn Iy'-;-----;- ........,logp+1JI•~v]ogv...lO/lp_tv,]og"pv"=cp+1 4.Atheorem analogous to173,blltstarting fromaconvergent series, isthefollowing: Theorem ofIJini 23.It:EcnISaconvergent seriesofpositIVe lerms,and rn-I=cn+Cn+1+•••denotes lisrelnmnd"r afterIhe(n-I,thlerm.then ""~="" Cn {converges when IX<1 , ~ -.::.J r~'_1 (cn+rn~I-/-•••)"dIVerges when Cl~1. Proof. Thedivergent caseisagainquiteeasilydealtwith,since, forIX=1, en en+Iten+...+en+11: t',.+It----+...+--~>------ =1---;rn-t rn+k-l -1',._, I'n-l andforevely(fixed)n,thisvaluemaybemade>~-byasuitable choice ofk,asrJ,.--O.BySI,2theseriesmusttherefore thelldiverge, ForCl>1 thiswillaforltori alsobethecase,sincernis<1foreverysufficiently largen. If,however, IX<1,wemaychoo~e apositive IntegerpsothatIX<1 -..!..,p anditnowsllfflces -ngambecause rn<1forn>n,-toestablish theCOD­ vergence oftheseries wherer=~. NowI'ntendsmonotonely to0andconsequently ~(1'~-1 -I'~)iscer­ tainlyconvergent withpositive terms. Ittherefore sufftces to~howthat thatistosay (1-yP)<P(1-y) Butthelatterrelation isevident, since0.....:::::y:::::;1. 22IfweWtitee=e',eC'=e",•..,ech')=c(,,+1),•••anddenoteby[e(")J=e. thelargest integer contained in(~lel"),wemaysaythatthefactors inthe denominators ofthetermsofourseriesareall>1,ifnbetakentostart fromthevalue(ep+1). 23v.footnote 18,p.290. 294 ChapterIX.Seriesofpositiveterms. §40.Seriesofmonotonely diminishing positive terms. Ourprevious investigations concerned forthemostpartseriesof quitearbitrary positive terms.Thecomparison seriesusedforthecon­ struction ofourcriteria, however, werealmostalwaysofamuchsimpler nature; inparticular, theirtermsdecreased monotonely. Itisclearthat forsuchseriessimpler lawsaltogether willbecome validandperhaps alsosimplertestsofconvergt'nce maybeconstructed. Wehavealreadyshownin80thatifinaconvergent seriesEenthe termsdiminish monotonely tozero,wehavenecessarily nen->0,afact whichneednotoccurinthecaseofotherconvergent series(evenwith positive termsonly).Again,Cauehy's condensation test77belongs to theseriesweareconsidering. Wepropose toinstitute oneortwofurtherinvestigations ofthis kindand,inthetlrstinstance, todeduceforsuchseriesafewvery simpleandatthesametllneveryfarreaching criteria. Theircon­ vergence, asweshallsee,ISoftenverymuchmoreeasIlydetermined thanthatofmoregeneral typesofsene,;. 00 176. 1.Theintegral test~4.Let2,'anbea givenseriesofmonotonely n=1 diminishing terms.Ifthereexistafunction((x),positiveandmonotone decreasing forx2:I,forwhich fen)=an then2'anconverges if.andonlyil,thenumbersforeveryn. (k=2,i3,...).arebounded ~~. Proof. Since,for(k-1)<t<k,wehavef(t)2:aI,'and fork<t<k-+-1,f(t)::::;;ak,(kaninteger2.:2),Itfollows(by§Ill, Theorem 20),that k+l kff(t)dt<ak<ff(t)dt k k-l Assuming theseinequalities written downfork=2,3,...,nand added,weobtain n+1ff(t)dt<a~-+-as-+-...-+-an=s..-a1< 2n ff(t)dt. 1 14Cauchy: Exercices mathem, Vo!2 p221.Paris1827. 2~By70,4it1Sofcoursesufficient thatl'(n)~houldbeasymptotically proportional tothetermsa..,orthatI'(n)=Cl"anwithapositzve lowerlimit forCl".-Instead ofreqUiring that]"shOl/ldremain bounded, wecanof coursealsorequire thatJ'{(I)dt~houldconverge. Thetwoconditions (by 1 §19,Del.14)areexactly equivalent. §40.Seriesofmonotonely diminishing positive terms. 295 Fromtherighthandinequality itfollows, astheintegralsI"are bounded, thatsoarethepartialsumsoftheseries; fromtheleft handinequalIty theconverse isinferred. 1his,by70,provesallthat wasrequired. Supplement. Thedifferences (s"-In)atthesametimeformamonotone decreasing sequence withlimitbetween0andaI'-Infact,wehave n+l (sn-In)-(sn+l-1,,+1)=Jf(t)dt-a"+I:;;:::0; n whence thestatement follows, sincea1:;;:::s"-In:;;:::a1-I!>0, ThelImitinquestion istherefore certainly positive, Iff(t)isstrictly monotone decreasing, Examples andIllustrations. 1.Thistestnotonlyenables ustodetermine theconvergence ofnumerou<; series,butisalsofrequently ameansofconvenicntly estimating therapidity ofthclrconvergence ordivergence. Thuse.g.wecanseeatoncethatfor a:>Ithesenes '\~_1_ .4.Ja'n=1nsince"In=f~: =__1_(1-_!-)<_1_,t"a-I nu-Ia-I 1 mustconverge, whereas n n~~,whereI,.=I~t=Iogn_+oo, 1 mustdiverge, Hutwelearnfurther that,fora:>I , n+k+l ,,+k j'dtn;,kIIdt-< .4.J-<-- tu,'I'=n+l VClea'n+l n andtherefore 1 I I 1--·------<"n<--·--.a-I(,1+1),,-1 a-Inu-1 Fora=2,thisevalllation wa<;already established onp.260. Tnthesameway thesupplement to176givesafreshproofofthefactthatthediffereol'e [1+}-+...+_I.-lognJ2 n isthetermofalnonotone desccndmg" sequence tending toapositive limit between 0and1.ThiswasEuler's constant mentioned in12S,2. Similarly, thesupplement alsoshowsthatwhen0<a:<I,thedifference n 1 1Idt 1+-+ ...+-- -2" nata 1 i<;thetermofamonotone descending sequence withapositive limitlessthan1. Therefore, inparticular (cL44,6),for0<a<1: 1 1 1 nl-III+-+-+· ••+-':'V---,2aSa n"I-a: anditiseasilyseenthatthisrelation holdsequally whena:;;;;0, 296 Chapter IX.Seriesofpositive terms. 2.Moregenerally, from fdt1--1-__1__,ifatI,=a-I(Jogt)a-1 tlogt ••.IOg"p_1t·(Iogpt)" p logp+ltIifa= 1, wecanImmediatcly deduce, bythesamemethod, theknown conditions ot convergence anddivergence ofAbel'sseries. Wehavenowthreetotally distinct mcthods ofobtaming these.Thesupplement to176agaInaffords us goodevaluations oftheremamders inthecaseofconvergence, andofthe partialsumsinthecaseofdIvergence. 3.Iff(x)beposI/IVe foreverysuffIciently largex,andpossesses, for thosex's,adIfferential coefficient equaltoamonotone decreasing (alsoposi­ tive)function wllhthelimit0atinfInity, theratiof'(x)!f(x)isalsomono· tonedecreasing. Since x ft'Jt)-dt f(t)andff'(t)7(1)dt=logf(t), integrals zJf'(t)dtitfollows thatthe areeitherbothbounded orbothunbounded. Henceweconclude thattheseries 2f'(n)and2l'0Lfen) willeitherbothconverge orbothdiverge. Inthecaseofdivergence, when necessarily f(n)-++00,wehave 5'-.E(n) convergent whena>1• ..:;..J[f(nW Infact,here Jfl(tL l1_ [{(t)]a-Cl-Ir{(t)ja-l whencethevaltdity ofthestatement canbedirectly IOferrcd. -Thesetheorems areclosely connected withthetheorem ofAbel-Dlm. 2.Atestofpractically thesamescope,andindependent ofthe integral calculus initswording, IS Ermal0:0D"s test90• 177. Iff(x)isrelatedtoagivenseries.xanofpositive, monotonely diminishing terms,inthemannerdescribed intheintegraltest,and alsosatisfies theconditions therelaiddown,then 2a=2 (n){co~verges} ite:&f(ex){<-o<l} nfdwerges ((x)~1 foreverysufficiently largex. Proof.Ifwesuppose thefirstoftheseinequalities satisfied for x>xo'wehaveforthesex's eZ z zff(t)dt=feC{(eC)dt~-0f{(t)dt. ,.co 2:0 Zo 20Bulletin dessciences mathem., (1)Vol.2,p.250.1871. §40.Seriesofmonotonely diminishing positive terms. 297 Consequently .,.Z ::t .Z (1-17)Jf(t)dt<{}[Jf(t)dt-Jf(t)dt] xo eXo eX <1'J[Jf(t)dt-Jf(t)di] Xo z .X" <{fJf(t)dt. x" Thustheintegral ontheleft,andhencealsoff(t)dt, is,forevery T"x>xo'lessthanacertain fixednumber. Theseries2'anmustthere­ foreconverge, bytheintegral test. If,ontheotherhand,weassume thesecond inequality satisfied forx>Xl'wehave,forthesex's, eX r XJf(t)dt=J etf(et)dt~J f(t)dt. X, X, Acomparison ofthefirstandthirdintegrals showsfurther that eX eX)ff(t)dt~ff(t)dt. x Xl OntherighthandsideofthisinequalIty, wehaveafixedquantity r>0,andtheinequalIty expresses thefactthatforeveryn(>Xl) wecanassignknsothat(withthesamemeaning forInasin176) n+~n InH,,-In=Jf(t)dt~'1>O. n By46and~O,thenumbersIIIcannotbebounded andIantherefore cannot converge'll7. Remarks. 1.Erl1lalw!l's te~tbearsacertain resemblance toCauchy's condensation testItcontains, inparticular, likethelatter,thecomplete logarithmic com­ parison scale,towhichwchavethusafourthmodcofapproach. Infact,the behaviour oftheseries Z 1 nlogn..logp_ln(logpn)" isdetermined bythatoftheratic logp_tx.(logpx)'" (logp_l x)" 27ItISnotdifficult tocarryouttheproofwithout introducing integrals, butitmakesitrathermoreclumsy. 298 Chapter IX,Seriesofpositive terms. andAsthisratiotendstozero,whenIX>1,but__+00,whenIX::;;1,Ermakof/'s testtherefore provides theknown conditIOns forconvergence anddlvergenc'" oftheseseries,asasserted 28. 2.Wcmayofcoursemakeuseofotherfunctions instead ofe:t.Iftp(x) isanymonotone increasing positive function, everywhere differentiable, for whichtp(x)>xahv.lys, theseries ~:a,.willconverge ordiverge accordmg as wehave ~(x)f(tp(X»{<if<1 f(x)>1 forallsufficiently largex's WithErmakof/'s te~tandCauchy's mtegral test,wehavecommand over themostimportant testsforourpresent series. §41.General remarks onthetheoryoftheconvergence anddivergence ofseriesofpositive terms. Practically thewholeofthe19thcentury wasrequired toestab­ lishtheconvergence testssetforthinthepreceding sections andto elucidate theirmeaning". Itwasnottilltheendofthatcentury, andin particular byPringshcim's investigations, thatthefundamental questIOns werebrought toasatisfactory conclusIOn. Bytheseresearches, whichcovered anextlemelyextensive field,aseriesofquestions were alsosolved, whichwereonlytimidly approached before hIStime, although nowtheyappear toussosimpleandtransparent thatit seemsalmostinconceivable thattheyshouldhaveeverpresented any difficulty2l1, stillmoreso,thattheyshouldhavebeenanswered IIIacom­ pletelyerroneous manner. HowgreatadIstance hadtobetraversed beforethispointcouldbereached isclearifwereflectthatEulcr nevertroubled himself atallaboutquestions ofconvergence; whena seriesoccurred, hewouldattribute toit,without anyhesitation, the valueoftheexpression whichgaverisetotheseries 30.Lagrange in 177031wasstilloftheopinion thataseriesrepresents adefinite value,provided onlythatitstermsdecrease to032.Torefutethelatter 18Thisalsoholdsforp=0,ifweinterpret log_txtomeane"'. 19Asacuriosity, wemaymention that,aslateas1885and1889,several memoirs werepubltshed withtheobjeclofdemonstrating theexistence ofcon- vergent seriesIc"forwhich ~~3"1didDottendtoalimitI(Cf.139,3.) c"1 10Thusinallseriousness hededuced from--=1+x+x2+..'.thatI-x 12=1-1+1-1+-00 • 13=1-2+22-23+-.... Cfthefirstfewparagraphs of§59. 81V.(Euvres, Vol.3,p.61. 12Inthis,bowever. sometracesofasenseforconver~ence maybeseen, §41.General remarks onsenesotpositive terms. 299 a55umption C'xpressly byreferring tothefact(atthattimealready wellknown) ofthedivergence of.2..!..,appears tousatpresentn superfluous, andmanyotherpresumptions andattempts atproofcur· rentinprevious timesareinthesamecase.Theirinterest isthere­ foreforthemostparthistorical. Afewofthequestions raised,how· ever,wbetheransweledintheaffirmative ornegative, remain of sufficient interest forustogivearapidaccount ofthem.Acon· slderable proportion oftheseareindeedofatypetowhichanyone whooccupies himself muchwithseriesisnaturally led. Thesourceofallthequestions whichwepropose todiscuss re~idesintheinadequacy oftheCliteria.Thosewhicharenecessary andsufficient forconvergence (themaincriterion 81)areofsogeneral anature,thatinparticular casestheconvergence canonlyrarelybeascer­ tainedbytheirmeans.Allourremaining tests(comparison testsortrans­ formations ofcomparison tests)weresufficient criteriaonly,andtheyonly enabled ustorecognise asconvergent serieswhichconverge atleast asrapidlyasthecomparison seriesemployed. Thequestion atonce arises: 1.Doesaseriesexistwhichconverges lessrapidlythananyotherI178. This que~tlOn isalready answered, inthenegative, bythetheorem c17:>,4.Infact,when~cnconverges, sodoes~cn'=.2T'though, ",,-1 obviously, lessrapidlythan2:c ,asc:c '=r~-O.'I n n n-I ThequestIOn i.,answered almostmore5implyby].Hada11lard3:\ whotakestheseries2:c'e=2'(Y;-1 -,,/~).Sincec=r1-r,n u- tl nn- n theratiocn:cn'=Yrn-1-1-1/;:-O.Theaccented seliesconver· geslessrapidlythantheunaccented series. Thenextquestion isequally easytosolve: 2.Doesaseriesexistwhichdiverges lessrapidlythananyother? Hereagain,thetheorem ofAbel·Dini 173showsusthatwhen2,'dn diverges, sodoes2:dn'=.2~n,andhencetheanswer hastobein n thenegative. Infactasdn:dn'=Dn--I-00,thetheorem provides, foreachgivendIvergent senes,another whosedivergence isnotso rapid. Thesecircumstances, together withourpreliminary remark~, showthat 3.Nocomparison testcanbeeffective withallseries. Closelyconnected withthis,wehavethefollowing question. raised. andalsoanswered, byAbeIS!: 83Actamathematic a,Vol.18,p.319.1894. UJ.f.d.reineu.angew.!\lath..Vol.3,p.80.1828 300 Chapter IX.SenesofposItive terms. 4.Canwefindpositive numbersPn'suchthat,simultaneousZ", a)Pa-+O} .. .. { convergence} b)nn........ 0aresufftctent cond~twns ford'Pnan~a> zvergence 01everypossible series01positive terms? Itagainfollows fromthetheorem ofAbel-Dini thatthisisnot thecase.Infact,ifweputan=p~,a>0,theseries,2'annecessar- ilydiverges, andhencesodoes~an'=.2-':",wheresn=a1+...+an. 11 But,forthelatter,pa'=~-+O. 71n Sn Theobjectofthecomparison testswas,tosomeextent,thecon· structlOn ofthewidestpossIble conditions sufficient forthedetermination oftheconvergence ordivergence ofaseries. Conversely, itmightbe reqUIred toconstruct thenarrowest possible conditions necessary for theconvergence ordivergence ofaseries.Theonlyinformation we havesofargathered onthISsubject isthatan-+0isnecessary for convergence. Itwillatonceoccurtoustoask: 5.Mustthetermsanofaconvergent seriestendtozerowith anyparticular rapidity? ItwasshownbyPringsheim:i:; thatthisis notthecase.However slowlythenumbers Pnmaytendto+00,we caninvariably construct convelgl'ntserils2'cnforwlllch limPncn=+00. Indeed everyconvergent senesXcn',byasuitable rearrangement, will produce aseriesXcntosupport thISstatement:l6• Proof. Weassume giventhenumbersP",Il1creasing to+00, andtheconvergent series2:cn'.Letuschoose theindicesnl'n~,".., n~....oddandsuchthat (v=1,2,"..) andletuswritecn=C;"-l,fillingintheremaining cn'sWJththeterms ~ c/'c/'...intheiroriginal order.TheseriesXcnisobviously arc- arrangement ofL:cn'.But P"cn>v whenever nbecomes equaltooneoftheindicesn."Accordingly, as asserted, limPncn =+00. Theunderlying factinthisconnection issimplythatthebehaviour litasequence oftheform(Pncn)bearsnoessential relation tothatof ••Math.Annalen, Vol.35,P344.1890 16Cf.Theorem 82,3,whIchtakesintoaccount asortofdecrease on theaverage ofthetermsa... §41.General remarks onsenesofpos1tlve terms. 301 theseries ~en-i.e.withthesequence ofpartialsumsofthis series,-sincethelatter,though nottheformer, maybefunda­ mentally altered byarearrangement ofitsterms. 6.Similarly, nocondition oftheformHmP..dn>0tSnecessary forthedivergence of2,'dn,however rapidly thepositive numbers Pn mayincrease to+ooa7.Onthecontrary, everydivergent seriesL:dn', provided Itstermstendto0,becomes, onbeingsuitably rearranged, aseriesL:dn(stilldivergent, ofcourse) forwhichlimPndn=O. Theproofiseasilydeduced onthesamelinesasthepreceding. Thefollowing question goessomewhat further: 7.Doesascaleofcomparison testsexistwhichissufficient for allcases? Moreprecisely: Givenanumber ofconvergent series '"(1)'1(.,) ~(k)..:Jcn'~cn",•••,£.cn,••• eachofwhichconverges lessrapidlythanthepreceding, withe.g. c(k+l) ~(,,)--+oo, nforfixedk. wehaver(2)1"hc(2)--.2C(1) n<2'Wit"~n r(3)1c(3)>2Cn(2)""n<29"n ""~Thelogarithmic scaleaffordsanexample ofsuchseries.) Isitpos­ sibletoconstruct aseriesconverging lessrapidly thananyofthegiven seTtcs?Theanswer isintheaffirmative3s•Theactualconstruction ofsuchasenesisindeednotdifficult. Withasuitable choiceofthe indices 111,n2,••"'nI;'...,theseries (1) (1; (1)+ (~) ..J(2)+(3) Cn~Cl +C2+.··+c", C",+I+···'-C". C".+I+··· +c~~)+C~~)+l+... ISitselfofthekindrequired. Weneedonlychoose theseindicesso largethatifwedenotebyrn(k)theremainder, afterthenthterm,of theseries~c"(k), foreveryn~ni' ".,"(k+1) 1 r"<2k"C1k+1)>2(k)" cn Theseries2,'cnIScertamly convergent, foreachsuccessive portion of itbelonging tooneoftheseries~c"(k)iscertainly lessthanthe 87pylltgshelm, lococit.p.357 I'Forthelogarithmic scale,thisW;)SshcwnbyP.duBois-Reymond a.f. J.reinc 11.angew. Math.,Vol.76,p.88.1873).Theaboveextended solution isduetoJ.Iladamard (Actamath.,Vol.18,p.325.1894). 302 Chapter IX.SeriesofposItive terms. remainder ofthisseries,starting withthesameinitialterm,i.c 1<2k(h=2,3,•..).Ontheotherhand,foreveryfixedk, e"+c,,(/"- 00; Infactforn>nq(q>k)wehaveobviously CC~kj>2Q -k•Thisproves "allthatwasrequired. -Inparticular, therearesenescOllverging moreslowlythanalltheseriesofourlogarithmic scaleau. 8.Wemayshow,quiteassimply, that,givenanumber ofdi­ vergent series ~~dn(k),k=1,2,...,eachdiverging lessrapidly than thepreceding, with,specifically, d~k+l)-7-dn(k)-..0,say,therearealways divergent series ~dndiverging lessrapidly thaneveryone ofthe series ~d n(1,)• Antheaboveremarks bringusneartothequestion whether and towhatextentthetermsofconvergent seriesarcfundamentally distm guishable fromthoseofdivergent series.Inconsequence of7.and8.,wc shallnolongerbesurprised attheobservation ofStzeltjes; 9.Denoting by(El'E',J'•••)anarbitrary monotone descending se­ quencewithlImit0,aconvergent series~cnandadivergent series ~dncanalwaysbespecified, suchthatcn=End.,.-Infact,Ifen-..0 monotonely, P=~-..+00monotonely. Theseries nE" whosepartialsumsarethenumbers Pn'istherefore divergent. By thetheorem ofAbel-Dini, theseries ISalsodivergent. Buttheseries 2~c=2~Ed=='\'(-!.__1_]IS n n n 4..JPIt1)"+1' convergent by131.- Thefollowing remark ISonlyarc-statement inotherwordsof theabove: 10.However slowlyPn-+00,thereisaconvergent series ~cn andadivergent series2'dnforwhichdn=Pncn. Inthisrespect, thetworemarks duetoPringsheim, givenin5. and6.,maybeformulated evenmoreforcibly asfollows: 19Themissing initialt\lrmsoftheseseriesmaybea~sumed tobeeach replaced byunity. §41.General remarks onseriesotpositive terms. 303 11.However rapidly 2,'cnmayconverge. therearealwaysdivergent series.-1nrleell dim"rgent serieswith'lllOlwtonely tlilninishing termsorlimit0,-torwhich 1"d"°Im-= •_C" Thus2'd,.musthaveaninfimtenumber oftermsessentially smaller thanthecorrespondmg termsof.2:cn'Conversely: However rapidly2'dnmaydIverge, provided onlydn-0,there arealwaysconvergent series2'cnforwhichlim~n=+00... Wehaveonlytoprovetheformerstatement. Hereaseries2'dn oftheform 00 111Vd=,,,,c+c+...+c+--c+--C+-...+--C.."':"0" 1 1 12n,2n, 2'" 1 1 1 1+"3cn,+"3cn,+...+;rcn,+-"4cn•+... isoftherequired kind,iftheincreasing sequence ofindicesnI'n2,... bechosen suitably andthesuccessive groups ofequaltermscontain respcctinJy nI'(n.!-n1).(nJ-n2),•••!erms. Infact,inorderthat thiSsenesmaydiverge, Itissufficient tochoosethenumber oftermsm eachgroupsolargethattheirsum>1,andinorderthatthese· quence oftermsintheseriesbemonotone, ItISsufficient tochoose nk>nk-1solargethat(n k<Cnk_ 1(k=1,2,...;no=1)as1Salways possible, sl11cecn-+O.Astheratio~~hasthevaluek:1forn=nk, ditfollows thatlIm---'!=0,asreqUired.--en Intheprecedmg remarks wchaveconsidered onlyconvergence ordivergence perse.Itmightbehopedthatwi:hnarrower require­ ments,e.g.thatthetermsoftheseriesshould clIminish monotonely, acorrespondingly greater amount ofinformation couldbeobtained. Thus,a')weh;lveseen,foraconvergent series:Eenwhoseterms diminish monotoncly, wehavenc"-.0.Canmorethanthisbeasserted? Theanswer isinthenegative (cf.Rem.5): 12.However slowlythepositive numbers p..mayincrease to+-00, therearealwaysconvergent seriesotmonotonely diminishing terms torwhich np"c.. notonlydoesnottendtoO.buthas+-00torupperlimit40. 40Prmgsheim, loc.cit.Inparticular itwasmuchdiscussed whether for convcrg-ent seriesofpositive term'>. diminishing- monotont'1y, theexpression nlogn·cnmust-+Ojtheopinion washeldbymany,aslateas1860,that nlog1I.cn-+0wasnecessary forconverj{ence. 304 Chapter IX.Seriesofpositive terms. Theproofisagainquiteeasy.Choose indices n1<n2<... suchthat Pn~>4V(v=1,2,...) andwrite =c21=...=Cn=-cc--=-:, 1n,VPn, cn~_l ~1= .1 •=Cn=--=,vn,...)Pn" Thegroupsoftermshereindicated contribute successively lessthan !,_!-,...,~,... tothesumoftheseriesIc,sothatthissenes 2222" n ""illconverge. Ontheotherhand,foreachn=n"wehave nPncn=VPn' sothat,aswasrequired, TImn·Pn,cn=+oo. 13.Theseremarks mayeasilybemultiplied andextended inall possible directions. Theymakeitclearthatitisquiteuseless to attempt tointroduce anything ofthenatureofaboundary between convergent anddivergent series,aswassuggested byP.duBois­ Reymond. Thenotioninvolved isofcoursevagueattheoutset. But inwhatever manner wemaychoose torenderitprecise, itwillnever correspond totheactualcircumstances. WemayIllustrate thisonthe following lines,which 0bviously suggest themselves u. a)AslongasthetermsoftheseriesIcnandIdnalesubjected tonorestriction (excepting thatofbeing> 0),theratio ~niscapable n ofassuming allpossible values,asbesides theinevitable relation 1·en0lm-d= - nwemayalsohave--;-enhmy=+00. n Thepolygonal graphsbywhichthetwosequences (en)and(dn)maybe represented, inaccordance with7,6,cantherefore intersect atanin­ definite number ofpoints(whichmaygrowmoreandmorenumerous, toanarbitrary extent). ..Adetailed andcarefuldiscussion ofallthequestions belonging tothesub­ jectwillbefoundinPringsheim's workmentioned onp.2,and al~oinhiswntmgs IntheMath.Ann.Vo!'30andintheMunch. Ber.Vo!'2(;(18UH)and27(18U7), towhichwehaverepeatedly referred. §42.Systematization ofthegeneraltheoryofconvergence. 305 b)Byourremark 11,thisremains truewhenthetwosequences (cn)and(dn)arebothmonotone, inwhichcasethegraphsabovereferred toarebothmonotone descending polygonal lines.Itistherefore certainly notpossible todrawalinestretching totheright,withtheproperty that everysequence oftype(cn)hasagraph,nopartofwhichliesabovetheline inquestion, andeverysequence oftype(dn)agraph,nopartofwhichlies belowthisline,-evenifthetwographsaremonotone andareconsidered onlyfromsomepointsituated atasufficiently greatdistance totheright. 11.Notes]1and12suggest thequestion whether thestatements theremaderemainunaltered ifthetermsoftheconstructed seriesECn andEdnarenotmerelysinzply monotone asabove,butfullymonotone inthesenseofp.263.Thisquestion hasbeenanswered intheaffirmative byH.IIahn 42. §42. Syst~matization ofthegeneral theoryofconvergence. Theelement ofchanceinherent inthetheoryofconvergence as .developed sofargaverisetovariousattempts tosystematize thecriteria frommoregeneral pointsofview.Thefirstextensive attempts ofthis kindweremadebyP.duBois-Reymond 43,butwerebynomeansbrought toaconclusion byhim.A.Pringsheim 44hasbeenthefirsttoaccomplish this,inamanner satisfactory bothfromatheoretical andapractical stand­ point.Wepropose togiveashortaccount oftheleading features ofthe developments duetohim45. Allthecriteriasetforthinthesechapters havebeencomparison tests, andtheircommon sourceistobefoundinthetwocomparison testsof thefirstandsecondkinds,157and158.Theformer, namely (I) e, isundoubtedly thesimplest andmostnaturaltestimaginable; notso thatofthesecondkind,givenoriginally intheform (II)an+1.•C"+l an Cne,'J)-. ..II.IIalm,DberRelhcnmitmonoton abnehmenden Ghedern, Monatsheft f.Math.u.PhySlk, Vo!.a:J,pp.121-134-, 1923. 43J.f.d.reineu.angew.Math.Vo!.76,p.61.1873. 44Math.Ann.Vo!.35,pp.297-:J!H. 1890. 4.Wchaveallthemorereasonfordispensing withdetailsinthisconneXlOn, seeingPrill!(sheim's researches havebeendeveloped bytheauthorhimself ina verycomplete, detailed, andreadilyaccessible form. 306 Chapter IX.SeriesofposItive terms. Inconsidering theratiooftwosuccessive termsofaseriesweare already goingbeyond whatisdirectly provided bytheseriesitself. Wemighttherefore inthefirstinstance endeavour toconstruct further typesoftestsbymeansofothercombinations oftwoormoreterms oftheseries. ThIsprocedure has,however, notyielded anycriterion ofinterest inthestudyofgeneral typesofseries. Ifwerestrict ourconsideration totheratiooftwoterms,itis stillpossible toassignanumber ofotherformstothecriterion ofthe secondkind;e.g.theinequalities maybemultiplIed bytlIepositive factors anorcnwithout altenng theirsignificance. Weshallreturnto thispointlater.Except fortheserelatively unimportant transformations, however, wemustregard(I)and(Il)asthefundamental formsofall criteria ofconvergence anddivergence46•Allconceivable specialcom­ parison testswillbeobtained byintroducmg in(1)and(ll)allconceiv­ ableconvergent anddivergent series,and,ifnecessary, carrying out transformations ofthekindjustindicated. Thetaskofsystematizing thegeneral theoryofconvergence will accordingly involve aboveallthatofproviding ageneral surveyofall conceivable convergent anddivergent series. Thisproblem ofcoursecannot besolvedinaliteralsense,since thebehaviour ofeveryserieswouldbedetermined thereby. Wecan onlyendeavour toreduce ittofactors inthemselves easiertosurvey andtherefore notappearing sourgently torequire furthertreatment. Pringsheim shows-andthisisessentially thestarting pointofhis investigatIOns -thatasystematzzation ofthegeneraltheoryofconvergence canbefullycarried outwhenweassume asgiventhetotalityofall monotone sequences of(Positive) numbers increasing to+00. Suchasequence willbedenoted by(P,,);thus andP,,-+OO. Inprinciple, theproblem issolvedbythetwofollowing simple remarks: a)El"eI'Ydivergent series2'd"isexpressible intheform '"I(1..!!::Po+(Pt-Po)+...+(P..-P..-t)+... •t=oO (eachinoneandonlyoneway)intermsofasuitable sequenceat type(p,,).Also,everyseries01thisformisdivergent. <.Thus-sinee(asseenIn160,1,2)(11)isnconscqucnce of(I)-.it isultimately from(1)thatalltherestfollows. §42Systematization ofthegeneral tneoryofconvergence. 307 b)Et'cryconl'el'yent series 47~:c"isexpressible intheform "e..==(_1__1)+(_1__1)+...+(_I__1_)+... ,,~Il PuPt PI P~ PnP..+l (eachinoneandonlyoneway)intermsofasuitablesequence of type(Pn)'Also,everyseriesofthisformisconvergent. 48. Infact,whenthesestatements havebeenestablIshed, wehave onlytosubstitute, inthetwocomparison tests(1)and(ll), Pnt-1-Pn ----~-P,,'Pn11and respectively forcnandd",toobtaininprinciple allconceivable tests oftheflrstandsecondkinds:Allparticular criteriamustnecessarily followbymoreorlessobvious transrormation fromthetestssoob· tamedjforthisveryreason,theformercanneverpresent anything fundamentally new.Theybecome ofconsIderable importance, how­ cver,IIIthattheygivedeeperinsightintotheconnexion between the variouscriteriaandstatethelatterinacoherent form,andalsoapply theminpractice. Hereinliesthechiefvalueofthewholemethod. It wouldaccordingly bewellworthourwhllctodescribe thedetailsof theconstruction ofspecialcriteria exactly; butforthereasons given, weshallabidebyourplanofgivingonlyabriefaccount. 1.Thetypicalformsa)andb)mustberegarded asundoubtedly 180. thesimplest imaginable formsforconvergent anddivergent series. Butwecanobviously replace thembymanyotherforms,thereby altering theoutward formofthecnteria 111vanousways.Forinstance, bythetheorem ofAbet-Dini17:1, and diverge with.:E(p"-Pn-I),whileatthesametime,byPringsheim's theorem17t, and converge fore>o.Withafewrestrictions oflittleimportance, all divergent andconvergent seriesarealsoexpressible inoneofthese newforms. 2.Sincetheonlycondition tobesatisfled bythenumbers Pn' IDthetypicalformsofdivergent andconvergent serieswhichweare t7Unlessthetermsareall0fromsomestageon. 18ThepIoafsofthesetwostatements aresoeasythatweneednotgointo themfurther. 308 Chapter IX.Seriesofpositive terms. considering, isthattheyaretoincrease monotonely to-f-00,wemayof coursewritelogPmlog2Pm...orgenerally F(Pn)insteadofPmwhere F(x)denotes anyfunction defined forx>0andincreasing monotonely (inthestrictsense)to+00withx.Thisagainleadstocriteria which, though notessentially new,areformally sowhenthePn'sarespecially chosen. Itiseasytoverifythatthefirstnamedtypesofseriesdiverge orconverge moreandmoreslowly,asPn---++00moreandmoreslowly; byreplacing Pnsuccessively e.g.bylogPmlog2Pm...,wetherefore obtainameansofconstructing scalesofcriteria 49.ThecasePn=n naturally callsforconsideration onaccountofitspeculiar simplicity; the development oftheideasindicated aboveforthisparticular caseforms themaincontents of§§37and38. 3.Afurtheradvantage ofthismethod isduetothefactthatoneand thesamesequence (Pn)willservetorepresent bothadivergent andacon­ vergent series.Thecriteriatherefore naturally occurinpairs.E.g.every comparison testofthefirstkindmaybededuced fromthepairoftests: f;SPn~-Pn-l P,,·Pn-l anl::::::Pn--Pn--l-Pn-l andsimilarly forothertypicalformsofseries. 4.Therighthandsidescanbecombined toformasingledisiunctive criterion, ifweintroduce amodification, arbitrary incharacter 10sofaras itisnotnecessarily suggested bythegeneraltrendofideas,butotherwise ofasimplenature. Weseeatonce,forinstance, thattheseries and converge when IX>1anddivergewhen Q(<].Forthefirstoftheseseries theproofhasjustbeengiven;andthesecondhasallitstermslessthan thefirstifIX>1,whileifIX=1,andhenceforallIX>1,itisimmediately seentobedivergent. Thepairofcriteriasetupin3.mayaccordingly bereplaced bythefollowing disjunctive criterion: {Cl>1 with Cl~1 ..Theusualpassage fromPndirecttologPn•log2Pm•••,isagainquitean arbitrary step,ofcourse. Theorems 77and175,2renderthestepnatural, however. Between e.g.PnandlogPn'wecouldeasilyintroduce mtermedlary stages,for instance eVIO~Pn, whichincrease. lessrapidly thanPn.-infactlessrapidlythan anyfixedpositive powerofPn•however smallitsexponent, -yetmorerapidly thaneveryfixedpositive poweroflogPmhowever largeitsexponent. !l42.Systematization ofthegeneral theory ofconvergence. 309 and,IIIallessentials 50,alsoby: with{a>1 ((s::1 Itisremarkable thatinthecriteria ofconvergence arisingthrough thesetransformations, theassumption Pn-++00isnolongernecessary atall.Itissufficient that(Pn)shouldbemonotone. Infact,it(Pn)isboun- ded,theconvergence of2,'(Pn-Pn-I)' andhencethatof2;"IY ~~n=-!.-and .J:!,,-_=-~n-l forarbitrarya>0,follows fromthatof(p),as(P-a)aPn n n and(a-PI')arealsobounded sequences. Theseconvergence tests51thus pos'3ess aspecialdegreeofgenerahty, simIlartothatofKummer's 52cri­ terionofthesecond kind,mentioned below III7. 5.Fromthisdisjunctive criterion -asindeedingeneralfromany criterion -o.hersmayagainbededuced byvarious transformations, though thecriteria soobtained canbenewonlyinform.Forthese transformations wecanofcourselaydownnogeneral rule;newways mayalways befoundbyskillandintuitiOn. Thisisthereason for thegreatnumber ofcriteriawhichultimately remainoutside thescope ofanygivensystematization. Itisobvious thateveryincquality maybemultiplied byarbitrary positivc factors without altering itsmeaning; Similarly wemayform thesamefunction F(x)ofcithermember, provided F(x)bemonotone increasmg (inthestricter sense), -inpartiCUlar wemaytakelog­ arithms, roots,etc.ofeitherside.E.g.thelastdisjunctive criterion maytherefore beputintotheform orlog(P,.-Pn-I)-loga,.{~fJ>0 P,. <0 :,./~a,,_ ~~_{s::{)<1VPn-Pn-l~1 Weseeataglance thatbythismeansweobtainageneral frame· workforthecriteria ofthepreceding sections whichweresetupby assuming Pn=nor-logpn. 50TheeqUIvalence isnotcomplete, i.e.withthesamesequence (P,.)asbasis, thenewcriterion isnotsoeffective astheoldone;infact,thedivergence of .2Pn-I!.n-I,forinstance, maybeinferred fromtheoldcriterion, butnot Pn fromthenewone .,Prtngshelm: Math.Ann,Vol.35,p.342.1890 6.Journ.f.d.reineu.an~ew. Math.,Vol.13,p.78.1835 It (051) 310 Chapter IX.Seriesofpositive terms. 6.Substantially the5amcremarks remain valId".whenwesub· stitute Pp"-:-pp,,-tforcnandPn-P"-lfordflinthefundamental cri· nn-l terionofthesecond kindlII),orperform anyoftheothertypical substitutions forcnanddnthere.Inthiswayweobtainthemostgeneral formofthecriteriaofthesecondkind. 7.Wemayobserve(cf.Rem.4.)thathereagain,aftercarrying outasimple transformation, wemaysoframetheconvergence test thatitcombines withthedivergence tcsttoformasingledisjunctive criterion. Theconvergencc testrequires inthefirstinstance that,for everysufficiently largcn, or IfI I b P"-p"-11f . I'dlcrewerepacec"y'-PP--,tleormermequaltyreucesto n-n-l asp"cancels out,thetypicaltermsofadivergent seriesautomatically appear, sothattheconvergence testreduces to or e. Finally,ifwetakeintoaccount thefactthat~edn(e>0)diverges with~dn,thecriterion takestheform: e. Nowtheoriginal criterion iscertainly satisfied bytheac;sumption 1a,,+1 1........ 0----·--~e> .c" anC"+1- Itthus app~ars thatinthisform-slightly lessgeneral thanthe original form-oftheconvergence test,itisabsolutely indIfferent whether aconvergent seriesoradivergent seriesisintraduced ascomparison series.Hence,stillmoregenerally, thecn'sanddn'sintheaboveforms ofthecriterion maybereplaced byany(positive) numbers bn;thus wemaywrite: e. Exercises onChapter IX. Thisextremely general criterion isduetoE.Kummer63• Ontheotherhand, 1(/n+l 1 {~Q>0 dn--0;:-' tln+1~0311 IS1. represents adisjunctive criterion ofthesecondkindwhichimmediately follows, asthepartrelative todivergence ismerelyaslighttrans· formation of(II) Allfurtherdetailswillbefoundinthepapersandtreatiseby A.Pringsheim. Thesequences ofideassketched abovecanofcourse leadonlytocriteriahavingthenatureofcomparison testsofthefirst orsecondkinds,thoughallcriteriaofthischaracter maybedeveloped thereby. Theintegral test176andErmakotl's test177ofcourse couldnotoccurintheconSIderations ofthissectIOn, astheydonot possess thecharacter inquestIOn. Exercises onChapter IX. 133.Proveinthecaseofeachofthefollowing seriesthatthegiven indications ofconvergence ordivergence arecorrect: ~, 5)., e)~(x-+-1)(2x+1)(nx-+-1) ~(y+l)(2y+l) (ny+l)for{y>x>o x~y>O 68Itwasgivenbyl\"mmcr asearlyas1835(lourn.f.d.reineu.angew. Math,Vo!.13,p.172)thoughwitharestnctive condition whichwasfnstre­ cognized assuperfluous byU.DIn!in1867.Lateritwasrediscovered several timesandgaverise,aslateas1888,tov,olent contentions onquestIOns of pnority, O.Stolz(Vorlesungen tiberallgem. Arithmetik, Vol.1,p.259)wasthe firsttogivethefollowing extremely simpleproof,bymeansofwhichthe criterion wasfirstrendered fullyintelhgible: Directproof: Thecriterion isthatfromsomestageon a"b"-a"+l·b"+l ~(!all' Itfollows inparticular thattheproducts allb"diminish monotonely and therefore tendtoadefinite limit,,:::::: O.By131,..E2-(a"b"-a"+1b"+I)is - (! thusaconvergent senesofpositive termsAndasitstermsarenotlessthan thecorresponding termsof~.a",thisseriesisalsoconvergent. 312 Chapter X.Seriesofarbitrary terms. 134.ForeveryfixedP,theexpression [~ 1 -log+lnJ_vlogv...logpv P hasadefinite limitCpwhenn-.+00Iifthesummation commences with thefirstinteger forwhichlogpon>1. 13:5.Foreveryfixed(!in0<(!<1Itheexpression i;[1~1!-nnC!J,,=1v c:: hasadefinite limiti'C!whenn-.+00. 136.Ifxn-.;,Itfollows that [~~n~~+_p~~~~+_;;~q +..'+i?;n;~--]__{~logp', wherep,p',andqdenotegivennatural numbers. 1:17.If2:dnisdivergent, withdll-0,andiftheDII'sareitspartialsums wehave n 1.2d"D"'"'""'Z-D n.,,=1 13S.If~anhasmonotonely diminishing terms,itiscertainly divergent whenp.apn-all>0forafixedpandeverysufficiently largen. 139.If0<dll<1foreveryn,thetwoseries Zdn+l[(1-do)(1-d1)•••(1-d"W, .2 dn+_1 _ [1+do)(1+d1) •• ,(1+d,,)]e' areconvergent, forevery (!>O. 140.Giveadirectproof,without theuseofErmakof!'s testandwithout theheIpoftheintegral calculu,;, ofthecriterion __2n a~n{<1 lim~~-->2 forseriesofmonotonely diminishmg terms 141.Iftheconvcq;cncc ofascries2.'allfollows fromoneofthecriteria ofthelogarithmic scale164,H,then,ason-.+00, [nlognloglln."IOgkn].all_0 anddiminishes monotonely fromacertain stagcon,whatever thevalueofthe positive integer kmaybe. Chapter X. Seriesofarbitrary terms. §43.Testsofconvergence forseriesofarbitrary terms. Withseriesofpositive terms, thestudyofconvergence and divergence wascapable ofsystematization tosomeextent; inthe caseofseriesofarbitrary terms, allattempts ofthiskindhave tobeabandoned. Thereason liesnotsomuchininsufficient de- §43.Tebtsofconvergence torseriesatarbitrary terms. 313 velopment ofthethcory, asintheessence ofthematter itself. Aseriesofarbitrary tcrmsmay·converge, without converging abso­ lutelyl. Indeedthisispractically theonlycasewhichwillinterestus here,asthequestion ofabsolute convergence reduces, by8ii,tothe studyofaseriesofpositive terms.Wetherefore needonlyconsider thecaseinwhicheitherthesenesisactually notabsolutely conver· gentoritsabsolute convergence cannot bedemonstrated byanyof thepreviously acquired means.IfaseriesISconditionally conver· gent,however, thisconvergence isdependent onthemodeofsucces~ion ofthetcrmsaswellasontheirindividual values;anycomparison test whichwemightsctupwouldtherefole havetoconcern theseries asawhole,andnotmerely itstermsindividually, asbefore. ThIs ultImately meansthateachscrieshastobeexamined byitselfand wecannotobtainageneral method ofapproach validforthemall. Accordingly wehavetobecontcnt toestablIsh criteria witha morerestricted fieldofvalidity. Thechiefinstrument forthepurpose istheformula knownas o,,4fJel's partialsummation2•Ifao,at'...andbo'bl''"denote182. arbitrary numbers. andwewrite (n~0) thenforeveryn>0andeveryk~1, ,,+k n+'.I(t~1J~= 2,'A~(1J.-b~+I)-A,,·1J"+I+An+k·b"+k+l ••=,,+1 .=,,+1 ':>ysummation from).=n+-1tov=n+-k,thestatement atonce follows 3. oSupplements. 1.Theformula continues toholdwhenn=-1,183. ifweputA-1=o. 1Thecaseinwhichtheseriesmaybetransformed intoonewithposi. tivetermsonly,bymeansofa"finitenumber ofalterations" (v.82,4)orby achange ofsignofaUitsterms,ofcourserequires nospecialtreatment. 2Journf.d.reineu.angew. MathVol.1,p.314.1826. •Itissometimes moreconvenient towritetheformula intheform n+k n+k-l ~'a~b~~ _':'A.(b.'-b.+1)-A,.b"+1+A,,+kb,,+ •.v=nt-1 ,-nt1 314 ChapterX.Seriesofarbitrary terms. 2.Ifcdenotesanarbitrary constant, andAv'=Av+c,wehavealso: n+k n+kl:apbp=l:A,:(b"-b,,+!)-A';bn+!+A~+!.:.bn+k+! V--n+l vn+1 -forav=Av-A"-l=Av'-A:_1• Accordingly, inAbel'spartialsummation we"may"increase or diminish alltheAv'sbyanyconstant amount. Thisisequivalent toalter­ ingao. Abel'spartialsummation enablesustodeduceanumberoftestsof convergence forseriesoftheformEavb"almostimmediately 4.Inthe firstplace,itprovides thefollowing general IS4. 0Theorem. Theseriesl:a"b"certainly converges,if 1)theseriesEA"(b"-bv+!)converges, and 2)limAll.bp+lexists. p->+'" Proof. Abel'spartialsummation givesforn=-1: k k l:a"b"=EAv(b"-b,.+!)+A"b"+1' v~O "=0 foreveryk>0;makingk--++00,thestatement follows, inviewof thetwohypotheses. -Therelationjustwrittendownshowsfurtherthat s=s'+1 where Ea"bv=s,EAv(bv-b"+l)=s',limAllbp+!=l. Inparticular, s=s'if,andonlyif,1=0. Thetheorem doesnotsolvethequestion astotheconvergence ofthe seriesEa"bv,sinceitmerelyreducesittotwonewquestions; butthese areinmanycasessimplertotreat.Theresultisinanycaseafar-reaching onc,anditenablesusimmediately todeducethefollowing morespecial criteria,whicharecomparatively easytoapply. o1.Abet'stest5.l:Qvb"isconvergent ifl:avconverges and(b,.) ismonotone 6andbounded 7. •Wecanofcoursereduceany serie~tothisform,asanynumber canbe expressed astheproduct oftwoothernumbers. Success Inapplymg theabove theorem Willdependontheskillwithwhichthetermsaresosphtup. 6locoClt.-Abet'stestprovides asuffiCient condition tobesatl,lied by(bn), inorderthattheconvergence of1:anmayinvolvethatof1:anbn.J.Hadl/mard (Actamath.,Vol.27,p.177.1903)givesnecessary andsuffiCient conditions; cf. E.B.Elliot(Quarterly Journ.,Vol.37,p.222.1906),whogivesvariousrefinements. •Inanticipation oftheextensIOn tocomplex numbers (v.p.307)itmaybeem­ phasized alreadythatasequence ofnumbers assumed tobemonotone isnecessanly real. 1Inotherwords:Aconvergent series"may"bemultiplied, termbyterm,by factorsformingabounded andmonotone sequence. -Theorem 184andthecriteria deduced fromitalldealWiththequestion: Bywhatfactorsmaythetermsofa convergent senesbemultplied sothataconvergent seriesresults? Andbywhat factorsmustthetermsofadivergent seriesbemultiphed, sothattheresulting series maybeconvergent? §43.Testsofconvergence forseriesofarbitrary terms. 315 Proof. Byhypothesis (A,,)and(bn),(v.46),andhencealso(A"b"+l)' areconvergent. Ontheotherhand,by131,theseriesJ:(bv-bv+1)is convergent, andindeedabsolutely convergent, asitstermsallhavethe samesign,inconsequence ofthemonotony of(b,,).Itfollows, by87, 2,thattheseriesEAv(h.,-bv+!)isalsoconvergent, sinceaconvergent sequence iscertainly bounded. Thetwoconditions oftheorem 184are accordingly fulfilledandJ:avbvisconvergent. 02.Dirichlei's test8.Eavbvisconvergent IfEavhasbounded partialsumsand(bn)isamonotone nullsequence. Proof. Bythesamereasoning asabove,EAv(bv-bv+!)iscon­ vergent. Further, as(A,,)isbounded, (Anbn+1)isanullsequence if(b,,) is,i.e.itiscertainly convergent. Thetwoconditions of184areagain fulfilled. 03.TestsofduBois-Reymond9andDedekindlO a)Eavbvisconvergent ifJ:(bv-b.,+!)converges absolutely andEav converges, atieastconditionally. Proof. By87,2,EA,,'(bv-bv+!)alsoconverges, as(A,,)iscer­ tainlybounded. Sincefurther (bo-bI)+(bI-b2)+...+(b"_I-bn)=bo-bn tendstoalimitwhen 1l---*+cLJ,sodoesb"itself;limA"existsbyhypo­ thesis,andtheexistence oflimA"b,,+!follows. b)Eavb"isconvergent ifE(bv-bv+I)converges absolutely andEav hasboundedpartialsums,provided bn---*O. Proof. EAv(bv-bv+!)isagainconvergent andAnbn+l---*O. Examples andApplications. 1.Theconvergence of1:aninvolves, byAbel'stest,thatofEa",n 2.1:(-1)"hasbounded partialsums. Henceif(bn)isamonotone null sequence, 8Vorlesungen uberZahlentheorie, 1stedition, Brunswick 1863,§101. 9Antflttsprogramm d.Univ.Freiburg, 1871.-ThedesignatIOn above adopted forthethreetestsisratheraconventional one,asallthreearesubstantially duetoAbet.Forthehistoryofthesecritenacf.A.Pringsheim, Math.Ann.,Vol. 25,p.42:1.IHHi;. 10§14:1oftheworkreferred toinfootnotes.IS:> 316 Chapter X.Seriesotaroltr!lry terms. converges byDU'lchIet's test.ThisisafreshproofofLeibniz's criterion for serieswithalternately positive andnegative terms(82,5). 3.Givenpositive integers ko,kl,k~,•.,suchthatI(_l)knhasbounded partial s,'m~-forthistheexcessofthenumber ofevellintegers overthat ofoddintegers amongthenfirstexponents k"kg,•••,knhastoremainbounded asn-++00-theseries I(-l)knbn converges, if(bn)denotes anynullsequence. 4.IfIa"isconvergent, thepower 5eriesIa"x"isconvergent foro<x:<:;:;+1,sincethefactorsx"formamonotone andbounded sequence. ]fIa';-merely hasbounded partialsums,thepowerseriesatanyratecon­ verges foreveryxsuchthat0~x<1,sincex..thentendsto0monotonely. 5.Theseries ~:sinnxand-2,'cos.nxhavebounded partialsums,thefirst forevery(fixed)realxandthesecond forevery(fixed)realxnotamultiple of2:re.Thisfollows fromthefollowing elementary butimportant formula, vahd 11foreveryx=F2k11: sinn~sin(0:+(n+1)~) sin(0:+x)+sin(0:+2x)+•..+!>in(0:+nx)=-----'----- .XsIn--2 Theproofoftheformula isgivenin201.ForIX=0,weget .x.(1)xsmn-"sln n+-.• . 2 . 2 2smx+sm x+···+smnx= ----- ,. xsm2- :nandfora="2'(x=1=2k:re) (x-j-2kn).sinn~-.cos(n_L1)-~2 .-2 cosx+cos2x+...+cosnx=--------­.x S111"2 Fromthistheboundedness ofthepartial sumscanbeinferred atonce. ThusifI(b"-b..+1)converges absolutely andb"__0,weconclude from thecriterion abthat 2,'bnsinnxconverges toreveryx, Ibncosnxconverges /01'everyx+2k:re. Inparticular 12,thiSISthec.newhenb"diminishes monotor.ely toO. 6.Ifthebn'sarepositive, andifwemaywrite b"+l_1IXIf"-b--;----;--n t+dl whered>0and(/fn)isbounded, thenI(-l)nbnconverges 1/,andonlyIt.IX>O.In fact,ItIX>0,itfollows fromthesehypotheses thatbnb+1<1fromsomestageon, n i.e.(bn)decreases monotonely, andtheconvergence oftheseriesinquestion is therefore secured by2.,ifwecan5howthatbn-+O.Theproofofthisis similar tothatoftheparallel factin170,1:Jf0<0:'<IXIwehaveforevery sufficiently large V,sayV>m, -b"+l.---1-a' b--. • " v 11Forx=2k'It,thesumhasobviously thevaluensinIX,foralln's. 11MalmsUn, C.J.:NovaactaUpsaliensis (2),Vol.12,p.255.1844. §43.Testsofconvergence forseriesofarbitrary terms. 317 ,,-1(')b"<bm•II1_..'.:... 'V=m 'VWriting downthis togethcr, weobtaininequality for"=m,m+I,..e,n-1andmultiplying Fromthedivergence oftheharmonic series,itfollowsasin170,1thatb"__O. Inthecase c<<0,bnmllstforsimilar reasons increase monotonely from somestageon,sothat2(-1)n bncertainly cannotconverge. Finally, when c<=O.wededuceinprccisely thesamewayasonp.28!J,thatbncannottend100 andthcseriestherefore cannotconverge. a7.Ifaseriesoftheform'\'-"-suchseriesareknownasDLrichletLJnX seric~;weshall inve~tigate theminmoredetaillateron(§58,A)-iscon­ vergent foraparticular valucofx,sayx=xo.italsoconverges forevery :c>xo,for(~) isamonotone nullsequence. Thissimpleapplication of nXXo Abel'stest,byreasoning quitesimilar tothatemployed forpowerseries(93), leadstothethe0rem:EveryseriesDjthejorm /~,~!'.possesses adefiniteabsnssa-nX Djconvergence lwLththeproperty thattheserLesconverges wheneverx>Aand dIVerges whenever x<A.(Forfurther details, v.§58,A.) General Remarks. 186. (n=2,3,..•)(-1)"bn=--­nand1.Wehavealready mentioned thefactthatthemag-nitude ofthezndtv­ ,dualterminanarbitrary seriesisnotconclusive withregardtoconverg-ence. Inparticular, twoseries ~'anand~b.,whosetermsareasymptotically equal, . Ia"1I.e.suchtlatb~->-,neednotexhibit thesamebehaviour asregards con- n verg-en"e (cL70,4). Thuse.g.for (-1)· 1a,,=---+---nnlogn wehave an(-1)·,-=1+----__1. b" logn But2b"isconvergent and2a"divergent, since ~(an-b,,)diverges by79,2. 2.11theserLes ~.ani<non~absolutely convergent, (cLp.136,footnote 9),'Lts POSItweImdLt.negutLve terms,takenseparately, lormtwodivergent serIes.More precisely, letPn=anwhena":>0,and=°whena"~0,andsinlllarly letq"= -an whena"<0,and=0whena"2::;0.13Thetwoseries2Pnand::::q"areseries ofpositive terms,thefirstcontaining onlythepositive termsof2.'anandthe secondonlytheabsolute valuesofthenegative termsof~'a",ineithercase withtheplacesunchanged, whiletheirothertermsareallO.BoththeseseYLes aredivergent. Infact,aseverypartial sumof:::a"isthedifference oftwo suitable partialSlimsof2:p"and2q",itfollowsatoncethatif2p"and2q. werebothconvergent, sowould21a"Ibe(by70),contrary tohypothesis; andiftheonewereconvergent, theotherdivergent, thepartialsumsofIall !a..l+a.. UThusPn= -2----I 11"la"I-a".q"'='2 (051) 318 Chapter X.Seriesofarbitrary terms. wouldtendto-00or+00(according asIPn01:J;qnisassumed convergent), whichisagaincontrary tohypothesis. 3.Bythepreceding remark, aconditionally convergent series,orrather thesequence formedbyitspartial S\1ffiS,isexhIbited asthedifference oftwo monotone increasing sequences ofnumbers tending toinfinity 14•Asregards therapidity withwhichtheseincrease, wemayeasilyestablish thefollowing Theorem. Theparhalsumsof2:p"andIq.areasymptohcally equal. Infact,wchave P,+P2+...+P. a,+a.+...+an---._-- --1=--·---- -iq,+q9+...+qn q,+q2+...+qn sincethenumerator inthelatterratioremains bounded, whilethedenominator increases to+00withn,thisratiotendsto0,whichprovestheresult. 4.Therelative frequency ofpositive andnegative termsinacondlhonally convergent series2'anforwhi('hIanIdiminIshes monotonely issubject tothe following elegant theorem, duetoE.Cesaro'": Thelimit,Ifitexists,ofthe ratio!,nofp,..thenumber01posltweterms,toQmthenumberofnegatIVe termsa., Qll forv:Sn,ISnecessarily 1 §44.Rearrangement ofconditionally convergent series. Thefundamental dIstinction between absolutely andnon·absolutely convergent serieshasalreadybeenmadeclearin89,2.Thisis,that thebehaviour ofIlon·ab~olutely convergent seriesdepends essentially ontheorderoftheterms IDtheseries,sothatfortheseseriesthe commutative lawofaddition nolongerholds.Theproofconsisted ltl showing thatanon-absolutely convergent seriescould,byamerere­ arrangement intheorderofitsterms,betransformed intoadivergent series. ThiSresultmaynowbeconsiderably elaborated. Infactit maybeshewnthatbyasuitable rearrangement anyprescribed behav­ iour,asregards convergence ordivergence, maybeinduced. The theorem whichweobtainis 187. Riemann' srearrangement theorem. 11~a.isaconditionally convergent series,wemay,byasuitablerearrangement (v.27,3),de­ duceaseries ~.an'withanyone 01thefollowing properties: 16Itisbesttoavoid,asbeingfartoosuperficial incharacter, themode ofexpression whichmaybefoundinsomewritings: "theslimofacondition­ allyconvergent seriesisgivenintheform00-00." 16Rom.Ace.LinceiRend.(4),Vol.4,p.1331888.-Cf.aNoteby G.H.Hardy,Messenger ofMath.(2),Vol.41,p.17.1911,andonebyH.Rad~ mache." ~lath.Zeitschr., Vo!11,pp.276-288. 1921. §44.Rearrangement ofconditIonally convergent series. 319 a)toconverge toanarbitrary 16prescribed sums'; b)todivergeto+00orto-00; c)toexhibitasupperandlowerlimits0/itspartialsumstwo arbitrary numbers fLandx,withIl2.x. Proof. Itsuffices toprovec),sincea)andb)areparticular cases ofc),theformer forx=fL=s'andthelatterforx=fl=+00or =-00. Toprovec),let(xn)beanysequence tending toxand(fln)any sequence tending tofl,withfln>xnand17{Jol>O. LetusdenotebyPI'P~,...thetermsin~an=al+all+ . whichare2:0,intheorderinwhichtheyoccur,andbyql'q2'. theabsolute valuesofthosewhichare<0,againintheirproper order,thusslightly modifying thedefinition in186,2.Theseries 2'P"and2'q"onlydiffC'rfromthosein186,2bytheabsence ofa number ofzeroterms,andareaccordingly bothdivergent, withposi. tivetermswhichtendtoO.Weproceed toshowthataseriesof thetype PI+P2...+Pm,--q\-q2-•••-qk.+Pm,+1+...+Pm. -qk.+1-•••-qk.+Pm.+I+... willsatisfy alltherequirements. Suchaseries ISclearly are· arrangement ofthegivenseries,andisindeedonewhichleavesun· alteredtheorderofthepositive termsrelatively tooneanother and thatofthenegative termsrelatively tooneanother. Letuschoose theindices ml<m2"-_•••,k1<kJ<...,inthe aboveseries,sothat: 1)thepartialsumwhoselasttermisPm.hasavalue>Ill' whilethatending onetermearlieris<Ill; 2)thepartialsumwhoselasttermis-qk.hasavalue<Xl' whilethatendingonetermearlieris>y.1; 3)thepartialsumwhose lasttermisPm,hasavalue>fl2' whilethatendingonetermearlieris<P2; 16Rlcmann, B.:Abh.d.Ges.d.Wiss.z.GoUingcl1, Vol.13,p.97.1866-68. Thestatements b)andc)areobvious sllpplementary propOSitions. 10Thisisclearly possible inanynumber ofways.Infact,if,,=14with afinitevalues',say,take"..=s'--.!.andI4n=s'+-.!.,-taking 141evenlarger,n n ifnccessary. Ifx=14=+00(-00),takexn=n(-n)andfin="n+2.If, finally, x<EL,takeany(xn)and(/I,,)tending toxandp.;fromsomestage on,XII<fJ-n.andbyafinitenumber ofalterations, wecanarrange thatthis maybethecasefromthebeginning, andalsothat11-1>O. 320 Chapter X.Seriesofarbitrary terms. 4)thepartialsumwhoselasttermis-qk.hasavalue<X2,while thatendingonetermearlieris>x2; andsoon. Thiscanalwaysbearranged; forbytakingasufficient number of positive terms,thepartialsummaybemadeaslargeasweplease,and byallowing asufficient number ofnegative onestofollow,thepartial summayagainbedepressed belowanyassigned value.Ontheother hand,atleastonetermmustbetakenateachstage,since Xn<11-..;so everytermoftheoriginal seriesreallydoesoccurinthenewseries. LetEan'denotethedefinite rearrangement ofEansoobtained; thepartialsumsofEan'havetheprescribed upperandlowerlimits.In fact,ifforbrevitywedenotebyTl,T2,•••,thepartialsumswhoselast termsarePm"Pm"...andbyaI'a2,•••,thosewhoselasttermsan; -qk"-qk.,•••,wehave Iav-Xv1<qk"andIT"-11-vI<Pmv' SincePn---+0andqn~->-0,itfollows thatav-).-x.andTv---+11-,so thatxand11-certainly represent valuesofaccumulation ofthepartial sumsofEan'Nowapartialsumsn'ofEan',whichisneitheraa"nor aT",hasnecessarily avaluebetween thoseoftwosuccessive partialsums ofthisspecialtype;hencesn'canhavenovalueofaccumulation outside theinterval X•••11-,(ordifferent fromthecommon valueofxand11-if thesecoincide). Inotherwords, 11-andxarethemselves theupperand thelowerlimitofthepartialsums,q.e.d. Vanous researches ofananalogous naturewerestartedindIfferent dIrectIOns asaconsequence ofthiStheorem. M.Ohm 18andO.Schlormlch 10investigated theeffectofrean:angement onthespecialsenesI-~+~-~+ -...,inpar­ ticularthecaseinwhichppositive termsarcfollowed byqnegative termsthroughout (cf.Exercise 148).A.Pringsheim 20wasthefirst,however, toaimatgeneral results forthecaseinwhichtherelative frequency ofthepOSitive andnegative termsIn aconditIOnally convergent seriesISmodified according todefinite prescribed rules. E.Borel21Investigated theopposite problem, astowhat rearrangement~ Inacon­ dItIOnally convergent senesdonotalteritssum.Later,W.SierpirlSkz 2.showed thatIf1:an=sconverges conditionally ands'<s,theseriescanbemadetohave thesums'byrearrangmg onlythepositIvetermsmtheseries,leaving allthenegative termsWithunaltered placeandorder,whIlesimilarly Itcanbemadetohaveany sums">sbyrearranging onlythenegative terms.(TheproofISnotsoSimple.) §45.Multiplication ofconditionally convergent series. Weshowed inthepreceding section, thuscompleting thecon­ siderations of89,2,thatthecommutative lawofaddition nolonger holdsforserieswhichconverge onlyconditionally. Wehavealsoseen 18Antrittsprogramm, Berlin, 1839. 19ZCltschr. f.Math.u.Phys.,Vo!.18,p. 520.1873. 20Math.Ann.,Vo!.22,p.45,'}.1883. 21Bulletin dessCiences mathcm. (2),Vol.14,p.97.1890. 22Bull.internat. Nc.Sciences Cracovie, p.149.1911, §45.Multiplication ofconditionally convergent senes. 321 already (endof§17),inanexample duetoCauchy, thatthedis­ tributive lawdoesnotingeneral subsist, sothattheproduct oftwo suchseries~anandJEb"maynolongerbeformedaccording tothe elementary rules.Thequestion remained unsolved, however, whether theproduct series~COl(with COl=aob"+a1bn-1+...+a"bu)mIght notcontinue toconverge underlessstringent conditions for~a"=A and~bn=B, andtohavethesumA·B.In§17,itwasrequired thatboth:£anand2'bnshouldconverge absolutely. Inthisconnection, wehavefirstthe oTheorem ofMertens 23.Ifatleastoneofthetwoconvergent series188. };a"=Aalld};b"=Bconverges absolutely, ECnconverges alld=A.B. Proof. Wehaveonlytoshowthat,withincreasing 11,thepartial sums c"=Co+Cl-1-.••+Cn =auho+(aohI-I-(/1ho)+...+(aoh"+aIhn-1+...+anho) tendtoA.Baslimit.Wemayassumethat};anis,ofthetwoseries,thc onethatconverges ahsolutely. IfwedenotebyAnthepartialsumsof };a,llhyBnthoseof.Eh",wehavc Cn=ao·Bn+a1B"-1+...+a..BoI or,ifwcputBn=B+(3n' =An'B+(ao'(3"+al(3n-l+...+a"(30)' SinceA,,'B--A.B,itonlyremalllS toshowthatwhcnJEa"IS absolutely convergcnt and(3"-0,theexpressions wn=a,,'(30+an-1(31+...+au(3" formanullsequence. Butthisisanimmediate consequence of44,9b; wehaveonlytoputx"=(3..andYn=anthere.Thusthetheorem isproved. Finally, weshallanswerthequestion whether theproduct series 2'cn'11'convergent, necessarily hasthesumA·B. Theanswer isintheaffirmative, asthefollowing theorem shows: °Theorem ofAbeZ~14.Itthethreeseries2,'an'2:bnand189. ,2'cn=~(aobn+...+anbo)areconvergent, andAIB,andCare theirsums,wehaveA·B=C. 1.Proof. Thetheorem follows immediately fromAbel'slimit theorem (100)andwasfirstproved byAbelinthisway.Ifwe write Ea"x"=(1(x), ~bnx"=f'J(x), ~cnx"=fa(X), 11].f.d.reineu.angew.Math.,Vo!.79,p182.1875.-Anextension was givenbyT.].Stieltjes (Nouv.Anna1es (3),Vo!.6,p.210.1887). 9~].f.d.reineu.angew. Math.,Vo!.I,p.318.1826. 322 Chapter X.Seriesofarbitrary terms. thesethreepowerseries(cf.IS3,4) certainly converge absolutely for o~x<1,andforthesevaluesofx,therelation (a) f1(x).f'J(x)=fa(x) holds.Theassumed convergence of:Ea",:Eb"and:EcrIimplies, by Abel'slimittheorem lOO,thateachofthethreefunctions tendstoa limitwhenx-+1fromtheleft;and f1(x)-A=:Ea", f'J(x)-B =~b", Sincetherelation (a)holdsforallthevaluesofxconcerned, itfollows (by§19,Theorem 1)thatitmustholdinthelimit: A·B=C. -Wemayalsodispense withtheuseoffunctions andadoptthe following 2.ProofduetoCesaro'J~. Itwasshownabovethat C.=aoB.+a1B"_1+...+a.Bo' Fromthisitfollows that Co+Cl+...+C"=AoB"+A1B"_1+...+A"Bo' DIviding bothsidesofthisequality byn+1andlettingn-+00, weobtainCaslimitonthelefthandside(by43,2)andA·Bas limitontheright(by44,9a).HenceA·B=C,q.e.d. Inconsequence ofthisinterestIng theorem, withwhichweshall againbeconcerned lateron,anyfurther elaboration ofthequestion ofmUltiplication ofserieshasonlytodealwiththeproblem whether thesenes:E crIconverges. Intotheseinvestigations wedonot,however, propose toentcr26• Exampl esand Applications. :Jr "",(-l)n_ I 1 1I.Itfollows from-4=L;2---1=1--3-+-"-----+.",bythepre- n~On+." 7 cedingtheorem, that :Jr' 00(1 1 1 ) 16=n~(-1)"1.(2n+1)+3·(2n-1)+...+(2n+-lj-:-i • provJded theserJesthusobtained converges. 2~Bull.dessciences math.(2),Vol.14,p.114.1890. ,eTheorems ofthekindinquestion havebeenprovedbyA.PringsheJm (Math.Ann.,Vol.21,p.340.1883),andinconnection WIththelatter's work,by A.Voss(ibid.Vol.24,p.42.1884)andF.Caiori(Bull.oftheAmeric. Math.Soc., Vol.8,p.231.1901-2andVol.9,p.188.1902-3). -Cf.also§66ofA.Prings heim'streatise, Vorlesungen UberZahlen- undFunktionenlehre (Leipzig 1916), towhichwehavealready referred morethanonce.G.H.Hardy(Proc.Lon­ donMath.Soc.(2),vo!.6,p.410,1908)hasprovedapartIcularly elegant exampll' ofarelatedgroupofmuchmorefundamental theorems. (a) Cb)§45.Multiplication ofconditionally convergent series. 323 Now 1 1(11) (=2---,p-+,----;-;I)c-c("'~-n-+;-.-I---;2c-p·) =2ln+1)\2P+1+2n--2~p+f ' sothatthegeneric termofthenewseneshasthevalue ~~);(1+~+"'+2--;;1+1), Since21i~i tendsmonotoncly tozero,sodoesitsanthmetic mean _~_(1+J_+...+_1_)n+l 3 2n+l ' andthenewseriestherefore doesconverge byLezbmz's testS2,5 Wethus have "",(-I)"(1 1 ) ;n" ,.~on+T1+"3+...+2;i--t-l =16. 2.Inapreclsc1y sImilar manner, wededuce (v.120),bysquaring the 1 1serieslog2=1--2+3--+"" "',(_1),.--1(1 1)el-k+CI+~2+...+~f~=(log2)1. 3.Theresultobtained m1.prOVIdes afreshmodeofapproach tothe ""1Jl2equatIon.2k2="6'whzchhasoccupied usrepeatedly beforenow(v.136 ~=l andl:'i6)Toscethis,wefirstprovethefollowing Theorem. Let(ao'al'a.,...)beamonotone sequence 01positivi' numbers, lorwinch2.'all2IScol!Vcrgent. Thentheseries 3.1;(-I)"lip=A, ,,=1and""1.2-'(-I)na,,=s; ..=0""2JEa"a,,+p=lip, ..=0p=I,2,..., alsoconverge, with ""(c) .L)a..·=s2-2A. n=O Proof. SinceIa,,1converges, a..-+0;accordingly theseries1con­ vergesbyLelbmz's test.Asa..a,,+p:saniforeveryp>I,and2:aniconverges, thesenes2arealsoconvergent forp2':1.Further, asa"+p+1:sall+1"we havelip+1::;:.!Jp.Theseries3WIllaccordingly converge jf!Jp-+O.Now given8>0,wecanchoosemsothata;:'j.1+a;:'+2+...<~;foreverysuffi. cientlylargep,weshallthenhave 8 e lJp<aoal'+a1al'+1+...+amap+m+"2<ap(ao+a1+...+am)+"2<e. HencelJ"-+0andthesenes3alsoconverges. LetusnowfomIthearray 2((0-aoa,+aoaJ-aoa.+-. -Q,ao+uJ2-ata,+a1a~-+ . +a.ao-a,a,a.'-ala.+-...-asa"+aJa1-a.a.+Ua~-+.... ~.............. 324 Chapter X.Senesofarbitrary terms. andletS,.denotethesumoftheproducts±a;.al'forwhichlandftare:::;n. TheseobvIOusly fillupasquare Intheupperlefthandcornerofthearray,and S,.=(ao-a,+ -••.+(-J)nan)9-.Sg. Ontheotherhand,thesumofallthe(primary) diagonal:, whichcontain at leastoneproduct a;.a",belonging tothatsquare, isclearly ""Tn-~a/i 2(-~'+~9-+···+(--I)"~,.). 1'':=-{) Hence, toobtain(C),Itnowsuffices toprovethatTn--Sn-->-O.Bywriting outtheabovearrayinamoredetailed fashion, wesee,moreover, that (_I)nIT,.-S..)=2[<13an+1+a2a,,+9+"'] -2[aga,. f-l+aaa..+2+···J +2[a3an+l+a•a"I_3+···]- f-..• +(-I)n-l.2[(~"a"+1+a" 1-1a,,+3+---] +(-1)"[a';+1+a::+~+...J. Thiswewriteforbrevlty =~a,-ccg+cc.,-+.-.+(-1)"-1Cl"+(-I)"Pn, andasIX,~((2~•••2':an-0,wehave(et.SIc,H) ITn-S"I:::;(XlI-{J"Sb"I-{in; thus,aswasasserted, Tn-S..-~0andtherefore ~:a..3=S2-2A. 4If.3wenowtakea=-~- -thehypotheses areobVIOusly 1I.,In" .n2n+1' a fulfilled, andwehave ""1 .n9 n~(2n-+l)' =16+2(c51-c5.+~a-+·-·)' Butinthiscase,wehave,by133,1, "", 1 l( 1 1 ) ~p=n~o(~:fn+l)(2 n-+2p+:1)-2P1+:31-..•+2P--=-l foreveryp;:::::I,andhence 1;-__1 =:r~+1;(-1_2:.(I+}_+...+-_~). n=O(2n+1)216n=On+1 3 2n-I-I 2 Bytheequality (a)proved in1.,therighthandside=]-.Bythemethod ""1.n2usedtodeduce137from136,theequalityZk-2=-iffollows atonce k~1 Thefreshproofthusobtained forthisrelation mayberegarded asthe mostelementary ofallknownproofs,sinceitborrow!> nothing fromthetheory offunctIOns excepttheLelhniz series122.Themainideaoftheproofgoe9 backtoNicolaus Bernoulh2'. Exercises onChapter X. 142.Determine thebehaViour ofthefollOWing senes: a)""(_l)[~n] b)~,(_1)[\';;]'"~--- ,2;--nX-- .-..Jn•n=1 n=1(. ~c)Ix~I:;_X) d)~.,inx_....n' 97Comment. Ac.Impscient.Petropolitanae, Vo!.X,p.19.1738. Exercises onChapter X. 325 e),1;(-I)nsin~,nf),1;sing~,n m)~ansinnxcos'nx.h)2"sin(n!nx), k)'"SIO'nx .t:Jn 'g)2,'sin(n9x), (_I)n i)'".t:JX+logn' 1),-,(1+].~+."+-.!.-)~nnx ..:..J 2 nn' Inthelastsenes,(an)ISamonotone nullseql1ence. Theseriesg)doesnotconverge unlessx=-knitheseriesh)converges forallrational valuesofx,alsoe.g.for :I:=e,=(2k+l)e, =~,=sinl, =cos1,andfore11111111 x=-2-41-51+2-6I-'rI+28-i-+... andmanyotherspecial vall1esofx.Indicate valuesofxforwhichitcer­ tainly <illl'r!!:c~. .-1:1. 1;[-1_~-1--1--_1_J=log2,,,=1X+2n-1x+2nx+n forfvcryx:>0, 144.If(nan)and2,'n(an-an+1)converge, theseries2"analsoeon­ verg('~. 141>.a)If2"anand2"lbn-bn+,1bothconverge, orb),if2"anhasbounded partial l>lIlns,.2,'1bn-b"+1Iconverges andbn-+0,thenforeveryInteger p;:;::1theseries2,'a"b"Pisconvergent. 146.Theconditions ofthetest184,3areinacertainsensenecessary, aswellassufficient, fortheconvergence of~anbn:IfItberequired thatfor agiven(bn),~'a"bnalwaysconverges with.2,'an,thenecessary andsufficient conditIon isthat ~'Ibn-bn+,Ishouldconverge_ -Showalsothatitmakes lIttledifference inthISconnection whether wcrequirethat2,'1bn-bn+,Icon­ vergesormerelythat(b,,)ismonotone. 147.If2"a"converges, andIfPnincreases monotonely to+00insuch awaythat2"P"-1ISdivergent, wehave -1-0-PIa,+P.a.+',.+Pnan{~0,Im --- n ~O. 148.Leta"tendto0monotonely, andassumethathmnanexists.If weWrItej;(-l)na"=s,andnowrearrange thissenes(cf.Ex.[)1)soasto n=O havealternately Ppositive andqnegative terms: ao+a,+...+a.p_.-a,-aa-•••-au_,+agp+..,, thesums'ofthenewseriessatisfies therelation 1 Ps'=s+2-hm(na,,).log-q' 149.Anecessary andsufficient condition fortheconvergence ofthe product series ~c"=I(aob"+a,b"_1+...+anbu) oftwoconvergent series2"an,Ibn,isthatthenumbers n en=~:a.(b..+b.._1+...+b"-.+I) .=1 shouldformanullsequence. 326 Chapter XI.Seriesofvariable terms. Il'SO.If(all)and(bll)aremonotone sequences withlImitO.theCauchy's product seriesof2'(-1)"anand2'(-1)" bnisconvergent if,andonlyif,the numbers an=an(bo+bt+.,.+b..)andrn=b.\ao+at+...+an)alsoforma nullsequence. (_I)" (_I)" 1::»1.Thetwoseries.2--- and.2--(1--' IX>0,fJ>0,maybenU n multiplted together byCauchy's ruleIf,andonlyif,IX+fJ>1. 1l'S2.If(an)and(b,,)arcmonotone nullsequences, CaHcliy's product of theseries2'(-1)"anand2,'(-1)"bl!certainly converges If2,'a..bnconverges. Anecessary andsutficient conditIOn fortheconvergence oftheproduct series isthat;:; (anbll)l+eshouldconverge forevery (!>O. 1l'S3.If,foreverysufficiently largen,wecanwnte a"=n"o.(logn),,' .(log.n)'"•..tlogrn)"r, b"=nPo.(lognl' .(log.nIP••.•(log,njll', andif2,'bnconverges, wehave,provided anisnotequaltob"(oreveryn, (aob"+atbn_I+...+allbo)~all.(2by).,,=0 Chapter XI. Seriesofvariable terms(Sequences offunctions). §46.Uniform convergence. Thusfar,wehavealmost exclusively takenintoconsideration serieswhosetermsweregiven(constant) numbers. Itwasonlyin particularly simplecasesthatthevalueofthetermsdepended onthe choiceofadefinite quantity, orvariable. Suchwasthecasee.g.when wewereconsidering thegeometric series2:a"ortheharmonic series .2_1_;theirbehaviour wasdependent onthechoiceofaorof(x.AmorenU general example isthatofthepowerseries~a..x",whercthenumber xhadtobcgiven,beforewecouldattacktheproblem ofitscon­ vergence ordivergence. Thistypeofcasewillnowbegeneralized in thefollowing obvious way:weshallconsider serieswhosetermsdepend inanymanner onavariable x,i.e.arefunctions ofthisvariable. Weaccordingly denote thesetermsbyf"(x)andconsider seriesof theform~f"(x). Afunction ofx,inthegeneral case,isdefined onlyforcertain valuesofx(v.§19,Def.1);forourpurposes, itwillbesufficient to assume thatthefunctions f"(x)aredefined moncormore(openor closed) intervals Forthegivenseriestohaveameaning foranyvalue §46.Uniform convergel1ct=. 327 ofxatall,wehavetoreqUIre thatatleastonepointxbelongs to theintervals ofdefinition ofallthefunctionsfn(x).Weshall,however, atoncelaydownthecondition thatthereexistsatleastoneinterval, inwhichallthefunctionsfn(x)aresimultaneously defined. Forevery particular xinthisinterval, thetermsofthesenes2fn(x)arein anycasealldeterminate numbers, andthequestIOn ofitsconvergence canberaised.Weshallnowassume further thataninterval ] (possibly smallerthantheformer) exists,foreverypointofwhichthe series2:tn(x)isfoundtoconverge. oDefinition 1.AnintervalJwillbecalledaninterl'UZ ofconver·190. (/enceoftheseries2'fn(x)it,ateveryone 01itspoints(including one, both,orneitherofitsendpoints), allthefunctionsf"(x)aredefined andtheseriesconverges. Examples andIllustrations. 1.Forthegeometric series ~x",theinterval -1<x<+1isaninterval ofconvergence, andnolargerinterval ofconvergence existsoutside it. 2.Apowerseries::: a"(x-xo)",-provided itconverges atonepointat least,otherthanxo'-alwayspossesses aninterval ofconvergence oftheform (xo-ri...(xo+1'),inclusive orexclusive ofoneorbothendpoints.'Vhen l'is properly chosen, nofurther interval ofconvergence exi~tsoutsidethatone. 3.Theharmonic series2:_1_hasasinterval ofconvergence thesemi­ nX axisx>1,withnofurtherinterval ofconvergence outside it. 4.Asaseriesisnomorethanasymbolic expression foracertain se. quenceofnumbers, sotheseries ~.fn(x)represents nomorethanadifferent symbolic formforasequence of{unctions, namely thatofitspartialsums s"(x)=fo(x)+fl(x)+...+f..(x). Infmnclple. Itistherefore 1nlmaterial whether thetermsoftheseriesorItspartial slimsareassigned. aseachsetdetermines tl:totherumqllely. Thus, in principle, it alsodoesnotmatterwhether wespeakofinfinite seriesofvariable termsor ofsequences 01fUflCtions. Weshallaccordingly stateourdefinitions and theorems onlyforth,cas,ofseriesandleaveIttothestudenttoformulate themfor thecaseofsequences 0/functIOns B. 5.Fortheseries et)x'(x.x.)(X'x.) n~/"(x) ==1+x'+C+x'-1+x.+r+~.-1+x·+... 1Forthecaseofcomplex numbers andfunctions, wehaveheretosubstitute throughout thewordregionforthewordinterval andboundary pointsofthereKwn forendpoints oftheinterval. WIththismodification, thesign0hasthesamesig­ nificance inthischapter aspreviously. •Occasionally, however, thedefinitions andtheorems willalsobeapphed tosequences offunctions. 328 wchaveChapter XI.Seriesofvariable terms. Theseriesconverges foreveryrealx.Clearly, indeed, wehave a) b) c) 6.5"(x)--0,IfIxi<1, s..(x)__1,ifIxl>1,and 1ifIx1=1. s"(x)--2' Ontheotherhand, 5,.(x)=(2sinx)" defines aserieswithaninfinity ofseparate Intervals ofconvergence; for lim5"(x)obviously existsif,andonlyif,-~<sinx<i.i.c.if __'n<x<!!- 6=6or5'n 7:re-<x<-­6= 6 orifxliesinaninterval deduced fromthesebyadisplacement through an integral multiple of2'n.TheSUllloftheselies=0throughout theinterior oftheinterval and=1attheincluded endpoint. sin2xsin3x "7.Theseriessinx+-2--+-3-+---converges, by181'),.>.forevery . cos2x cos3xrealx;thesenescosx+--2-+-3--+...converges foreveryreal x*2kn. Ifagivenseriesoftheform2'fn(x)isconvergent inadeter- minate intervalJ,therecorresponds toeverypointof]aperfectly definite valueofthesumoftheseries.Thissumaccordingly (§IB, Def.1)isitselfafunction ofx,whichisdefined orrepresented by theseries.Whenthelatterfunction isthechiefcentreofinterest, it isalsosaidtobeexpanded intheseriesinquestion. Inthissense, wewrite co F(x)=.2(n(x). n=O Inthecaseofpowerseriesandofthefunctions theyrepresent (v.Chapters VandVI),theseideasarealready familiar tous. Themostimportant question tobesolved,whenaseriesofvariable termsisgiven,willusuallybewhether, andtowhatextent,properties belonging toallthefunctionsf..(x),i.e.tothetermsofthegiven series,aretransferred toitssum. Eventhesimpleexamples givenaboveshowthatthisneednot bethecaseforanyoftheproperties whichareofparticular interest inthecaseoffunctions. Thegeometric seriesshowsthatallthefunc­ tionsfn(x)maybebounded, withoutF(x)beingso;thepowerseries forsir:x,x>0,showsthateveryf..(x)maybemonotone, without F(x)beingso;example 5showsthateveryfnex)maybecontinuous, §46.UDlform couvcrgenct=. 329 withoutF(x)beingso,andthesameexample illustrates thecorres­ ponding factfordifferentiability. Itiseasytoconstruct anexample showing thattheproperty ofintegrability mayalsodisappear. Forinstance, let {=1foreveryrational xexpressible asafraction withdenominator 5"(x) (positive and)::::n, =0foreveryotherx. ThensI'(x),foreachn,-andconsequently f"(x),foreachn,-isinte­ grable overanybounded interval, asithasonlyafinitenumber ofdiscon­ tinuities insuchaninterval (cf.§19,theorem 13)Alsolims"(x)=F(x)exists foreveryx.Infact,ifxisrational, say=.:L(q>0,pandqprimetooncq another), wehave,foreveryn>q,s"(x)=1andhenceF(x)=1.If,on theotherhand,xisirrational, 5,.(x)=0forevery 11andsoF(x)=O.Thus ~;I;.(x)=lims"(x)defines thefunction F() {=1forarationalx, x=0foranirrational x. Thisfunction isnotintegrable, foritisdiscontinuous3foreveryx. Evenbythesefewexamples, weareledtoseethataquite newcategory ofproblems ariseswiththeconsideration ofseriesof vanable terms.Wehavetoinvestigate underwhatsupplementary con­ ditionsthisortheotherproperty 01thetermsfn(x)istranslerred to thesumF(x).Itisclearfromtheexamples citedthatthemerelactof convergence doesnotsecurethis,-thecausemustresideinthe modeofconvergence. Aconcept ofthegreatest importance inthis respectisthatknownasuniform, convergence ofaseries2'fn(x)inone ofItsintervals ofconvergence orinpartofsuchaninterval. Thisideaiseasytoexplain, butitsunderlying natureisnotso readilygrasped. Weshalltherefore firstillustrate themattersomewhat intuitively, beforeproceeding totheabstract formulation: Letif,,(x)converge, andhaveforsumF(x),inanintervall, a~x<b; "=0 wcshallspeakofthegraphofthefunction,,=s,.(x)=fo(x)+...+f..(x)as beingthenthcUnJeofapproxunallOn andofthegraphofthefunction" =F(x) a'Vcmaymodifythisdefinition alittlebytakings"(x)=1forallrational x'swho"edenominators arcfactors ofn',and=0elsewhere; therational x's inquestion comprise, foreachn,adefinite number ofothervaluesbesides theintegers ~nusedabove. 'Vethenobtainaslims..(x)thesamefunction F(x)asabove. Inthiscase,however, boths..(x)andF(x)mayberepresent­ edintermsofaclosedexpression, bytheusualmeans; infact,wehave s..(x)=lim(cos'nIxx)",andtherefore k....... F(x)=lim[lim(cos2nl.nx)kJ. "....""k-~"" Thiscuriol1s example ofafunction, discontinuous everywhere, yetobtainable byarepeated passage tothelimitfromcontinuous functions, isdueto Dlrichlet. 330 Chapter XI.Seriesofvanable terms. asthelimIting curve.Thefactoftheconvergence ofif"(x)toF(x)inJ n=O thenappears toimplythatforincreasing n,thecurves ofapproximation lie closerandclosertothelimiting curve. This,however, isonlyaveryimper­ fectdescription ofwhatactually occurs. Infact,theconvergence inJimplies only,inthefirstinstance, thatatfachIndIvIdual pomtthereisconvergence; allwecansay,tobeginwith,istherefore thatwhenanydefinite abscissa x issingled out(andkeptfixed)thecorresponding ordinates ofthecurves of approximation approach, asnincreases, theordinate ofthelimiting curvefor thesameabscissa. Thereisnoreasonwhythecurvey=s"(x),asawhole, shouldliecloserandclosertothelimiting curve.Thisstatement soundsrather paradoxical, butanexample willimmediately makeitclear. Theserieswhoscpartialsumsforn=I,2,..,havethevalues nx s,,(x)=l+n Jx9' certainly converges inthcinterval 1::Sx<2.Infact,inthatinterval, nx 10<5"(x)<n9x~~n· outtheinterval, foranyn(however large).Thelimiting curvcistherefore thestretch1:5x<2ontheaxisofx.Thenlll curveofapproxImation liesabovethlSstretchand,bytheaboveinequahty, ata distance oflessthan~fromthelimiting curve, throughout thewholeojthen Forlargen's,thedistance allalongth.curveistherefore Interoal1::Sx<2. verysmall. Inthiscase,therefore, mattersaremuchasweshouldexpect; theposition is entirely altered ifweconsider thesameseriesintheinterval 0<x::;1."Ve stillhavelim5"(x)=0ateverypointofthisinterval 4,sothatth;;-limiting curve isthecorresponding pO!tionofthex-axis.Butinthiscasethe nibapproximation curvenolonger liesclosetothelimiting curvethrough- 1Forx=n-'wehavealways 1 5"(x)=:r'sothat,foreveryn,theapproximation curveintheinterval from oto1hasahumpofheight ~!IThegraphofthecurvey=54(x)hasthe following appearance: - •Infact,forx>0weFig.4. 1have0<5"(x)<nxasbefore, ;.e.<•for 1evcryn>--;forx=0,5"(x)=0evenpermanently.ex §46.Uniform convergence. 331 Thecnrve,,=SjO(x).however, corresponds morenearlytothef(lllowing graph: Fig.5. Forlargern's,thehumpinquestion -without diminishing inheight -be­ comescompressed nearerandnearer totheordinate-axis. Theapproximation curvesprings moreandmoresteeplyupwards' fromtheorigintotheheight-}, 1whichitattains forx=n-'onlytodropdownagainalmost asrapidly to within averysmalldistance ofthex-axis. Thebeginner, towhomthisphenomenon willappear veryodd,should takecaretogetitquiteclearinhismindthattheordlOates oftheapproxima­ tioncurves donevertheless, foreveryfuedx,ultimately shrinkuptothepoint allthex-axis, sothatwedohave,forevery!jxedx,limSOl(x)=O.Ifxis givenafixedvalue(however small),thedisturbing humpofthecurvey=s"(x) WIllultimately, 1.e.forsufficiently largen's,besituated entirely totheleftof theordinate through x(though stilltothel'ightofthey-axis) andonthis ordinate thecurvewillagainhavealready dropped veryclosetothex-axis•. Therefore theconvergence ofourserieswillbecalledtlmfoym inthe interval 1<x;;;2,butnotintheinterval 0<x=;1. Wenowproceed totheabstract formulation: Suppose If,'(x) possesses aninterval ofconvergence ];itisconvergent foreveryindiv­ idualpointof],forinstance atx=xo;thismeansthatifwewrite F(x)=sn(x)+rn(x) andassumee>Oarbitrarily given,thereisa number nosuchthat,foreveryn>no' Ofcoursethenumber no'aswasalready emphasized Cv.10,rem.3),de­ pendsonthechoiceofe.Butnonowdepends onthechoice0/Xoalso. Infactforsomepointsof]theserieswillingeneral converge more 6Attheorigin,itsslopeis5n'(0)=n. 8Ifwetake,say,x=lO~Oandn=1000000, theabscissa ofthehighest pointofthehumpisfOO~000'andatourpointxthecurvehasalready dropped . 1 toaheIght<1000' 332 Chapter XI.Seriesofvariable terms. rapidlythanforothers? Byanalogy with10.3,we3halItherefore writeno=no(I',xo);ormoresimply,dispensing withtheindex0and withthespecial emphasis onthedependence one,weshallsay: Given B>0andgIvenxintheinterval], anumbern(x)canalways beassigned, suchthatforeveryn>n(x), Ir,,(x)I<B. Ifwenowassumen(x)-stillforthedefinite given B-chosen, sayasaninteger, assmallaspossible, itsvalueisthenuniquely definedbythevalueofxjassuchitrepresents afunction ofx.Ina certainsense,itsvaluemaybeconsidered asameasure 01therapid· ity01convergence oftheseriesatthepointx.Wenowdefineas follows: 191. °Definition ofuniform convergence (ptform).Theseries ~/"(x) convergent intheintervalJ,issaidtobeuniform7y convergent inthe szeb·interval]' 01],ilthelunction n(x)delinedaboveisbounded in]', foreachvalueHojE.-SUppOSlllg wethenhaven(x)<Nin] ­ thisNwillofcoursedepend onthechoiceof1',likethenumbers n(x)themselves -wemayalsosay: °2nd(principal) formofthedefinition. Aseries~In(x),con· vergentintheinterval],issaidtobeunilormly convergent ina sub-interval ]'01],il,givene,asingle numb~r N=N(e)canbe assigned independently 01x,suchthat I'I'n(x)I<E, notonly(aslormerly) loreveryn>N,butalsoloreveryxin]'. Wealsosaythattheremainders r"(x)tenduniformly to0in]'. Illustrations andExamples. I.Uniformity ofconvergence invariably concerns awholeinterval, never anisolated point8. 2.Aseries:Ef"(x)convergent inanintervalJdoesnotnecessarily COll­ vergeuniformly inanysub-interval ofJ. 3.Ifthepowerseries ,2'a..(x-xo)"hasthepositIve radiusrandifo<e<r,theseriesisuniformly convergent intheclosedsub·mterval J'of 7Thestudentshouldcompare, forinstance, therapidity ofconvergence ofthegeometric series:Ex"(Le.therapidity withwhiehtheremainder diminishes asnincreases) forthevaluesx=l~Oandx=19:0' BIf,thatistosay,theabove-mentioned measure oftherapidity ofcon· vergence evinces noundulygreatirregularities intheinterval]'. -Inpar· ticularcases]' mayofcourseconsistofthecomplete intervalJ. 8Moregenerally, itmayhavereference tosetsofpointsmorethanfilllfe innumber. §46.Uniform convergence. 333 itsinterval ofconvergence, defined by-e<x-xo<+(!.Infact,asthepoint x=Xo+f:!liesintheInterior oftheinterval ofconvergence ofthepowerseries, thelatterisabsolutely convergent atthatpoint. Butif~:a"e"converges absolutely, wecan,given6>0,choose N=N(6)sothatforeveryn>N Ian+1 , •(!"+1+Ian+2I.(!"+9+...<If• Also,sinceIx-XoI<(!foreveryxinI',wehave Irn(x)I:::;Ia"+1I·e"+1+Ia"+2I·(!"+2+..'. Thusforn>N,wecertainly haveIrn(x)I<li,whatever theposition ofxin J'maybe. Theresultwehaveobtained isasfollows oTheorem. Apowerseries ~all(x-xo)"ofpos~hve radmsrconverges unl· farmIymeverysub·mterval oftheformIx-XoI<e<rofitsmterval ofcon­ vergence. 4.Theaboveexample enables ustomakeoursdves understood, ifwe formulate thedefinitIon ofuniform convergence alittlemoreloosely, asfollows: ~'r"(x)'ssaidtobeunIformly convergent In]'.If,tispOSSible tomakea statement aboutthevalueoftheremainder, ,ntheform"Ir"(x)I<8",valldfor allpOHtions ofxs~mtllta1leo"sly. .~sin11x.fI f I f5.Thesenes £,--2- ISun!ormyconvergent oreveryvaue0z; n=ln for,whatever theposition ofxmaybe, 1 1Ir"(x)I<(n+1)2+(n+2)2+...J whence therestmaybeinferred by4. (j.Thegeometric seriesisnotuniformly convergent inthewhole interval ofconvergence - 1<x<-I-1.For xn+1 rn(x)~xn+1+xn+a+...=1-;-x however largeNmaybechosen, wecanalwaysfindanrn(x)withn>Nand 0<x<1,forwhiche.g.rn(x)>1. If,formstance, wechooseanyfixedn>N,thenasx->-1-0wehave Hencern(x)>Iforallxinadefinite interval oftheform Xo<x<1. 7.Theaboveclearsupthemeaning ofthestatement: z:In(x)is1lOtuni­ formlyconvergent inaportion./' ofitsinterval ofconvergence. Aspecialvalue ofE,saythevalue En>0,exists,suchthatanmdex 11greater thananyassigned Nmaybefound,sothattheinequahtyIrn(x)I<Eoisnotsatisfied forsomeSUit­ ablychosenxm./'. 8.Withreference tothecurvesofapproximation y=sn(x),ourdefinition clearlyimplies that,withincreasing n,thecurveshouldliearbitranly closetothe limltmg curvethroughout theportionwhichliesabove./'.If,foranygiven E:>0, wedrawthetwocurvesy=F(x)±E,theapproximation curvesy=sn(x)will ultImately, foreverysuffiCiently largen,cometoheentuely wlthmthestnpbounded bythetwocurves. 334 Chapter Xl.Seriesofvariable terms. 9.Thedistinction between uniform andnon-uniform convergence, andthe greatsignificance oftheformer inthetheoryofinfinitesenes,werefirstre· co~nized (almost simultaneously) byPh.L.v.Seidel(Abh.d.Mitnch. Akad., p.383,1848)andbyG.G.Stokes(Transactions oftheCambridge Phil.Soc., VoI.8,p.533.1848).Itappears, however, fromapaperbyK.Weierstrass, un· published till1894(Werke, Vol.I,p.67),thatthelattermusthavedrawnthe distinction asearlyas1841.Theconcept ofUniform convergence didnot become common property tillmuchlater,chiefly through thelectures of Welerstrass. Otherformsofthedefinition ofuniform convergence. Cardform. ~:f..(x)issaidtobeuniformly convergent inrtt, inwhatever waywemaychoosethesequence 10(X,.)intheintervalJ', thecorresponding remainders 'rn(X,.) invariably formanullseqllencell• Wecanverifyasfollows thatthisdefinition isequivalent tothe preceding: a)Suppose thattheconditions ofthe2ndformofthedefinition are fulfilled. Then,givene,wecanalwaysdetermine NsothatI1'..(x)I<'" foreveryn>Nandeveryxin]';inparticular 1r,.(x..)1<eforeveryn>N; hencer..(x..)--O. b)Suppose, conversely, thattheconditions ofthe3rdformareful­ filled.Thusforevery(x..)belonging toJ',1'..(x..)--0 .Theconditions ofthe2ndformmustthenbesatisfied also.Infact,ifthiswerenot thecase,-ifanumberN=N(e)withtheproperties formulated there didnotexistforevery B>0,-thiswouldimplythatforsome speciale,say8=80'nonumber Nhadtheseproperties; aboveany numberN,however large,therewouldbeatleastoneotherindexnsuch that,forsomesuitable pointx=X,.inJ',I"..(x,,)I~EO'Letn1bean indexsuchthatIr",(x,,)I~EO'Aboven1therewouldbeanother index n2,suchthatI1'".(x".)I~Eoforasuitable corresponding point X"2'and soon.Wecanchoose(x,,)in}'sothatthepointsx" l'xn.'•••belongto (x,,),inwhiehease 10Thesequence neednotconverge, butmayoccupyanypositioninJ'. 11Should eachofthefunctIOnsIr"(x)IattamamaXImum m/',wemay choosex"inparticular sothatIr..(xn)I=MaxIr..(x)I;ourdefinition thustakes thespecialform:If..(x)issaidtobeuniformly convergent inI'ifthemaxima MaxIr..(x)Iin.!'formanullsequence. IfthefunctionIr..(x)Idoesnotattainamaximum inJ',ithas,however, a definiteupperboundfL".Wemayalsoformulate thedefinition inthegeneralform: oForm3a.If..(oX·)issaidtobeuniformly convergent in/'iffLn-,.O.(Proof?) §46.Uniform convergence. 335 willcertainly nutformanullsequence, contrary tohypothesis. Ourassump­ tionthattheconditions ofthe2ndformcouldnotbefulfilledisinadmissible; the3rdformofthedefinition iscompletely equivalent tothe2nd• Intheprevious formsofthedefinition, itwasalwaystheremainder oftheserieswhichweestimated, theseriesbeingalreadyassumed tocon­ verge.Byusingportionsoftheseriesinsteadofinfiniteremainders (v.81) thedefinition ofuniform convergence maybestatedsoastoincludethat ofconvergence. Weobtainthefollowing definition: o4thform.Aseries1:fn(x)issaidtobeuniformly convergent inthe interval.l' If,given I>>0,wecanassignanumberN=N(I»depending onlyone:,andindependent ofx,suchthat Ifn+!(x)+fn+2(x)+...+fnH(x)I<E foreveryn>N,everyk2'::1andeveryxin./'.Foriftheconditions of thisdefinition aresatisfied, thenitfollowsfirstly(by81)that1:fn(x) converges foreachfixedxin}'.Intheinequality, wemaymakektend to<1'.),andwefindthatIrn(x)I<e:foreachxin./'.Conversely, if Irn(x)Is;::e:foralln>Nandallxin.l',thenforallthesen,allk2I, andallxin.l',wehave Ih+l(x)+...+h+k(x)I=Irn(x)-rn+k(x)I<2E. Thisshows,however, thatiftheseries1:fn(x)satisfies theconditions of the4thform,italsosatisfiesthoseofthe2'1(\form,,andconversely. -We mayfinallyexpressthisdefinition inthefollowing form(cf.81a): o5thform.Aseries1:fn(x)issaidtobeuniformly convergent inthe interval.l' If,whenpositiveintegerskl,k2,k3,•••andpoints Xl'x2,xa,..• of.l'arechosenarbitrarily, thequantities [fntl(xn)+h+2(xn)+...+fntkn(xn)] invariably formanullsequence 12. Further Examples andIllustrations. 1.Thestudentshouldexamme afreshthebehaViour oftheseriesEfn(x),with192. s(x)=11'>:... n1-+n2x2 a)intheinterval 1;;:;x~2, b)mthemterval 0;::;;x:51(cf.theconsiderations onpp.330-1). 2.Fortheseries 1-+(x-1)-+(x2-x)-+...-+(xn-xn-1)-+. 12By51,wemightevenwriteUVn+l(xn)-+...-+fVn+kn(xn)]fortheabove, wherethevn'sareanyintegers tending to-+00.Exactly asin81,wemayspeak ofasequenceofportions, exceptthatherewemaysubstitute adifferent valueofx ineachportion. Thestatement wethenobtainis:AseriesEfn(x)issaidtobe uniformly convergent inj'ifeverysequence ofportionsoftheseriesformsanull sequence. Similarly: Asequence offunctions Sn(x)issaidtobeuniformly conver­ gentin'/'ifeverydifference-sequence isanullsequence. 336 Chapter XI.Seriesofvariable terms. wehaveobviously sn(x)=x".TheseriesaccordIngly converges intheinter valJ:-1<x<+1,inparticular inthesub-interval J':0<X;;S1.Here {-aF(x):1for0<x<1. forx=1. Theconvergence inthiSmterval isnatuniform. ItisnotsoevenInJ":o<x<1;forhere""(x)=F(x)-s"(x)= -x".Wehaveonlytochoose inJ" (hence InJ')thesequence ofpoints 1x"=1- - (n=I,2,..)n (1)"1 tohave I'n(xn)=-1 -n-+-e'sothattheseriescannotconverge uni- formly13. -Thismaybemadecleargeometrically byexamining theposition ofsuccessive curves ofapproximation, a~illustrated bytheaccompanying figure; Forlargevaluesofn,thecurve:y=s"(x) (1,1)remains, almostthroughout thewholeinterval quiteclosetothex-axis,whichrepresents the Itl11lting curve.Justbeforetheordinate x=+1, itrisesabruptly untilItreaches itstermInal POInt(I,1).However largeavaluemaybe assumed forn,thecurve :y=Sn(x)Willnever remain closetothehmiting curvethroughoul theentire Hmterval.J" (or.I), 3.Inthepreceding example, wecould almost expect aprzonthattheconvergence wouldnotbeuniform, asF(x)itselfhasa Fig.6. "jump" ofhClght1attheendpoint oftheinterval. Thecasewasdifferent withtheexample treated onp.330.Anexample sImilar tothelatter,butevenmorestnkIng, isthe following: Consider theseriesforwhich -)nz2 Sn(xl=n x e 2 (n=I,2, ...). Forx=0,wehave Sn(0)=0,foreverynjforx=l=0,thenumber e-tz'is positive and Ic~sthanI,sothat(by3S,1)Sn(x)-+o.Oursenesistherefore convergent foreveryxanditssumISF(x)=0,i.e.thelimllmg curvecoin­ cideswiththex-axis. Theconvergence isnotintheleastumtorm, however, ifweconsider aninterval containing theorigin. Thus,forXII=J~n-, -t..z•_/-tVn"1I(xn)=F(x,,)-s,,(x,,)=-nx,,·e R=-V n·e-=-7' whichcertainly doesnot-+o.Theapproximation curves haveasimilar 13Forx,,=(I-;.),weevenhaveI',,(x,,)-+I. 10Inspiteofthis,itiseasytoseethatforeveryfixedx(in0<x<1) thevaluess"(x)diminish to0asnincreases, sothattheabruptrisetothe height1occurstotherightofx,however nearxmaybetakento+I,provid­ edonlylhatnischosen sufficiently large. §46.UnIform convergence. 337 appearance tothoseInFigs.4and5,with[hismodIficatIon, thattheheightof thehumpnowincreases indefinitely withn;thisisbecause" (1)dn- s•.In=Ve---++OO' 4.Wc1lUstemphasize particularly thatuniform convergence doesnot require erlchofthefunctionsrn(x)tobeindIvidually bounded. Theseries _~_+1+x+x·+.."forinsta~e, isumformly convergent in0<x<-~-,with thesumx(1~x)'sincetheremainders havethevalue I"n(x)I=11~xI<'2n~l• ThefirsttermofthIsseries(asalsothelimiting function) isnotbounded in theinterval inquestIon. (Cf.,however, theorem 4.below.) Withaviewtocalculation withuniformly convergent series,It isconvenient toformulate thefollowing theorems specially, although theproofsaresosimplethatwemayleavethemtothereader: oTheorem 1.Ifthepseries~'rnt(x),2'rn,,-(x),...,2'r'Rp(x)are, simultaneously, uniformly convergent inthesameintervalJ,(pisa definite wholenumber), theseries2'rn(x)forwhich rn(x)=ctrnt(x)+c"-fn'J(x)+'"+cprnp(x) isalsouniformly convergent inthatinterval, ifCl'c2,••'.cpdenote anyconstants. (1.e.:Uniformly convergent seriesmaybemultiplied byconstant factorsandthenaddedtermbyterm.) oTheorem 2.If2:rn(x)isuniformly convergent inI,soisthe series.2:g(x)rn(x),whereg(x)denotesanyfunction definedandbounded intheinterval]. (1.e.:Auniformly convergent seriesmaybemulti­ pliedtermbytermbyabounded function.) °Theorem 3.Ifnotmerely.2:rn(X).but2'lrn(x)Iisuniformly convergent inI,thensoistheseries.2:gn(x)rn(x),provided thatwhenm issuitably chosen, thefunctions gm-t1(x),gm+2(x),.."areuniformly bounded inI,-i.e.provided wecanfindanintegerm>0anda numberG>0suchthatign(x)I<GforeveryxinIandeveryn>m. (1.e.:Aserieswhichstillconverges uniformly whenitstermsaretaken inabsolute valuemaybemultiplied termbytermbyanyfunctions allbutafinitenumber ofwhich,atmost,areuniformly bounded ~nI.) 15Thepointforwhichx=~isactually themaximum pointofthecurve.In "~SIl(X), asmaybeinferred froms~=(n_ngxg)6-inz·=o. 338 ChapterXI.Seriesofvariableterms. oTheorem 4.IfEfn(x)converges uniformly in.I,thenforasuitable mthefunctions fm+1(x),fm+2(x),..•areuniformly boundedinJandcon­ vergeuniformly toO. oTheorem 5.Ifthefunctions gn(x)converge umformly to0in.l,so dothefunctions Yn(x)gn(x),wherethefunctions Yn(x)areanyfunctions definedin.Iand-withthepossibleexception ofafinitenumberofthem­ uniformly boundedin./. WemaygiveasamodeltheproofsofTheorems 3and4: ProofofTheorem 3.Byhypothesis, given E>0,wccande­ termine nu>msothatforevery 11>nuandeveryxin./, If,.+l(x)I-+IIn+2(x)I-+...<-;;. Forthesamen'sandx'swethenhave Ign+I!n+1-+···1;£Ign+11·lfn+l1 -+...<G(If,'1-11 -+...)<E. Thisprovesallthatwasrequired. ProofofTheorem 4.Byhypothesis, thereexistsanmsuchthat, foreveryn~mandeveryxin./,Irn(x)I<t.Henceforn>mand everyxin./, Ifn(x)I=Irn-dx)-r"(x)I;£Irn-1I-+IrnI<1, whichprovesthefirstpartofthetheorem.Ifwenowchoosenu>mso thatforeveryn~noandeveryxin.!,Irn(x)I<lE,(Ebeingpreviously assigned) thesecondpartfollowsinquiteasimilarway. ,..~ §47.Passage tothelimittermbyterm. Whereas wesawonpp.328-9thatthefundamental properties of thefunctions f"(x)donotingeneralholdforthefunctionF(x)repre­ sentedbyEfn(x),weshallnowshowthat,roughly speaking, thiswill bethecasewhentheseriesisuniformly convergent 16. Wefirstgivethefollowing simpletheorem, whichbecomes particularly important inapplications: 193. 0Theorem 1.!ftheseriesEfn(x)isun~formly convergent inan intervalandifitstermsfn(x)arecontinuous atapointXoofthisinterval, the function F(x)represented bytheseriesisalsocontinuous atthispoint17. 18Wemay,however, mentionatoncethatuniformconvergence stillonlv represents asufficientcondition inthefollowmg theorems andisnotingeneral necessary. ~ 17IfXoisanendpointoftheintervalJ, onlyone-sidedcontinuity canofcourse beassertedatXoforF(x),butofcourseonlythe corresponding one-sided con­ tinuityneedbeassumedatx.forIn(x). §47.Passage tothelimittermbyterm. 339 Proof. Givene>0,wehave(inaccordance with§19,Def.6b) toshowthatanumber ~=r5(e)>0existssuchthat IF(x)-F(xo)I<e:foreveryxwithI::-XoI<3 intheinterval. Nowwemaywrite Bytheassumed factofuniform convergence, wecanchoosen=mso largethat,foreveryxintheinterval,Irm(x)I<;.Then IF(x)-F(xo)I~Isrn(x)-sm(xo)I+~e. Theintegermbeingthusdetermined, srn(x)isthesumofafixed numberoffunctions continuous atxo'andistherefore (by§19,Theorem 3) itselfcontinuous atxo'Wecanaccordingly choose 15sosmallthat foreveryxintheinterval forwhichIx-XoI<r5,wehave Ism(x)-srn(xo)I<~. Forthesamex'swethenhave IF(x)-F(xo)I<Il, whichestablishes thecontinuity ofF(x)atxo• oCorollary. 1f2fn(x)=F(x)isuniformly convergent inaninterval, andifthefunctionsfn(x)areallcontinuous throughout theinterval, thensoisF(x). Inconnection withexample 3of191,2,wehaveintheaboveafresh proofofthecontinuity ofthefunction represented byapowerseriesinitsin­ tervalofconvergence. Ifweusethelim-definition ofcontinuity (v.§19,Def.6)instead ofthee-definition, thestatement ofthetheorem maybeputintothe form: rz, rz, lim(.Efn(x»=2,.'(limj~(x». ~-+-a-o,1=0 n=O '1:-)0011"0 Inthisformitappears asaspecialcaseofthefollowing muchmore elaborate theorem: oTheorem 2.WeassumethattheseriesF(x)=i;fn(x)isuni-19-1. n=O formlyconvergent inthe0peninterval18Xo.••Xlandthatthelimit, whenxapproaches Xofromtheinterior oftheintervall9I limfn(x)=an lZ'-)oXo 18Xomaybe>or<x,,Whether theseriesremains convergeut atxo' andindeedwhether thefunctions fn(x)aredefined thereatall,isimmaterial forthepresent theorem. 19Wearetherefore concerned here,asalsointhetwosubsequent state­ ments,withaone-sided limit. 340 Chapter XI.SeriesofvarIable terms. 00 exists. Theseries2,'allthenconverges andlimF(x),whenx-Xoin n=lJ theabovemanner. exists.Moreover, ifwewrite2:an=A,wehave limF(x)~A, 2::.....xo or,otherwise. 00 Cl' Hm(~'ln(x»=..l'(limf ••(x». ce-+:ro'1=0 n=O.r-)o-Zo (Thelatterformisexpressed shortlybysaying:Inthecaseofuni­ formconvergence, wemayproceed tothelimittermbyterm.) Proof. Givene>0,firstchoosenI'(v.4thformofthedefini­ tion191)sothatforeveryn>nl'everyk:21andeveryxinour interval, Letusforthemoment keepnandkfixed,andmakex-xo'By §19,Theorem 1a,itfollows that Ian+l+anH+...+an+kI<e. Andthisistrueforeveryn>nlandeveryk21.HenceXanis convergent. Letusdenotethepartialsumsofthi.,seriesbyAnand itssumbyA.ItiseasytoseenowthatF(x)-.A.If,foragivene, noisdetermined sothat,foreveryn>no'wenotonlyhave Irn(x)I<~,butalso then,fora(fixed)m>no' [F(x)--AI =I(sm(x)-Am)-(A-Am)+ron(x)I~Ism(x)-AmI+-;-+;. Asx-xoinvolvessm(x)-Am•wecandetermine dsothat foreveryxbelonging totheinterval, suchthat0<Ix-XoI<6. Forthesex's,wethenalsohave IF(x)-A1<8. whichprovesallthatwerequired. If(xn)ischosen arbitrarily intheinterval ofuniform convergence, it follows from F(X,.)=S"(xn)+r"(x,,) aodr"(x,,)....0(v.191,3""form)thatthesequences F(x,,)andsn(xn)willin­ variably exhibitthesamebehaviour asregards convergence ordivergence, and thatiftheyconverge, thelimitswillcoincide. Wemaycontrast thiswiththecase §4-7.Passa~etothelimittermbyterm oftheseries,already seentobenon-umformly convergent, whosepartial ~ums ares.,(x)=-1+n--.;.--;;.Ifherewetakexn=1 •wehaveF(xn)=0,i.e.itiscon-n"x~ n vergent withthelimit0,whereas sn(xn)=t,i.e.italsoconverges. butwiththe limitt.Thetwosequences donothavethesamebehaviour. Theorem 3.TheseriesF(x)=Efn(x)isassumeduniformly con-195. vergentintheinterval.!. andallthefunctions fn(x)aresupposedz'ntegrable overtheclosedsub-interval /':a~x;;:;;b,sothatF(x)isalsocontinuous inthatsub-interval. ThenF(x)isalsointegrable overj'andtheintegralof F(x)overtheinterval/, maythenbeobtainedbyterm-by-term integration, i.e. b b b !F(x)dx or![n§/~l(x)JdX=n~o[! };.(x)dx]. a Q Q (Moreprecisely: Theseriesontherighthandsideisalsoconvergent and hasforitssumtherequiredintegralofF(x). Proof. Given E>0,wedetermine msolargethatforeveryn>m andeveryxina.••b, Irn(x)I<4(h~a)' Since Srn(x)isthesumofafinitenumber ofintegrable functions, itis itselfintegrable overj'.By§19,theorem 11,wecantherefore divide theinterval.!' intoppartsi1,i2,•••,i'Dsuchthat,ifu.denotestheoscil­ lationofSm(x)ini.,wehave Nowtheoscillation ofrm(x)iscertainly<2(b~~' bythemannerin whichmwasdetermined. Alsotheoscillation ofthesumoftwofunctions isnevergreaterthanthesumoftheoscillations ofthetwofunctions. So forthesamesubdivision i1,i2,•••,i'Doftheinterval a•••b,wehave p EivUv<E, v=l whereUvdenotestheoscillation ofF(x)iniv'Thus(againbv§19,theorem 11)F(x)alsoisintegrable over/'. Furthermore, asF=Sn+rmwehave, foreveryn:?:m, I!F(X)dX- !s"(X)dxl=I!Tn(X)dxl<~ <c, -thelatterby§19,theorem 21.Nows"(x)isthesumofafinitenumber offunctions; applying§19,theorem 22,wetherefore atonceobtain IiF(x)dx-v~oif.(x)dx1<e. 1:1 (051) 342 Chapter XI.Seriesofvariable terms. 196.b Tlus,however, impliestheconvergence of.:EJf~(x)dxandtheiden· a tityofitssumwiththecorresponding integral ofF(x). Matters arenotsosimpleinthecaseofterm-by-term differen­ tiation. In190,7, wesaw,forinstance, thattheseries .f;SInnx 11=1 n converges foreveryx,andsorepresents afunction F(z)defined forevery realx.Thetermsofthisseriesare,without exception, continuous anddiffer­ entiable functions. Ifwedifferentiate termbyterm,weobtaintheseries QC ~cosnx, n=1 whichisdivergent 20foreveryx.-Evenifaseriesconverge,> uniformly foreveryx.asforl1lstanee theseries 1.;sinnX 11=1ng (cLExample 5,191,2), theposition ISnobetter,sinceondifferentiating term bytermweobtal1l aserieswhichdiverges e.g.forx=o. Thetheorem onterm·by-term differentiation mustaccordingly be ofadifferent stamp. Itrunsasfollows: co Theorem 4.Given 21aseriesEin(x)whosetermsaredifferen. 11=0 tiableintheinterval]==a...b,(a<b);iltheseries .f;fn'(x), 11=0 deducedIromitbydillerentiating termbyterm,converges uni10rmIy in],thensodoesthegivenseries,provided itconverges atleastat onepoint01].Furthe"ilF(x)andcp(x)arethelunctions represented bythe,twoseries,F(x)isdillerentiable, andwehave F'(x)=cp(x). Inothe,words,withthegivenhypotheses, theserzesmaybedille,· entiatedte,mbyterm. gOTheformulae established onp.357give.foreveryx+2kn, 1 . ~G+~x- +cosx+cos2x+...+cosnx=---'----'-- 2' .x2SlO2" glAsregards theconvergence oftheseries,noassumption ismadein thefirstinstance. ~47.Passage tothelimittermbyterm. 343 Proof. a)Letcdenoteapointof](existent byhypothesis) for which ~r"(c).converges. Bythefirstmeanvaluetheorem ofthe differential calculus (§19,theorem 8) ,,+k nI-k.2(f~(x)-f"(c)=(x-c)..2f,,'(~), ~=n+1 ~~n+1 where ~denotes asuitable pointbetween xandc.Given t;>0,we can,byhypothesis, choosenosothatforeveryn>no'everyk~1, andeveryxinj, 1nlk I.2r:cx)<b~a'.=n1-1 UnderthesamecondItions, wetherefore have I,,=~k1(r"(x)-r~(c)I<t;. Thisshowsthat2;(r,,(x)-f,,(c)), andhenceEf,,(x) itself,isuniformly convergent inthewholeinterval ]andaccordmgly represents ade· finitefunction F(x)inthatmterval. b)NowletXobeaspecialpointof]andwrite ("(xo+~-f"(xo)=g~Ch), (Y=0,1,2,...). Thesefunctions aredefined foreveryh::z:.0forwhich Xo+hbelongs to].Asabove,wemaywrite ig"(h) n=Oandwefind,asIDn+k n+k.2g~(h)=27f,,'(xo+{}h) .=n1-1 ~=n+1 a),that(0<{}<1) converges uniformly forallthesevaluesofh.Thisseriesrepresents thefUDction F(xo+h)-F(xo) h Bytheorem 2,wemayleth-.0termbyterm,andwcconclude that F'(xo)exists,WIth F'(xo)=l'(Iimgn(h))=l'rn'(xo)' n=Oh~O n=O Thissignifies thatF'(xo)=g;(xo),asasserted. Examples andRemarks. 1.If2"an(x-xo)nhastheradius'>0andifO<I?<', theseries 2"nan(x-xo)"-1convergesuniformly foreveryIx-XoI<e.Bytheorem 4,the givenpowerseriesaccordingly represents afunction whichisdifferentiable Cor everyIx-XoI~(!.Foranyparticular x,withIx-XoI<1',whichwemay choosetoconsider, wecandetermine f/<I'sothatIx-XoI<(!<1'.The 344 Chapter XI.Seriesofvariable terms. function represented byIan(x-xo)"therefore remains differentiable Eltevery pointoftheopenintervalIx-Xo1<1'. aJsinnx2.ThefunctIOn represented by.2--3- isdifferentiable foreveryx n=1n anditsderived function is.i:cos:x.(Ct.Example 5,191,2.) n=1n 3.Thecondition ofuniform convergence iscertainly sufficient inall fourtheorems. Butitremains questionable whether itisalsonecessary. a)Inthecaseofthecontinuity-theorem 1oritscorollary, thISiscert­ mnlynotso.Theseriesconsidered in192,2and4haveeverywhere-con­ tinuous termsandrepresent everywhere-continuous functions themsclvc~. Yet theirconvcrgence wasnotuniform. Theframing of71ecessary andsuffIcient conditions isnotexactly easy.S.Arzel.l(Rendiconti Accad.Bologna, (1),Vol.19, p.85.1883)wasthefirsttodosoinasatisfactory manner. Asimplified proofofthemaintheorem enunciated byhImwillbefound IIIG.VlVanh (Rendlcontl delcirc.matem. diPalermo, Vol.30,p83.1910).Inthecasein whichthefunctions In(x)areposihve, IthasbeenshownbyU.D,mthatUDl­ formconvergence isalsoneces!>ary forthecontinuity ofF(x).Cr.Ex.1.'i8. b)ThefactthatIIItheorem 19~onterm-by-term integrntion uniform convergence isagamnotanecessary condition mayalsobevenfted byvarious examples. Taking theseriesifn(x)discussed onpp.330-1,whosepartial n=1 sumsare andwhosesumisF(x)=0,weseeatoncethat n1 1 1 .L;Jf..(x)dx=fsn(x)dx~0=fF(x)dx. ,-=10 0 0 Thusterm-by-term integration leadstothecorrect result. Inthecaseofthe 1 series192,3, however, inwhichwealsohavefF(x)dx=0,term-by-term o integration gives,onthecontrary,nil _{n.2.rf..(x)dx=fSII(X)dx=1-e~1. ..=00 0 Inthiscase,therefore, term-by-term integration isnotallowed. §48.Testsofuniform convergence. Nowthatweareacquainted withthemeaning oftheconcept ofuniform convergence, weshallnaturally inquire howwecande­ termine whether agivenseriesdoesordoesnotconverge uniformly inthewholeorapartofitsinterval ofconvergence. However difficult itmaybe-andweknowitoftenisso-todetermine themereconvergence ofagivenseries,thedifficulties willofcourse beconsider~bly enhanced whenthequestion ofuniform convergence §48.TestsofunIform convergence. 345 isapproachcd. ThetcstwhIch i~themostimportant forapplications, be· causeItistheeasiesttohandle, isthefollowing: oJYeier:strass' test.Ifeachofthefunctionsf"(x)isdefinedand197. bounded intheinterval].-say Ifn(x)I<rn throughout] -andiftheseriesZr..(ofpositiveterms)converges, the series2'f~(x)converges uniformly in]. Proof.Ifthesequcnce (xn)ischosenarbitrarily in],wehave Ifn+l(x,,)+fnH(xn)+...+fnH"(xn)I:s::i'n+l+J'nH+...+i'n+k,,' BySI,2,therighthand ~idc-+0whenn-+00;hencesodoes theleft.By191,5tllform,.2,'fn(x)istherefore uniformly conver­ gent 111J. Examples. 1.Intheexample 191,3wehavealready madeuseofthesubstance of WClcrslrass' test. 2.Theharmonic series"'~,whichconverges forx>l, isuniformlyLJnX convergent onthesemi-axis x~1+il,whereilisanypositive number. In fact,forsuchx's, whereIr"converges. Thisprovesthestatement. Thefunction represented bytheharmonic series-knownasRiemann's {-functIOn anddenoted by{(x)-istherefore certainly continuous forevery 21 x:>1. 3.Dlfferentiatlllg theharmonic seriestermbyterm,wededucetheseries ~ _i~gn n-=lnX • Thisagainisulllformly convergent inx~1+il>1.Infact,foreverysuffi· logncielltlylargen,~<1(by3S,4);forthesen'sandforeveryx~1+il, n wethenhave IIOI!nI<1logn1----,;r =n1+J(2'nJ/~<~T-t.\-;2=r... Riemmm's ~-function isaccordingly differentiable foreveryx>1,andItsderivative isrepresented bytheseries(-). 4.If1:anconverges absolutely, theseries 1:ancosn xand1:ansinn:It areuniformly convergent foreveryx.sincee.g.Iancosn xI;:::;an='Yn'These senesaccordmgly definefunctions continuous everywhere. Inspiteofitsgreatpractical importance, Weierslrass' testwill necessarily beapplicable onlytoarestricted classofseries, since it 22Infact.ifweconsider aspecialx>I,wecanalwaysassume Il>0chosen sothatx>1+Il. 346 Chapter Xl.Seriesofvariable terms. 198.requires inparticular thattheseriesinvestigated shouldconverge absolutely. Whenthisisnotthecase,wehavetomakeuseofmore delicate tests,whichweconstruct byanalogy withthoseof§43.The mostpowerful meansforthepurpose isagall1Abel'spartialsummatIon formula. Onlinesquitesimilartothosealready followed, wefirstob­ tainfromitthe °Theorem. AseriesoftheformZan(x).bn(x) certainly converges n~O uniformly intheintervalf,if,inf, 00 1),J;A...(bp-b..+l)isuniformly convergent (asaseries)and ..=0 2)(All·bn+l)isuniformly convergent (asasequence)~:S. Herethefunctions An=An(x)denotethepartialsumsof2'an(x). Proof. Asformerly -wehavemerelytointerpret thequant­ itiesa..,b..andA..asnolongernumbers, butfunctions ofx-we firsthave n+k ,,+k ,J;a..b..=,J;A,,'(bp-b..+l)+(A"H·bn+k+l -An·bn+l)' "~n+1 ,,=n+1 Lettingxandkvaryinanymanner withn,wehaveontheleft2 sequence otportions oftheseriesIa..b",andonthenghtthecorresponding onerelative totheseriesIA..(b"-b,,+l)'andadifference-sequence ofthesequence (An'bn+l)'Sincebyhypothesis thelattersequences alwaystendto0 (v.191,5thform),itfollows thatsodoesthesequence ontheleft. This(againby191,5),provesthestatement. Exactly asin§43,theabovetheorem, whichisstillverygeneral incharacter, leadstothefollowing morespecial, butmoreeasilyman· ageable tests24: °1.Abel'stest.Ia..(x).b,,(x)isuniformly convergent inf, ifIa..(x)converges uniformly inJ,iffurther, foreveryfixedvalue ofx,thenumbers bn(x)formarealmonotone sequence 25andif,for 23an'bn,Anarenowalwaysfunctions ofxdefined intheinterval/;only forbrevityweoftenleavethevariablexunmentioned. -Forthenotionofthe umform convergence ofasequenceoffunctions cf.190,4. 24Forsimplicity's sake,wenamethesecriteriaafterthecorrespondmg ones forconstant terms.-Cf.p.:::15,footnote 8. 25Cf.footnote to184,1. §48.Tcst~ofunitorm cOllvcrgence. 347 everynandeveryxin],thefunctions b"(x)arelessinabsolute valuethanoneandthesamenumber 28K. Proof.Letusdenotebyan(x)theremainder corresponding to thepartialsumAn(x); i.e.l'a,,(x) =An(x)+an(x).Intheformula n=O ofAbel'spartialsummatIon, wemay(bythesupplement 183)sub· stitute-a"forA",andwcobtain ,,"k n+k ~a,,·b.= -..1'a.'(b"-b"+1)-(a,,+k·bn+1<+1-C:,,·b"+1); 1'=,,"1 "=n+1 ittherefore againsuffices toshowthatboth,2'c:"(b"-b"+1)and (a".bn+1)converge uniformly inJ.However, thean(x/s,asremainders ofaunIformly convergent series,tenduniformly to0andtheb"(x)'s remain<Kinabsolute valueforeveryxin];itfollows that(a,,'b"+1) alsoconverges uniformly to0in].Ontheotherhand,Ifwecon· sidertheportions nI-k Tn=2)a"(x).(b"(x)-b"+1(x)),,-=,,+1 wecaneasilyshowthatthesetenduniformly to0in],-thereby completing theproofoftheuniform convergence in]oftheseries underdiscussion. Infact,if(X"denotes theupperboundofa"(x)in], a.-O (v.form3a).Thusifenisthelargest ofthenumbers iXn+l' an+2,•••,thisEnalso-0and n+AI:Z:,I<e,,'~Ib"-b"+ll<Fn'lbn+1-bn+H11<2J(.F...1'=,,+1 involves thefactthatTn-0uniformly in]. °2.Di'l'icltlet's test.1;an(x).bn(x)isuniformly convergent in], n-O ifthepartialsumsoftheseries ~an(x)areuniformly bounded 26inJ andifthejunctions b"(x)converge uniformly to0in],theconver­ gencebeingmonotone joreveryfixedx. Proof. Thehypotheses and192,5immediately involve the uniform convergence (againto0)of(An·bn+1).If,further,K'denotes '"Thebn(x)'sform,forafIxedx,asequence ofnumbers bo(x),b1(x),..•; forafixedn,howcver, bn(x)isafunctIon ofx,definedin].Theabove IIS­ sumptlOn may,then,beexpressed asfollows: Allthesequences, forthevarious valuesofx,shallbcumformly bounded wIthregardtoallthesevaluesofx; inotherwords, eachoneisbounded andthereisanumber J{whichis s~mllllaneously aboundaboveforthemall.Oragain: Allthefunctions defined in]forthevarious valucsofnshallbeuniformly boundcd withregard to allthesevalucsofn;i.e.eachfunct:on isbounded, andanumber1\eXists whichsimultaneously ('xcecds themallinabsolute value. 348 Chapter Xl.Seriesofvariable terms. anumber greater thanalltheIAn(x)"sforeveryx,wehave R+kI R+k.2A...(b..-b..+!)<K'·.2Ib..-b..+ll<]('.Ibn+7<+1-b"+1I·v=ntl v=n+l Inwhatever wayxandkmaydependonn,therighthandsidewill tendto0bythehypotheses, hencealsotheleft.Thisprovestheuni· formconvergence in]oftheseriesunderconsideration. Themonotony oftheconvergence ofbn(x)forfixedxhasonly beenusedineachoftheseteststoenableustoobtainconvenicnt upperestimations oftheportionsZIb..-bV+1I.By~lightlymodifying thehypotheses withthesameendinview,weobtain 03.TwotestsofduBois-Reymonrl andDedelrlnd. a)TheseriesZa..(x).bv(x)isuniformly convergent in],ifboth Zavand2,'Ibv-bV+1Iconverge uniformly in]andif,atthesame time,thefunctions bn(x)areuniformly bounded in]. Proof. Weusethetransformation R+k n+k .2av·bv= -.2"'".(b"-b"+l)-("',,+k·b,,+k+l -"'".bn+1).,,=ntl v=ntl Astheremainders "'"(x)nowconverge umformly to0,wehavc,for every'V>m,say,andeveryxin],I"'"(x)I<1.Henceforevery n2m, theexpression ontheright-evenifxandkaremadetodepend onn,inanymanner -nowtendsto0asnmcreases, henceso doestheexpression ontheleft.That"',,'bnt1tendsuniformly to0 111]follows, by192,5,fromthefactthat"'n(x)doesandthatthe bn(x)'sareuniformly bounded in]. b)Theseries}}av(x).h,.(x)tSuniformly convergent in.7~fthe seriesElbv--bv+1Iconverges uniformly inJ,andtheseriesEQvhasuni­ formlybounded partialsums,provided thefunctions bn(x)-.)-0uniformly in./. Proof.Fromthehypotheses, itagainfollowsatoncethatAnb"~l converges uniformly (to0)in].Further, ifK'oncemoredenotes a number greater thanalltheIAn(x)I'sforeveryx, \R+k I n+k v=~lA ...(b..-b..+1)I<]('~=Il~b..-b..+11, whence, onaccount ofourpresent hypotheses, theuniform convel" gencein]oftheseries.2:A..(b..--b..+1)mayatoncebeinferred. §48.Testsotumtorm convergence. 349 ExampiesandIllustra lions. 1.InapplIcatIOns, oneorotherofthetwofunctions an(x)andbn(x)199. \\111oftenreducetoaconstant, foreveryn;itwIllusually betheformer. Nowaseriesofconstant terms:::avmust,ifitconverg-es, ofCOUlsebere· garded asu,nzformly convergent ~nevery ~nte1'Val ..for,itstermsbeingindependent ofx,soareItsportions, and any upperestimatIon valIdforthelatteris valid ~psofactoforeveryx.Similarly thepartlalSlImsofaseriesofcom,tant terms2:av,ifbounded, mustbeaccounted unzformly bounded 1IleveryIllterval 2.Lct(an)beasequence ofnumbers with~anconvergent, andlet bn(x)=X".Theseries2,'anxnISumformly converg-ent In°~X~1,forthe conditIOns ofAbel'~testarefulfIlled IIIthbinterval. Infact,:::a",asre­ marked inI.,ISuniformly convergent; fluther,foreveryfixedxintheInter­ val,(xn)ISmonotone andIxnI~1.-Bythetheorem 19-1onterm·by-term passage tothelimit,wemaytherefore conclude that lim(Sa"x")=2.' (IIma,.x"),Le. z~1-0 ~~1-0 =Xa,.. ThisgivesafreshproofofAbel'slimittheorem100. 3.Thefunctions bn(x)~_1_alsoformasequence bounded uniformlyn< IIIJ(namely, agaIn ~1),andmonotone foreveryfIXedx.Hence,asabove, wededucethat ifXa,.denotes aconvergent seriesofconstant terms. (Abel'slImittheorem orDHlc"let series.) 4.Letan(x)=cosnx or=sinnx, andb,,(X)=-.!_, et>O. Theseriesn" (et>0), or00cosnx.2)an(x).b,.(x)=2) n=1 n~lna thensatIsfytheconditions ofDirichlet's testineveryinterval oftheform 21S~ xS2TT-D,whereDdenotes apositive number<TT. If(bn)denotes anymonotolu -umformly, nullsequence,Infact,byISil,5, thepartialsumsof2,'an(x) intheinterval (wemaytakeK=~\-") andbn(x) sin'2"d because bndoesnotdepend onx. itfollows forthesamereasonthatareuniformly bounded tendsmonotonely to0, andXb"sinnx areunzformly convergent inthesameintervals (cf.185,5). -Alltheseseries accordingly represent functions whicharedefined andcontllluous ,.forevery 17Orinintervals obtained fromtheabovebydisplacement through anintegral multiple of27f. ••EveryfixedXof2k7fmayindeedberegarded asbelonging toaninter­ valoftheaboveform,ifJissuitably chosen(cf.p.343,example I,andp.345, footnote). 12- (Gal) 350 Chapter XI.Seriesofvariable terms. x,*2kfT.Whether thecontinuity subsists attheexcluded pomtsx=2kTTwe cannotatoncedetermine, -noteveninthecaseoftheseriesEbnsinnx,although itcertainly converges atthesepoints(cf.116,4). §49.Fourier series. A.Euler's formulae. Amongthefieldstowhichwemayapplytheconsiderations developed inthepreceding sections, oneofthemostimportant, andalsooneofthe mostinteresting initself,isprovided bythetheoryofF'ourierseries,and moregenerally bythatoftrigonometrical series,intowhichwenowpro­ posetoenter 29. Byatrigonometrical seriesismeantanyseriesoftheform 1 002ao+£(ancos11x+hnsin11x), n=1 withconstant 30anandhn.Ifsuchaseriesconverges 10aninterval of theformc<x<c+2TT,itconverges, inconsequence oftheperiodicity ofthetrigonometrical functions, foreveryrealx,andaccordingly represents afunction definedforallvaluesofxandperiodic withtheperiod2TT.We havealready comeacrosstrigonometrical seriesconvergent everywhere, forinstance, theseries,occurring afewlinesback, 00• 00£_~mn.:: (X>0;icosn x (X>1;etc. n=lnOC J n=1n'X, WehaveneverbeeninapOSitIOn, sofar,todetermine thesumofany oftheseseriesforallvaluesofx.Itwillappearverysoon,however, that trigonometrical seriesarecapable ofrepresenting themostcurious types offunctions -suchasonewouldnothaveventured tocallfunctions atallinEuler'stime,astheymayexhibitdiscontinuities andirregularities ofthemostcomplicated description, sothattheyseemrathertorepresent apatchwork ofseveralfunctions thantoformoneindividual function. ,.Moreorlessdetailedandextensive accounts ofthetheoryaretobefound inmostofthelargertextbooksonthedifferential calculus (inparticular, that referred toonp.2,byH.v.Mangoldt andK.Knopp,Vo!'3,8thcd.,Part8,10-1-1). Forseparate accounts, wemayrefertoH.Lebesgue, Le90nssurless~riestngono­ metriques, Paris1906,andtotheparticularly elementary Introduction tothe theoryofFouner's series,byM.Bocher,AnnalsofMath.(2),VD!.7,pp.81-152. 1906.Aparticularly detailedaccountofthetheoryisgivenbyE.W.Hobson,The theoryoffunctions ofarealvariable andthetheoryofFourierseries,Cambndge, 2nded.,VD!.I,1921,andVo!'2,1926.Thecomprehensive worksofL.Tonelli, Senetrigonometnche, Bologna 1928,andA.Zygmund, Trigonometrical series, Warsaw 1935,arequitemodern treatments; thelittlevolumebyW.Rogosinski, Fouriersche Relhen, Sammlung Goschen 1930,isparticularly attractive andcon­ tainsawealthofmatter. 1 aDItisonlyforreasonsofconvenience that2aDiswritteninsteadofaD. §49.Fourier senes. -A.Euler's formulae. 351 Thusweshallseelateron(v.21?a)thntc.g. .{=Oforx=k:rr, (k=O,±1,±2,...),bllt cosinnx2-- (2k+I)n-X •n=1n= 2 for2kn<x<2(k+1)n; thefunction represented bythisseriesthushasagraphofthefollowing type: Fig.7. Similarly, weshallsee(v.209)that 1=0forn=k:Tt,but y,<;in(~n+~)x=:for2kJr<:l:«~k+l):Tt, and n-':::o2n+1=-: for(2k+1):rr<z<2(k+1):rrj thusthefunction represented bytheseneshasagraphofthetype: ~,: I I ,.----.------- -~--------+--------~--------- ------Z.iZ -.1l: ()I .7l 2.n: I Fig.8. Ineithercase,thegraph ofthefunction consists ofseparated stretches (unc1osed ateitherend)andofisolated points. However, thecircumstance thatSimpletrigonometrical seriessuch astheabovearecapable ofrepresenting functions whicharethem­ selvesaltogether discontmuous and"pieced together", isprecisely what waschieflyresponsible forthethorough revision towhIchtheconcept offunction, andthencethewholefoundation ofanalysis, cametobe subjected atthebeginning ofthe19thcentury. Weshallseethat trigonometrical seriesarecapable ofrepresenting mostoftheso-called "arbitrary functions" 31;inthisrespect, theyconstitute afarmore powerful instrument inhigheranalysis thanpowerseries. 31Ofcoursetheconcept ofan"arbitrary function" isnotsharply defined. Thetermusually denotes afunction whichcannot beassigned bymeansof !lsingleclosedformula (Le.oneavoiding theuseoflimiting processes) IDterms 352 Chapter XI.Seriesofvariable terms. Wewillmention onlyincidentally thattherangeofthisinstru­ mentisbynomeansrestricted topuremathematics Quitethecon­ trary:suchserieswerefirstobtained intheoretical physics, inthe courseofinvestigations onperiodIC motIon, i.e.chieflyinacoustics, optics, electrodynamics, andthetheory ofheat;Fourier, inhis Theorie delachaleur (1822)instituted thefirstmorethorough study ofcertaintrigonometncal series,-although hedidnotdiscover any ofthefundamental resultsoftheirtheory. Whatfunctions canberepresented bytrigonometrical seriesandby whatmeanscanweobtaintherepresentation ofagivenfunction, sup· posingthistobefeasible? Inordertoleaduptoasolution oftIusquestion, letusfirst assume thatwehavebeenabletorepresent aparticular function{(x) byatrigonometrical seriesconvergent everywhere: Onaccount oftheperiodicity ofthesineandcosinefunctions, {(x)isthennecessarily periodic withtheperiod2:rc,anditissuff!· e!Cnt,therefore, toconsider anyinterval oflength2:rc.Wechoosethis interval, forallthatfollows,tobeo::s:::x<2:rc,-whereoneofthecnd· pointsmay,moreover, beomItted. Thefunction {(x)isthenrepresented inthisinterval byacon­ vergent seriesofcontinuous functions. Weknowthat((x)maynone thelessbedIscontinuous, although italsowillbecontinuous ifthe seriesinquestion converges uniformly intheinterval Forthemoment, wcwillassume thistobethecase. Withthesehypotheses, weobtainarelationship between ((x)and thecoefficients anandbnwhichwasconjectured byEuler: oftheso-called elementary functions alone,-i.e.inparticular, itdenotes a function whIch ISapparently bUiltupfromseparate portions ofsimplefunc· tlOnsofthistype,likethefunctions givenasexamples inthetext,orthe following, defined foreveryrealx: ((x)'=k J(x)=k+(x-k)1fl flx)=x-k {Xo((x)=ink::::x<k+1 (k=0,±1,±2,...) forirrational x forrationalx, etc.Cf.,however, the"arbitrary" function expressed bymeansoflimiting processes onp329,footnote. Notuntilitwasfoundthatevenaperfectly "arbitrary" function suchasthesecouldberepresented byasmgle(relatively simple) expreSSIOn, asforinstance byourtngonometrical seriesorbyother limitmg processes, -didanynecessity ariseforregarding itasbeingactually onefunction, instead ofamerepatchwork ofseveral functions. §49.Fourier series.-A.Euler'sformulae. Theorem 1.Theseries 1 .."2ao+2,'(a"cosnx+b"sinnx) fI=1353 200. isassumed uniformly convergent32intheinterval 0~x<2n,with thesumf(x).Thenforn=0,1,2,...,wehave :I", bn=~ff(x)sinnxax. o forp4=q forp=q>0 forp=q=02", an=~Jf(x)cosnxax, u (EulerorEuler-Fourier formulae) 33. Proof. Asisknownbyelementary considerations, the formulae 34holdfore"eryintegralpandq(2':0): a)rcospx.cos qxdxJ--~1=2n 2'" b)Jcospx.sinqxdx =0 ofollowing c)["Sinpx.sinqxdx{~forP4=qandp=q=0 forp=q>O. Letusmultiply theseriesforf(x),whichisuniformly convergent in0<x<2n,bycospx;byU'2,2theumformity oftheconver genceisnotdestroyed, andafterperforming themultiplilation we mayaccordingly (v.195)integrate termbytermfrom0to2:n. Weimmediately obtain: Q1=_!..aojncospxdxforp=0.:< 20Jf(x)cospxdxo 2n =apJcospx•cospxdxforp>O. o 31Inconsequence oftheperiodicity ofcosxand!>InJ:,itisthen,spso facto.uniformly convergent foreveryx. 13Thisdesignation isapurelyconventional one;historical remarks are givenbyH.Lebesgue. lococit.,p.23;A.Sachse. Versuch einerGeschichte der trigonometrischen Reihen, Inal1g.-Diss., Gottingen 1879;P.duBOls-Repnond, in hisanswertothelast-named paper;aswellasveryextenSively byH.Burh­ hardt,Trigonometrische ReihenundIntegrale bisetwa1850(Enzyklop. d.malh. Wiss.,Vol.Il,1,Parts7and8,1914-15). HWehaveonlytotransform theproduct ofthetwofunctions inthe integrand intoasuminaccordance withtheknownaddition theorems, (e.g.cospx·cosqx=~[cos(p-q)z+cos(p+q)xl),inordertobeableto integrate straight away. 354 Chapter XI.Seriesofvariable terms. I.eIIIeithercase 2,.. a=~f{(X)COSPXdX;Pn o fortheremammg termsgive,onintegration, thevalueO.Inthesame way,multiplying theassumed expansion of{(x)bysinpxandthen integrating, weatoncededuce thesccond ofEHler's formulae 2,.. bp=~S{(x)sinpxdx. o Thevalueofthistheorem isdiminished bythenumber or assumptions required tocarryouttheproof. Also,itgivesnoindi­ cationhowtodetermine whether agivenfunction canbeexpanded in atrigonometrical seriesatall,or, ~fitcan,whatthevaluesofthe coefficients willbe. However, thetheorem suggests thefollowing modeofprocedure: Lct{(x)beanarbitrary function defined intheinterval 0~x<2:n, andintegrable inRiemann's senseintheinterval. Inthatcasethe integrals mEuler's formulae certainly haveameaning, by§19, theorem 22,andgivedefinite valuesforanandbn.Wethererare notethatthesenumbers, exist,onthesinglehypothesIs that{(x)is intcgrable. Thenumbers ;ao'a1'a2,.,.andb},bp...thusdefined byEuler's formulae willbecalledtheFoul'ier constants orl"uu1'le1' coefficients ofthefunction ((x).Theseries 1 DC2aO+Z(a,.cosnx+bnsinnx) n=l maynowbewritten down,although thisimplies nothing asregards itspossible convergence. ThiSserieswillbecalled(without reference toitsbehaviour ortothevalueofitssum,IfeXistent) tlteJl'ourier .'lcriesgcncl'fltccl fly,orbclouylng to,{(x),andthisisexpressed symbolically by 1 CZ)f(x)'""2ao+I(ancosnx+b..sinnx). n=1 Thisformula accordingly implies nomorethanthatcertain constants an'bn,havebeendeduced from{(x)(assumed onlytobeintegrable) bymeansofEuler's formulae, andthatthentheaboveserieshasbeen writtendown 35. 3:;Thesymbol"","hasofcourse noconnection herewiththesymbol introduced in40,Definition 6,for"asymptotically proportional", Thereisno fearofconfusion. §49.Fourier series. -A.Euler's formul!\.f. 355 Fromtheorem 1.andthemanner inwhichthisserieswasderived, wehave,itistrue,somejustification forthehopethattheseriesmay converge andhave(x)foritssum. Unfortunately, thisisnotthecaseingeneral. (Examples willbe metwithveryshortly,) Onthecontrary, theseriesmaynotconverge inthewholeinterval, norevenatanysinglepoint;andifitdoesso, thesumisnotnecessarily (x).ItisimpossIble tosayoff-hand whentheoneortheothercasemayoccur;ItISthlScircumstance which prevents thetheoryofFourier seriesfrombeingentirely asimplesubject, butwhich,ontheotherhand.renders itextraordinarily fascinating; forhereentirely newproblems arise,andwearefacedwithafunda­ mentalproperty offunctions whichappears tobeessentially newin character: theproperty ofproducing aFourier serieswhosesumis equaltothefunction Itself.Thenexttaskisthentoelucidate the connection between thisnewproperty andtheoldones,-viz.con­ tinuity,monotony, differentlability, I11tegrability, andsoon.Morecon­ cretelystated,theproblems whicharisearetherefore asfollows: 1.IstheFourier seriesofagiven(integrable) function {(x)con­ vergent forsomeorallvaluesofxin0<x<2n? 2.11itconverges, doestheFourier series01{(x)haveforits sumthevalue01thegenerating function? 3.IftheFourier seriesconverges atallpoints01theinterval a<x<13,istheconvergence uniform inthisinterval? Asitisconceivable thatatrigonometrical expansion of(x)might beobtained byothermeansthanthatofEuler's formulae, wemay alsoraisethefurtherquestion atonce: 4.IsitpOSSIble (orafunction whichiscapable ofe;tpansion in atrigonometrical seriestopossessseveral suchexpansions, -inpar­ ticular, canitpossess another trigonometrical expansion besides the possible Fourier expansion provided byEuler's formulae? Itisnotveryeasytofindanswers toallthesequestions; in factnocomplete answer toanyofthemisknownatthepresent day. Itwouldtakeustoofartotreatallfourquestions inaccordance with modern knowledge. Weshallturnourattention chiefly tothefirst two;thethirdweshalltouchononlyincidentally, andweshallleave thelastalmostentirelyoutofaccount 36. 36Itshouldbenoted,however, thatthefourthquestion isanswered under extremely ~eneral hypotheses bythefactthattwotrigonometrIcal serieswhich converge in0~x:52'TTcannot represent thesamefunction inthatinterval without beingentirely identical. Andiff(x),thefunction represented, isintegrable over0...2'TT,itsFourier coefficients areequaltothecoefficients ofthetrigono­ metrical expansion; cf.G.Cantor(1.f.d.remeu.angew.Math.,Vol.72,p.139. 1870)andP.duBoif-Reymo1ld (MUnch. Abh.,Vol.]2,SectIOn I,p.117.1870). 356 Chapter XI.Seriesofvariable terms. \Viththedesignations introduced above,thecontentofTheorem 1 maybeexpressed asfollows: Theorem 1a.Ifatrigonometrical seriesconverges uniformly ino:sx<27T(i.e.forallx),itistheFourierseriesofthefunction repre­ sentedbyit,andthisfunction 37admitsofnootherrepresentation byatrigono­ metrical seriesconverging uniformly in0~x<27T. ThefactthattheFOl/rier senesofanintegrable function doesnotneces­ sarilyconverge willbeseenfurtheron;thatevenwhenItdoesconverge, Itneed nothavef(x)forItssum,ISObVlOUS fromthefactthattwodifferent functlOns 11(x) andI.(x)mayverywellhaveIdentically thesameFourier constants; infacttwo intcgrable functlOns havethesameintegral (andtherefore thesameFaurier con­ stants; 1.e.thesameFourier series), Iftheycoincide, forinstance, forallrational valucsofx,\\Ithout coincldmg everywhere (v.§19,theorem 18).Thefactthat inaninterval ofconvergence theseriesneednotconverge ulllformly isshownby .sin11xtheexample already usedabove; forthesenes.E--- converges everywhere 11 (v.185,0),andIftheconvergence wcrcuniform, sayinthemterval -I)::;::;x;;:::I),a0,It\\ouldhavetorepresent acontinuous function inthatmterval, by193. ThI~ISnotthecase,however, asweIllenuoned beforeonp.351andWIllprovc latl,ronp.:l75. Thesefewremarks sufficetoshowthatthequestions formulated abovearenotufasimplenature. Inanswering them,weshallfollow thelineadopted byG.Lejeune-Dirichlet, whotookthefirstnotablestep towards asolution oftheabovequestlOns, inhispaperSurlacom'ergence desseriestrigonometriques 38. B.Dirichlet's integral. Weproceed toattackthefirstoftheproposed problems, namely, thequestion ofconvergence: IftheFourier series ~-ao+.E(ancosnX+bnsinnx)generated by agivenintegrable functionf(x),-i.e.withcoefficients giveninterms off(x)byEuler'sformulae, -istoconverge atthepointx=xo,its partialsums 1 n • Sn(xo)=-200+v:l(avcosvXo+bvsmv xo) musttendtoalimitwhenn-++00.Itisoftenpossible todetermine whether ornothisisthecase,byexpressing sn(xo)intheformofadefinite integral asfollows: a,ThiSfunction isthen(by193,Corollary) everywhere continuous. asJourn.f.d.remeu.angew.Math.,Vol.4,p.167.1829. §49.Fourierseries.-B.Dinchlet's integral. 357 Forv>I,thefunction aycosvXo+bysinvXoisrepresented by38 2~ 2~ =[~ff(t)cos"tdt]cos"xo+[~ff(t)sin"tdt]sinuo o 0 2", =~-Jf(t).COS"(t-xo)dt. u Thus 2)l' 2.n Sn(Xo)=2I ;n;J((t)dt+~ff(t)cos(t-Xo)dt+... 1I U 2", +~ff(t).cosn(t -xo)dt o 2~ 1JfI J =nfCt)·L"2+cos(t-Xo)+cos2(t-xo)+'"+cosn(e-xo)dt. u Wenowtaketheimportant stepofreplacing thesumofthe(n+1)terms inbrackets byasingleclosedexpression. Wehaveindeed 40forevery ~9=2kn,forevery IXandallpositive integral m's, COli(a+.~)+(OS(a+2z)+...+cos(a+'flU) sill(<<+:!m-n-7)-sill(a+~)= 2slll-=­:! sillIII~'C(lS(f(+;uTI ~) = , sill; ..Inordertodistinguish theparameter ofintegration fromthefixed l'ointxo'wehenceforth denotetheformerbyt. ODProof.Iftheexpre~slOn ontheleftisdenoted byCm,wehave z m Z 2sin"2'Cm=2:2sin2cos(ex+vz) 1=1 =;:E[-sin(ex+ 2v-I~-)+sin(ex+ 2v+ 1;)] =-sin(ex+ ;-)+sin(ex+2In+l~-) =2sinm~_.cos(ex+;n-+l;). Moreover theaboveformula continues toholdforz= 2k;n;,provided weattrib­ metotheratioontherighthandsidethehmiting valueforz-+2k:r. i.e.thevaluemcosex.201. 358 l:bapter XI.Seriesofvariable terms. fromwhichmanyanalogous formulae maybededuced asparticular cases41.Takinga=0,Z=t-xo'm=n,weobtain 1-2+cos(t-xo)+...+cosn(t-xo) Accordingly42, (a) 2.a=0gives:Finally, wemaytransform thisexpression somewhat. Thefunction {(x)needonlybedefined intheinterval°<x<2nandintegrable overthisinterval. Thelatterproperty remams unaltered ifwemerely modifythevalueof{(2:n)(d.§19,theorem 17).WeWIllequateit toreO)anddefine{(x)further, foreveryxsuchthat 2kn<x<2(k+l)n, (k=±l, ±2,...), by: {(x)=f(x- 2kn). tlForsubsequent use,wemention thefollowing: ~-+asubstituted foragives: z(--2&-). sinm2·sina+m+l sin(a+z)+sin(a+2z)+ ...+sin(a+mz)= ,. zsm"2 . z--zsmm2cosm+1-­2cosz+cos2z+ ...+cosmz=-- ~----. zsm-2 z------ Z 71:• sinm2-'sinm+1"2 3.a=-2gIves:sinz+sin2z+"'+Slllmll= --j. z5m2 4.z=2x,a=r-x, give:---- sinmx·cos(r+Inx)cos(r+x)+cos(1'+3x)+...+cos(1'+2m-I.x)=-----;-----jSInx 5.I=2x,a=%+r-x, give:.. .--- sinmx· sin(r-I-mx)sm(r+x)+sm(r+3x)-I-•••+smer+ 2m-I .x)= . .sInx .2Fort=xo,liSwcob!>crveu oncebefore, weshouldattribute tothe sine-ratIO thelimiting valuefort--xo'here(2n+1). §49.Fourier series.-B.Dirichlet's integral. 359 Ourfunctionf(x)isnowdefined forallrealvaluesofxandwehave arranged forittobeperiodic withperiod2n.Nowforanyfunction cp(x)periodic withperiod2n,wehave(by§19,theorem 19),what­ everthevaluesofcandc'maybe, fI P+9n:fcp(t)dt=fcp(t)dt. a+~n2n c+2n 2nfcp(t)dt =fcp(t)dt =frp(c'+t)dt and u 0 Astheintegrand in(a)isnowafunction ofthistype,wehave Ifwesplitupthisintegral intothepartsrelative totheintervals otonandnto2n,substituting -tfortinthesecond, thelatter becomes -2" I fsm(2n+1)-:r -21 :rc((xo-t)'---.-,-- dt, -n: sm-2- fl 1.e.bytheaboveremark withregardtofcp(t)dt a +n: I 1J sin(2n+1)2- 2n{(xo-t)·--.-1--dt, o.. sm-~f andweaccordingly obtain ~ , ()1ff(xo+/)+f(xo-t) sin(2n+ll2"" dt S"Xo=n 2'. t ' o sm-f Substituting 2tfort,weareultimately ledtotheformula ()2If(:J:il+2tl+t'(To-2t) sin(2n+l)t Its"Xo='i" 2 '--si-n-t-- ( , o ThisisDirichlet's integral43,bywhichthepartialsumsoftheFourier seriesgenerated byf(x)maybeexpressed. Wemaytherefore state, asourfirstimportant result,thetheorem: Theorem 2.InorderthattheFourierseriesgenerated byafunc­ tionf(x),integrable (hencebounded) andperiodicwithperiod2n,may 43Wedesignate as.Dlflchlet's inlegrab allintegrals ofeitherofthe twoforms202. /Jfet)si~11Idt q>smI nor/J fsinlit nq><')-t-d" 360 Chapter XI.Seriesofvariable terms. :rr ~ ~ff(x"+2t)+J:.(xo-~.sm(2.n+1)tdt n 2 smt uconverge atapointxo'itisnecessary andsufficient thatDirichlet's integral shouldtendtoa(finite)limitasn--+00.Thislimitisthenthe sumoftheFourier seriesatthepointxo' LetusdenotethIssumbys(xo)'Thesecondquestion (p.355), concerning thesumoftheFourier series,whenconvergent, maybe included inourpresentconsiderations andourresultmaybeputina formstillmoreadvantageous inthesequel,byexpressing thequantity s(xo)intheformofaDirichlet integral also.As 1 sin(2n+1)-~-+cost+cos2t+...+cosn t=----~, 2 2'tSill2- wehave f2,",sin(2n+1)~ --t--- dt=1r, o2sin2 (b)or,effecting thesametransformations asbeforewiththegeneral integral, :rr ., ~Jsm(2n+!)t dt=1. n smt o Multiplying thisequation 44bys(xo),wefinallyobtain,bysubtraction from 202, 1T., Sn(xo)-s(xo)=~f[[(xo+_2t);-j(Xo~~ -s(xo)]~m(~i::IUdt. t) Ourpreceding theorem maynowbeexpressed asfollows: 203. Theorem 2a.InorderthattheFaurierseriesgenerated byafunction f(x),integrable andperiodic withperiod217,shouldconverge tothesum s(xo)atthepointxo,itisnecessary andsufficient that,asn--++00,Dirichlet's integral UThisequation mayalsobeobtained from202,bysubstituting f(x)==]; thisgivesao=2and,foreveryn~;I,an.-b".cO.i.e.Sn(x.)=1foreveryn andevery Xo' §49.Fourier series.-13.Dinchlet's integral. 361 shouldtendto0,wherejorbrevitywehaveput [!(x_-1:2t);f(x:-::2t) -s(x)] =rp(t;x). Although thistheorem bynomeanssolvesquestions 1and2insuch amanner thattheansweringivenconcrete casesliesreadytohand,yet itfurnishes anentirely newmethod ofattackfortheirsolution. Indeed thesamemaybesaidwithregardtothethirdofthequestions proposed onp.3fjfj,fortheorem 2amayatoncebemodified tothefollowing: Theorem 3.Ontheassumption thatthepartialsumss"(x)converge tos(x)ateveryp()intoftheinterval (X::::::x-===:{3,theywillconverge uniformly tothzslimitintheinterval, zf,andonlyij,theintegral, depending onx. Tr -2' ~J(tx)·2in(2~-I--1)t_d t 1Trp, sInt u tendsumformly to0asn->--+-00in(X~x~{3,thatistosay ~f,given E>0,wecapassignN=N(E)sothatthirintegralislessthanEinabsolute valueforeveryn>Nandeveryxinex~x;S{3. Beforewemakeuseoftheorem 2toconstruct immediate testsof convergence forFourierseries,weproceed firsttotransform andsimplify thistheorem invariousways.Forthispurpose, webeginbyproving the following theorems, whichapparently leadusratheroffthetrack,but alsoclaimconsiderable interest inthemselves. Theorem 4.Ifj(x)isintegrable over0...2'IT,andif(an)and(bn) '"areitsFourier constants, then~(an2+bn2)converges. 12--=1 Proof. Theintegral 2/t 11J[f(t)-.E(a,.cosv t-I-bvsinvtFd t o I'1 is?:0,asitsintegrand isnevernegative. Ontheotherhand,itis 211 2/tJ[f(t)]2dt- 2.E[avJf(t)cosvtdt] -2.E[b vJj(t)sinvtdt] o u u +J[E(avcosv t+bvsinvt)]2d t o =J[f(t)]2d t-2'IT.Ea~-2'IT27b~--/-'ITLa~+7T~b; o 2/t=J[f(t)J'd t-'IT27(a~+b~), o whereeachsummation isextended fromv=1tov=n.Sincethis expression isnon-negative, wchave 0 ~~l(a~+ b;)<~1[f(t)]2dt. o 362 Chapter XI.Senesotvariable terms. ThusthepartialSilmsoftheseries(ofpositive terms)inquestion are bounded andtheseriesisconvergent, asasserted. Theabovecontains inparticular Theorem ii.TheFourier constants (a,,)and(b,,)ofanintegrable function formanullsequence. Fromthis,wemaydeduce quitesimplythefurther Theorem 6.It'lp(t)isintegrable intheinterval a<t<b,then "An=Jt~(t)cosntlU----+0, u. h .B..=Jt/'(t)sinntlU----+O. a Proof.Ifaandbbothbelong tooneandthesameinterval of theform2kn<t<2(k+l)n, wedefinef(t)='lp(t) ina<t<b andfIt)=0attheremaining pointsofthefirst-named interval. forevery otherrealt,fCt)isdefinedsoastobeperiodic withthepenod2n.Then b 2~ An=J1p(t)cosn tdt=Jf(t)cosn tdt=nan a 0 andsimilarly Bn=nbn,whereanandbndenotetheFourier constants ofthefunctionfU).Bytheorem 5,AnandBntherefore -+O.Ifa andbdonotfulfiltheabovecondition, wecansplituptheinterval a<t<bintoafinitenumber ofportions, eachofwhichsatisfies the conditlOn. A"andBnthenappearasthesumofa(fixed)finitenumber ofterms,eachofwhichtendsto0asn-+oo. HenceAnandBndo thesame 45. Thisimportant theorem willenableustosimplify theproblem of theconvergence ofDirichlet's integral 46. Supposing c5chosen arbitrarily with0<()<-;-,thefunction 1 (.)2-[f(xo+2t)+f(x o-21)]-S(x o) .11(t)=~~ =-------;----;-----r smI sint ..Thisimportant theorem appears intuitively plausible ifweimagine the curve" ='Jl(I)cosnttobedrawnforlargevaluesofn:Weisolateasmallinterval 0:•••fJinwhich'P(I)hasanalmo~tnegligible oscillation (ispractically con­ stant)andproceed lochoose nsolargethatthenumber ofoscillations of cosntisfairlylarge lOtheinterval iinthatcase,thearcofthecurve "='P(t)cosntcorresponding to0:•••{Jwillenclose positive andnegative areasinapproximately equalnumbers andofapproximately thesamesize, sothattheintegral isalmostO. ••Ofcoursetheorem 6maybeprovedquitedirectly, without firstproving theorem 4.Thelatteris,however, anequally important theorem intheth£'ory, eventhough, asithapp£'ns, weshallnotneedItagaininthesequel. §49.Fourier series.-B.Dirichlet's integral. ISintegrable III15StS;.Hence,forfixedr5,363 2 c) fV'(t)sin(2n+l)t.dt-toO. 6 TheDirichlet integral oftheorem 2awilltherefore tendto0as limitasn-to00,if,andonlyif-forafixed,butinitselfarbitrary, valueofb>0 -thenewintegral 6 ~f(t·x)~in(2~+~~!dtnrp'0 Sillt o tendsto0asnincreases. Nowthelatterintegral onlyinvolves the valuesoff(xo±2t)in0<tS<5,i.e.off(x)inXo-2<5<xSXo+2b. Sinceb>0maybeassumed arbitrarily small,thisremarkable result contains atthesametimethefollowing Theorem 7.(Ri<'lIIann's theorem.47) Thebehaviour otthe204. Fourierseriesoff(x)atthepoint Xodependsonlyonthevaluesot f(x)intheneighbourhood otxu'Thisneighbourhood maybeas­ sumedassmallasweplease Inordertoillustrate tIllSpeculiar theorem, wemaymention the following consequence ofit:Consider allpossible functionsf(x)(inte. grablein0...2n)whlchcoincide at apointXooftheinterval 0...2n andIIIsomeneighbourhood ofthispoint,however small,possibly varying withtheparticular function. ThentheFourier seriesotall thesefunctions -however muchtheymaydifferoutside theneigh. bourhood inquestion -must,atXoitself,eitherallconverge orall diverge, andintheformercasetheyhavethesamesums(xo)(which mayor maynotbeequaltof(xo))' Afterinsertlllg theseremarks, weproceed tore·formulate the criterion obtained above, whichwemayhenceforth substitute for theorem 2: Theorem 8.Thenecessary andsufficient condition tortheFourier seriesotf(x)toconverge atXotothesums(xo)'isthattoranar- bitrarily chosenpositive c5<i,Dirichlet's integral d ~f(t·x)sin(2.n+l)tdt:n:rp'0 Sint o shouldtendto0asnincreases 48. t7OberdieDarstellbarkeit einerFunktion durch clUetngonometrische Reihe,Hab.·Schrift, Gottingen 1854(Werke, 2nded.p.227). 48Asregardsuniformity ofconvergence, wecan a~~ertnothing straight away,sinceweareIgnorant astowhether theintegral (c)aboveconsidered, whichtendsto0asnincreases, foreveryfixedxo'willdosouniformly for everyxofaspecified interval onthex·aXlS.Actually thisisthecase,butwe donotpropose toenterintothequestion further. 864 Chapter XI.Seriesofvariable terms. Thereisnodifficulty inshowing thatthedenominator sintIn thelastintegrand maybereplaced byt.InfactthedIfference be· tweentheoriginal integral andtheonesoobtained, i.e.theintegra) d ~fcp(t;xo)[Si~'-+J·sin(2n+l)t.dt, o automatically tendsto0asnincreases, bytheorem 6,-because _.1__ ~_iscontinuous andbounded49,andhenceintegrable, in0<t<d.Sin' , - Thuswemayfinallystate: 20:>. Theorem 9.Thenecessary andsulticient condition fortheFourie1' seriesofafunctionrex),periodicwiththeperiod2:Jlandintegrable over o...2:n,toconverge tos(xo)atthepointxO'isthatforanarbit1'arily chosenpositived(<-i),thesequence ofthevaluesoftheinteg1'al d 2f(t)sin(2'It+1)tdtitP;Xo t o formsanullsequence. He1'ecp(t;xo)hasthesamemeaning asfn theo1'cm 2a.Inanotherform,thecondition isthat,givenc>0,we canassign c5<iandN>0,sothat:;Oforeveryn>N, dI~flp(t;xo)sin(2~_-t!L~dtI<8. o C.Conditions ofconvergence. Ourpreliminary investigations haveprospered sofarthatthe firsttwoquestions ofp.355maynowbeattacked directly. Bythe above,thesearecompletely reduced tothefollowing problem: Givenafunction q;(t),integrable in0~t<d,whatfurthe1' conditions mustthisfunction fulfilinorderthattheintegrals 51 ,1 Jk=~f cp(t).sin/'dt o 1 1 11-sin'6"t-+.... ..Infact,-- -=----= Intheinterval, andthusitselfsint t t.sint1-+... tendsto0ast--+O. 60Thestudent shouldmakeitquitecleartohimselfthatthesecondfor­ mulation isactually equivalent tothefirst,although <5needonlybedetermined atterthevalueofEhasbeenchosen. sink'. 61For'=0,weattribute to-,- IIItheintegrand thevalueIt, §4t!.Fourier series.-C.CondItions ofconvergence. 365 shouldtendtoalimitaskincreases, andwhat,inthatcase,isthe value01thislimit? 52 Sinceinthisintegral, 15hasafixedbutarbitrarily smallvalue, theanswertothisquestion depends only-cf.Riemann's theorem 7 ­ onthebehaviour ofcp(t)immediately totheright010,sayinan interval oftheform0<t<151«l5).Wemayaccordingly inquire also:Whatproperties mustcp(t)possessimmediately totheright 010,inorderthatthelimitinquestion mayexist? Alargenumber ofsufficient conditions forthishavebeenfound, ofwhichweshallonlyexplain two,thegreatgenerality ofwhich renders themsufficient formostpurposes. Thefirstofthesewas establIshed byDZ:richlet intheabove-named paper(v.p.356)andwas thefirstexactcondition ofconvergence inthetheoryofFourierseries, inwhichDtrichlet's workisaltogether fundamental. Thesecond is duetoU.Diniandwasdiscovered in1880. 1.fJirichlet's rule.Ifcp(t)ismonotone totherightof0,--206. i.e.inaninterval01thelorm0<t<d](<l5)-thentheltmZ:tin question exists,andwehave "I·T ).2f sinIftIm.Jk= Im-p(t).--dt=g:o, k~+'" I.~+'" :r: to where CPodenotes the(righthand)limiting valuelImcp(t),whichcer- tainlycxZ:stswiththeassumptions made53. 1-++0 Proof. 1)Inthefirstplace, x CD limr~nldt=fsin~dt=~. Z-~+"'. t 1 2o 0 Theexistence oftluslimit,i.e.theconvergence oftheimproper in­ tegral,followssimplyfromthefactthat,givene>0,andanytwo 3valuesx'andx"both>-,wehave(by§19,theorem 26) 8 :r" x" fSi~1dt=[_cos'JZ"_fcOS1dt t 1z' I~' ~ ~ hence IfZ'~in1I1 1 fZ'~' 11---dt~-+- + -<3·-=8.t-x'x" 19 3 x' Z 52Thereisnosimplification Inobserving thatitwouldsufficetorkto tendto+00through oddintegral values. 63Infact,usqJ(I)isIntegrable, itiscertainly bounded, andbyhypothesis itismonotone in0<t<.)1'-Furtherm(ite tponeednot=tp(0). Chapter XI.Seriesofvariable terms, Now,aswesawonp.360, equation (b),theintegrals ~._fSin(2n+I)'d In- sinIe o 3tforn=0,1,2,. ,.,areall="2' Ontheotherhand,thenumbersTherefore wealsohave 1C ~ i'=J(-J----!.)sin(2n+1)t.dtn smII o (d.thedevelopments onp,364)formanullsequence, bytheolem 6, Accordingly wealsohave n "2."__',..,_Jsin(2n+I)td ;11; In--In--In- Ie'-2" o Since,however (v.§lU,theorem 25), (2n+1)i i"=f~in!dt nt' o thisimplies thattheabove-named limithasthevaluei. 2)By1),aconstant K'existssuchthat '"If~i:IdtI<K' o foreveryx~0,andtherefore aconstant K(=2K')existssuchthat bIJSi;tdtI<K a foreverya,bsuchthat0Sa<b. 3)Suppose Bgiven>0andchoose apositive clS"1'so.that Irp(c5')-rpol<:8~' Writing /I' ~frp(t)sinektde=lh', o wethenhavelie-1h'tending to0ask-++00,bytheorem 6,andwe §49.Fourier series. -C.Conditions ofconvergence. 3G7 canaccordmgly choosek'solargethatIIk-Ik'I<iforeveryk>k'. Further, ~' ~, (d)JI=~f((t)--).sink/dt+!.fsink/dt=],"+IIII•,":ntp tpot JtfJJo t le le o 0 Forthesecond ofthesetwoquantities, wehave kd' ~ I",2fSint2fSintd k=-;'Po.e---;tpo'-i-t=tpo o 0 andwemayaccordingly choose ko>k'solargethat 11'"Iek-tpo<a- fareveryk>ko'ForIk",thefirstofthetwoquantities ontheright of(d),weusethesecond meanvaluetheorem oftheintegral calculus §19,theorem 27),whichgives,forasuitable non-negative b"<b', d' r It=~f[tp(t)-!Po]·sin/tdt=~[tp(b')-IJ'o]JSin/tdt. o d" Trol' fSin/Thelatterintegral=-/-dtandtherefore remains<Kinabsol· "d" utevalue,by2).Accordingly IJ"I<!.~--.K<!-k=:n:3l\ 3' Combining thethreeresultsofthisparagraph, bymeansof Ik=(]k-Ik')+Ik"+It', weseethat,givene>0,wecanchoosekosothat,foreveryk>ko' Ilk-IpoI<Ilk-Ik'l+lItI+Ilk'"-IpoIs::3·i=e. Thereby thestatement iscompletely established. 2.Diui'srule.11limIp(t)=Ipoexists,andifloreverypositive t-~+O 'l<c5,theintegrals dJI'I'(t)t-rpoIdt T whIchisimproper at0,d integralJIrp(tlt-rpoIdt, I)6,Moreshortly:Ifthe=CPo'remain lessthanafixedpositive number 54,thenHmlkexistsand k--+-I-00 hasamenning. 368 Chapter XI.Seriesotvariable terms. 3.L1pst:1tit:::'s rule. suchthat55Proof. When l:decreases to0,theaboveintegral increases mono· tonelybutremains bounded; Ittherefore tendstoadefinite hnlltas l:--.0,whichwedenote forbreVityby ,I fI'P(t)-'PoIdtt . u GivenE>0,wemaychoose apositive 1/<i5sosmallthat d' fI'P(t)t-'PoIdt<i. o Writing, asintheprevious proof, ,r Ik'=~fep(t)SintktdtandIk'=It-+Ik'". u thedifference(Ik-Ik')tendsto0,bytheorem 6,andwemaychoose k'solargethatIIk-Ik'l<i-foreveryk>k'.Further, aswesaw before,withasuitable choiceofku>k'wealsohave d' II~'"-CPoI=I~CPoJSlntktdt-CPuI<i- o foreveryk>ko'Fmally, d' d' IItI=I~f(cp(t)-CPo]·sintktdt1<fL'P(t)i'Po1dl, u 0 1.e.when0'issuitably chosen,IitIalsoremains<i.Thus,pre­ ciselyasbefore}weconclude that,foreveryk>ko, IIk-CPoI<E. whichprovesthevalidity ofDini'srule. Wemayeasilydeduce fromitthetwofollowing conditions. Ittwopositivenumbers Aand0:exist, IcP(t)-CPoI<A•la toreveryIin0<I<0,thenIk--.CPu. Proof. d d fl'P(t)-<PoIdt<Afta-1dt<A•i: t "' .. T bbThe"LJpschitz-condition", I'P(I)-'PoI<A.taast-0,itseltimplies thatlim'P(t)='Poexists. J...+u §4!J.Fouricr series.-C.Conditions ofconvergence. 369 sothatforeverypositive 'l:<<5theformerintegral remains lessthan afixednumber andinconsequence ofDint'sruleI"-+CPo,asrequired. 4thrule.Ifcp'(0)exists 56andtherefore limcp(t)='Po=cp(0) exists,thenlk-+CPo' 1++0 Proof. Theexistence of lim<p(t)-rp(0) 1-»-+0 t implies theboundedness ofthisratioinaninterval oftheform 0<t<(51'i.e.thefulfilment ofaLipschitz-condition witha=1. Hencelk-+To'asasscrted. Thefollowing corollary totheseconditions isimmediately ob­ tainco' Corollary. Ifcp(t)canbesplitHPintothewmottwoormore functions, eachofwhichsatisfies theconditions ofoneofthefourrules above,thenhmcp(t)=CPoagainexists,andtheDirichlet integralslkt-»-+O otthefunction rp(t)tendtoro' Theabovcrulesmayatoncebetransferred totheFvttrier series ofanintegrablc function(x),whichweassume fromthefirsttobe givenin0<x<2:reandtobeextended toallotherrealvaluesofx bytheequatIOn (x±2:re)=(x). InorderthattheFottrier scriesgenerated by(x)should converge toasums(xo)atthepomtxO'theintegrals ,I J-!f(t·)sin(2n+1)tdtn-:ntp,Xo t u must,bytheorem 9(205),formanullsequence, where,asbefore, 1tp(t;xu)="2[(xo+2t)+(xo-2t)]-s(xo). Thisformofthecriterion shows,overandaboveRiemann's theorem 204,thatneitherthebehaviour of(x)immediately totherightofxc' northatimmediately totheleftofxc'haveinthemselves anyinfluence whatever onthebehaviour oftheFourier seriesof(x)atxo'What isimportant isthatthebehaviour ot(x)totherightofXoshould standinacertainrelation tothatontheleft01xo'namely, suehthat thefunction 1cp(t)=rp(t;xo)="2[(xo+2t)+(xo-2t)]-s(xo) 56Itsufficesthat<p'(0)shouldexistasther;f(ht11l11lddifferential coefficient (v. §19,Def.HI),asInfactthepOSSIble valuesofcp(t)fort~0donotcomeIntoaccount. 370 Chapter XI.Seriesotvariable terms. shouldpossessthenecessary andsufficient properties 57fortheexistence ofthelimitofDirichlet's integralslk(206)relativetoqJ(t). Itisnotknownwhattheseproperties are.Thefourconditions givenabovefortheconvergence ofDirichlet's integrals furnish us, however, withthesamenumber ofsufficient conditions forthecon· vergence, ataspecialpointxO'oftheFourierseriesofafunction f(x). Eachoftheseconditions requires, inthefirstinstance, thatthefunction 1Ip(t)=tp(t;xo)="2[f(xo+2t)+f(xo-2t)]-s(xo) shouldtendtoalimittpo'Acommon assumption foralltherules whichweareabouttosetupisaccordingly thefollowing: Thelimit (g) limi[f(xo+2t)-f(xo-2t)J '....+0 mustexist.Thevalueofthislimit,bytheorem 2,willthenalsobe thesumoftheFourier seriesoff(x)atxO'ifthelatterconverges. Thi~convergence isensured ifthefunction tp(t)=tp(t;xo)=i[f(xo+2t)+f(xo-2t)]-s(xo)' considered asafunction oft,fulfilsoneofthefourconditions given above.Atthesametime,thevaluetpointhoseconditions must,by theorem 2a,beO.Weaccordingly assume thatthetwofollowing conditions aresatisfied: 207. ptassumption. Thefunction f(x)isdefinedandintegrable (hence bounded) intheinterval0<x<2nanditsdefinition isextended toall realvaluesofxbymeansoftherelation ((x)=f(x+2kn), k=±l, ±2,... 2",1assumption. Thelimit limi[f(xo+2t)+f(xo-2t)], '....+0 where Xodenotesanarbitrary realnumber,butiskeptfixedthroughout, exists 58, anditsvalueisdenotedbys(xo),sothatthefunction 1tp(t)=tp(t;xo)=2[f(xo+2t)+f(xo-2t)]-s(xo) hasarighthandlimitlimrp(t)=O. 1~+O Withthesejointassumptions, wehavethefollowing fourcriteria fortheconvergence oftheFourierseriesoff(x)atthepointxo: 67Definee.g.f(x)asentirelyarbitrary totherightofxo(butintegrable in aninterval ofthefonn Xo<x<Xo+Il)and,inXo-Il<x<XOIletf(x)= 1-f(2Xu-x)say.TheFourlerseriesoff(x)atXuisconvergent withthesum~. (Proof,forinstance, bymeansofDlrichlet's rule208,1below.) .sThetwo-sided limitthennecessarIly alsoeXIsts. §49.Fonrier series.-C.C:onllitions ofconvergence. 371 1.Diricltlet's rule.Ifep(t)ismonotone inaninterval ofthe208. form0<t<c51'theFourierseriesoff(x)converges atXoandt"tssum59 isequaltos(xo). 2.Dini's rule.Ifforafixed(otherwise arbitrary) positivenum· be'6theintegrals remainlessthanafixednumberforevery 'I:suchthat0<T<c5,the Fottrierseriesoff(x)converges atXoanditssumiss(xo)' 3.Lipscllitz's rule.Thesameistrue,ifinsteadofrequlrmg thattheintegrals shOltldbebounded, westipulate thattwopositive numbersAandIXshouldexist,suchthat,foreverytsuchthat0<t<c5, 4thrule.Thesameistrue,ifinsteadoftheLipsfhitz-condition werequirethatep(t)shouldpossessarighthanddifferential coeffi­ cientatO. Theapplication oftheserulesismadeconsiderably easierbythe following corollaries: Corollary 1.Thefunctionf(x)alsofulfilstheassumptions 1and2 anditsFourier seriesconverges atXotothesums(xo)'iff(x)can besplitupintothesumoftwooranyfixednumber offunctions, eachofwhichsatisfies thesetwoJointassumptions (forasuitables) andinsomeneighbourhood ofXofulfilstheconditions ofoneof theaboverules. Corollary 2.Similarly, itsufficestostipulate inplaceofassump­ tion2thateachofthetwo(onc-sided) limits limf(xo+2t)=f(xo+0) t.....+oandlimf(xo-2t)=f(xo-0) t.....+o shouldexist,andthatthetwofunctions epl(t)=f(xo+2t)-f(xo+0)andep2(t)=f(xo-2t)-f(xo-0) shouldeach,individually, satisfytheconditions ofoneofthefourrules. TheFourier seriesoff(x)isthenconvergent atXoandhasthesum 1s(xo)=2[f(xo+0)+f(xo-0)]. Oneortwospecialcases,which,however, areofparticular im­ portance inapplications, maybementioned inthefollowing further mrollaries: 69Incaseitconverges at:1:0,theFauncr seriesofafunction {(x)satis­ fyingtheassumptions 207accordingly hasthesumf(x~)if,andonlyif,the limits(Xo),whoseexistence isstipulated inthesecondassumption, =f(xo)' Similarly inthecaseofthefollowing rules. 372 Chapter XI.Seriesofvariable terms. Corollary 3.If(x)satisfies thenrstassumption andismonotone bothtotherightandtotheleftofxo'thelImitsmentioned inthe preceding corollary exist,andtheFourier seriesof(x)converges atXo tothesums(xo)={[f(xo+0)+{(xo-0)].-Hence, stillmore particularly: Corollary 4.TheFourier seriesofafunction(x)whichsatisfies the firstassumption willconverge atthepoint Xoanditssumwillbethe value(x)ofthefunction atthatpuint,if{(x)iscontinuous atXoand monotone oneithersideofxo' Corollary O.If{(x)satisfies thefirstassumption, andthetwo limits((xo±0)exist;if,further, boththe(one-sided) limits timf(xo+h)-f(xo+0) 11-..+0 handlimf(xo-h)-f(xo--0) lI~+O h exist;thentheFourier seriesof(x)willconverge atXoandwill 1havethesums(xo)=-2-[{(xo+0)+{(xo-0)J. Corollary 6.TheFourier seriesofafunction{(x)whichsatisfies the firstassumption willconverge, andwillhaveasitssumthevalueofthe function, atanypoint Xoatwhich(x)isdifferentiable. §50.Applications ofthetheory ofFourim" series. Asweseefromtherulesofconvergence developed above, extremely general classcs offunctions arerepresented bythclrFourier series. Thiswepropose toillustrate byanumber ofexamples. ThefunctIon {(x)tobeexpanded mustalwaysbegiveninthe interval 0<x<2:nandmustpossesstheperiod2:n:(x±2n)=(x). Thecorrespondll1g Fourier seriesisthen,ingeneral, obtallled inthe form 1 00"2ao+2)(ancosnx+bnsinnx). n~l Inparticular cases,thesine-orcosine-terms maybeabsent. Infact, if(x)isanevenfunction, (-x)={(2:n-x)=(x), (thegraphof(x)issymmetrical withrespect tothestraight lines x=k:n,(k=0,±1,±2,...),andtherefore 2n n2n n·bn=f(x)sinnxdx =f+f=0,o 0n asisevident ifwereplacexby2TT-xinthesecondofthesetwo partialintegrals. TheFourier seriesoff(x)thusreduces toapure §50.Applications ofthetheoryotFOllrier ~erie8. cosine-series. If,ontheotherhand,((x)isanoddfunction. f(-x)=f(2n-x)= -{(x),373 (thegraphof((x)issymmetrical w1threspect tothepointsx=k:rc, k=0,±1,±2,...).andtherefore 2:r n·a"=f{(x)cof>nxdx =0. o asisequally evident. ThusheretheFourier seriesof((x)reduces toapuresineseries. Thereareaccordingly threedifferent waysinwhichanarbitrary givenfunction F(x),whichisdefined andintegrable ina<x<b, maybeprepared forthegeneration ofaFourier serie<;. l,tmethod. Ifb-a~2n,aportIOn oflength 2nIScutoutof theinterval(a,b),saya<x<a+2n,andtheorigin 1Stran"ferred tothepointa;wethusobtainafunction {(x)defined in°<x<2n Itisthendefined forthewholex-axis 60bymeansofthecondition of periodicity /(x±27T)= /(x).Ifb--a<27T,define/(x)tobecon­ stant=F(b)inb:Sx<a+27Tandproceed asbefore61. 2ndmethod. Precisely asabove,defineafunction /(x)in0~x:c;:7T (not217)bymeansofF(x),put/(x)~/(217-x)in17<X:s217,and thendefine/(x)forallfurtherx'shythecondition ofperiodicity. 3rdmethod. Define/(x) asahovefor0<x<17,put/CO)=f(17)= 0,butputf(x)= -/(27T-x)in17<X<217;thenagaindefinelex) forallfurtherx'sbythecondition ofperiodicity. Thethreefunctions whichaIt:obtained bythesemethods from agivenfunction F(x),andwhicharenowsuitable forthegeneratIOn ofaFourier series,weshalldistmguish asf1(x),f'J(x),fa(x).Whereas f'J(x)WIllcertainly giveapurecosinesenesand({(x)apuresine­ series,f1(x)willlead,asarule,toaFourier serie<;ofthegeneral form(unless, infact,f1(x)isitselfalready anoddoranevenfunction). Sinceourrulesofconvergence enable ustorecognize thecon· vergence onlyatpoints Xoforwhich lim~[{(xo+2t)+f(xo-2t)] :--.+0 exists, itwillbeadvisable tomodify ourfunctions further atthe 6.Ifb-a>2TT,aportion ofthecurvey=F(x)isleftoutoftherepre­ sentation altogether. IfwewishtoaVOIdthiS,weneedonlyaltertheunitofmeasure­ mentonthex-axissothattheinterval ofdefinition ofF(x)hasthelength2TT; . b . b-a..I.e.wesustltutea+ -2-;-xLorx. 61Orelsegivetheintervalofdefinition ofF(x)theeX.lctlength2'TTbymodi­ fyingtheumtofmeasurement onthex-axis. 13 lG51) 374 ChapterXI. Serie~ofvariableterms. 209.junctions 2h17bywriting f(O)=f(2Il17) --.,Em~,[f(x)-[-f(217---x)] x<~I U~ whenever thislimitexists. (Thisiscertainly thecaseforfa(x),and provides theconditionfa(0)=fa(2k17)=0.)Ifthislimitdoes notexist,thefunctional valuef(2k17)doesnotcomeintoaccount, aswithourresoun.es wecannotdiscover whether thcFourierseriescon­ vergesthereornot.-Forcorresponding reasonswehavealreadyput fa(17)=0above. Wenowgoontoconcrete examples. 1.Example. F(x)==a+O.Here f1(x)==f~(x)==a,whilewchavetoput j0forx=0andx=n, fa(x)=~a"0<x<n, "n<x<2n. Dirichlet's conditions areevidently fulfilled ateverypoint(inc1usive of thejunctions), foreachofthethreefunctions. Theexpansions obtained mustaccordingly converge everywhere andmustrepresent thefunctions themselves. Forf1(x)andf'J(x),however, theyaretrIvial,asthey reducetotheconstant term ~ao=a.Forfa(x),however, weobtain: 21l J1 2:&: a b=~ffa(x)sinnxdx=~Jsin nxdx-!!-fsinnxdx =2aJsinnxdx,n1l 3t :re :if o 0 " 0 i.e. 10forevenvaluesofn, bn=4aforoddvaluesofn.nn Theexpansion accordingly is 4a[ . sin3x.sin5x]fa(x)=-:rr-SillX+-g-T-5-+... or+:in0<X<n, . + sin3x+sin;'x+ I)BlDX-3- -1)- •••= at0andatn, :It• 2-Tmn<x< n. Thisestablishes thesecond oftheexamples givenonp.351,and provides thesumofthiscuriousseries,ofwhoseconvergence wewere :re:ren .Forx=2'T'3'weobtainspecIal wearefanuliar IIIanentirely different§50.Applications ofthetheoryofFourier series. 375 already aware(v.IS5,5)62. series,withthefirstofwhich connection (v.122): 1 1 1 n 1 --3+-5---.or+-... =4"' 1 1 1 1 1 n 1-+-57-n+13+17- -++...=3' 1 1 1 1 n 1~~+-7--11-+13-+... =--.... 213 2.Example. F(x)~'ax, (aet0).Here {axIn0<X<2n,il(x)=anat0andat2n, {axinO<x<n,r.(x)= -- 2 a(2n-x)inn<x<2n, \axin0<x<"ll, fa(x)=0at0andat"ll, - a(2n-x)IDn<x<2"ll. Afteraneasycalculation, theexpansion offl(x)gives: (a). + sin2x+sin:1x+sin-lx+SIDX--:3- --S----4- ••• =\7t' 02xla'ntO<X< 2.71', oandat2.71', whichestablishes thefirstoftheexamples ofp.351.Similarly, the expansion off~(x)gives Ib)' sln2x+sin:JX sin-lx+,SlllX---:3-- --S-----ol- -•••!~Xin0:::::;X<.71', =0 at:r, 1in:r<X<2:r110. 2"X-.71'210. Or,moreshortly,rxin-TT<X<TT,- 2-10at±TT. 62Thisandthefollowing examples arealreadyfound,forthemostpart,in Euler'swntmgs. ManyothershavebeengivenbyFourier,Legendre, Cauchy, Frul/ani, Dirichlet andothers. Theyarecollected together, inaconvenient formforrefer­ ence,inH.RlIrkhardt, Trigonometnsche ReihenundIntegrale bisetwa1850, Enzyklopadie d.math.Wiss.,Vol.IIA,pp.902-920. 376 ChapterXI.Seriesofvariableterms. 211.Thefunction f2(x),however, provides theexpansion: (c)~+co83x_t-C085x+ _{~'-7t4Xino~x:<::;7t, l' 3' 5'...-nx3n'T-8""in7t;Sx:£27t. Thefirstoftheseexpansions givesforx=~theknownseriesfor~;the ·24 third,forx-=U,givestheseries,alsopreviously knowntous(137), I I I 17" 1+3'+52+'fi+...=8' fromwhichwemayimmediately deducetherelation I I I 17' 1+2'+3'+4'+...=-6' previously established (136,156and189)inanentirely different way83. -Oncomparing thetworesults,weobtaintheremarkable factthatin o<x<7Tthefunction xiscapableofthetwoFourierexpansions x={7T-2[~~~~+'in22x+sins3x+..-]and 17_~[c:os~+co~3x+cos5x+...J 2"P 3' 5' . Withaviewtopenetrating stillfurtherintothesignificance ofthese results,itiswelltosketchthegraphsofthefunctionf(x)andafewof thecorresponding curvesofapproximation. Thiswemustleavetothe reader,andweshallonlydrawattention tothefollowing phenomenon: Theconvergence oftheseries210eisuniform forallx's;notso thatoftheseries210aandb,sincetheirsumsarediscontinuous, the 83Afifthproof,quitedifferent again,isasfollows: TheexpansIOn 123is uniformly convergent in0:<::;x;;;;I,bythestipulations madein123,together with199,2.Puttmgx=smt,weseethattheexpansion oc (--~)sm,n+lt t=n::~-I)n n-2n+I isuniformly convergent in0~t:$~andmaytherefore beintegrated term-by­ termoverthatmterval. Now nil} f'-·W+ld2·4...(2n)_ I .smt t=---- - , o 3·5...(2n+I)(-W(2n+I)(-,~!) thisisshewnbyarecurrence process, orbywritingcost=zandusmgExample 117b.Henceatonce ?'"~=E__1__ 8n~o(2n+1)2' Thismethod wasessentially givenbyEuler.(C£.thenotereferred tointhefoot­ note38to156.) §50.Applications ofthetheoryofFourier series. 377 firstat0,thesecondat7T'.Intheformercase,theapproximation curves lieclosetothezigzaglinerepresenting thelimiting curvealongthe wholeofitslength,whilstin(a)and(b)thecorresponding stateofaffairs doesnotandcannotoccur(cf.216,4). 3.Example. F(x)=cosexX(exarbitrary 64,but=l=0,±1,±2,...). a)Wefirstformthefunctionf2(x),andaccordingly define {cosexx inO~x<7T' f2(x)=cosex(27T'-x)in7T'Sx~27T'; thusf2(x)isafunction continuous everywhere, whichbyDirich/et's rule willalsogenerate aFourierseriescontinuous everywhere, whichrepresents thefunction, andisnecessarily apurecosine-series. Herewehave " 7T'an=2fcosexXcos11Xdx-=f[cos(ex+n)x+cos(ex-n)x]dx; o u hence,asexwasassumed nottoheaninteger, (2exsmex1T7T'a= -1)"--~-~-.• n cx.2-n" Therefore thefunctionf2(x)in0~x<27T',orinotherwordsthefunction cosexxin-7T'~X~+7T',isrepresented bytheseries: sin(17t[12(1 2(1 ]COSIXX=-7t- ---'-12 C09X+-'--2'cos2x-+....212(1(1"- (1'-- • Forx=7T',weobtainfromthistheexpanSIOn 117,previously deduced fromentirely different sources: COS/X1T I2/X 20: 7T'----=7T'cotex7T'=-+--+-.-.-I-....sm/X1T /X/X2-}2 IX--2· Wethusenterthesphereofthedevelopments of§21.Ofcoursethe otherseriesexpansions therededuced mayalsobeobtained directlyfrom ournewsource. Thus212givesforx=0 1TI2" 2/X 2/X sin/X 1T=IX-~.---=-T'+~2~2'-;0----..::12+-.... Subtracting thecotangent expansion obtained justbefore,wefurtherobtain andsoon. b)Ifwenowsimilarly construct anoddfunctionfa(x)fromF(x)~ cosexx,wchaveIcosexXin0<x<7T', fa(x)=10at°andat7T', -cosIX(27T'-x)in7T'<X<27T'. 84Because otherwise thecosine-expansion wouldbecometrivial. 378 ChapterXI.Seriesofvariableterms. Here IXmayalsoassumeintegralvalueswithout reducing theresultto atrivialone.Thecoefficients bnareobtained fromintegrals whosevalue iseasilyworkedout,andtheyleadtothefollowing expansions, validin 0<x<17: 213. IX)forIX=l=0,±1,±2,••• 1+cosor."[2 . I-6•3+10.5+ ]coset.X."-' "1i~.~.smx-3'=,x'smx5'=-~2smx.•. l-COSor.,,[ 4. 8. ]+--,,-- 2'-='-~' SIO2x+4'-=~'sm4x+...; fJ)forIX=±p=~integer 14[1. 3.3 ] Of• ---smx+----SIO.X+...IPISeven"12_p' 3'_p' , ,cospx=4[2.24. A ]'f.dd...,.-,SIOX+-4"---~.sm'lex+...,1PIS0 •"..-p --p Fromalltheaboveseries,innumerable numerical seriesmaybe deduced bytakingparticular valuesofxandIX. 4.Thetreatment ofF(x)=sin(Xxleadstoquitesimilarexpansions. 5.IfthefunctionF(x)= -log(2sin-g)isarranged forthegenera­ tionofapurecosineseries,weobtaintheexpansion, valid.in0<x<17, cos2xcos3x ( .x)214. Cosx+-2-+-3-+...= -log2sm2 . Ithas,however, tobeshewnbyaspecialinvestigation thattheresult holdsinspiteofthefactthatthefunction isunbounded intheneigh­ bourhood ofthepoints0and217,andtherefore isnot(properly) inte­ grable.(Cf.§55,Vbelow,wherethiswillfollowquitesimplyinanother way.) 6.Example. F(x)=e'"+e-rx."(X=l=0,istobeexpanded ina cosineseries.Wehavetherefore totake j(x)={F(x)in0:-x~17 2F(217-x)III17::::::x<217. Afterworking outtheextremely easyintegrals givingthecoefficients am weobtain "e=+e-rx<1 a a2Ui. 2'erxn-e:xn=2a -a'+l'cosx+a'+2'cos2X-+...I whichisvalidin-17~X~+17.Ifwesubstitute e.g.x=17andwrite tfor2(X17forsimplicity, weareled,afterafewsimpletransformations, totherelation, -validforeveryt=P0 - -![--.!-_.-!.__lJ=2l'_----;1~- t1-e-tt2 n~1(211,,)2+t2• §50.Applications ofthetheoryofFourierseries. 379 i.e.toan"expansion inpartialfractions" ofthisremarkable function; itsexpansion inpowerserieswecanatoncededucefrom§24-,4,where tet-/-1thefunction -----wasconsidered, -forourfunction reduces to2et-1 thelatterbymultiplying byt2andadding1. Various remarks. Theveryfactthattrigonometrical seriesarecapableofrepresenting extremely generaltypesoffunctions renuers theque,tIon astotheItmitsofthIScapacity doublyinteresting. Aswasalready remarked, necessary andsujJiclent comhtlOns forafunctIOn toberepresentable byitsFouriersenesarenotknown. Onthecon­ trary,wefindourselves obligedtoconsider thiSasafundamental property offunctions, newofitskind,forallattempts tobUildituplhrectly bymeansoftheotherfun­ damental properties (continuity, differentlabIllty, integrabIlity, etc.)havesofar failed. Wemustdenyourselves thesatisfaction ofsupporting thisstatement In alldetailsbyworking outrelevant examples, butweshouldnevertheless hketo putforward afewofthefactsInthisconnectIOn. 1.Oneoftheconjectures whichwIllnaturally bemadeatfirstsightISthat216. allcontinuous functiolls arcrepre~entable bytheirFourier senes. Thisisnotthe case,asduBois-Reymond wasthefirsttoshowbyanexample (Gott.Nachr.1873, p.(71)··. - oIf~{---- OL-_-~-:f;:---";;:9':::7~-t-:!-.!f=----~;;!9"~-'-----'1o!::g;-----;;=~:-----:O:;;-;'--;;'--~ Fig-.9. 2.Ontheotherhand,toassumethefunction differentiable aswellascon­ tinuou~ ismorethanisnecessary, asisshownbyWeierstrass' 66example ofaUni­ formlyconvergent trigonometrical series,viz. n~tncos(bn11"x)(0<a<1,bapositive integer,ab>1+~11"), whichaccordingly istheFourierseriesofitssum(v.200,1a),butwhichrepresents afunctIOn thatiscontinuous butnowhere dIfferentiable. 65Wcnowhavesimplerexamples thanthatmentioned above.E.g.L.Feier hasgivenaveryclearandbeautiful example(J.f.d.reineu.angcw.Math.,Vol. 137,p.I.lUll9). 66Abhandlungen zurFunktionenlehre, Werke,Vo\.2,p.22:1.(Firstpubltshed 1875.) 380 ChapterXI.Seriesofvariableterms. 3.Whether continuous functions existwhoseFourier seriesareeverywhere divergent ISnotatpresent known. 4.Aspecially remarkable phenomenon isthatknownasGibbs'phenomenon 67, whichwasfirstdiscovered (by:l.W.G1,bbs)mconnectIOn withthesenes210a: ThecurvesofapproximatIOn y=sn(x)overshoot themark,sotospeak,mthe neighbourhood ofx=O.Moreprecisely, letusdenotebyentheabscissa ofthe greatest maximum 60ofy=sn(x)between 0and7Tandlet"Inbethecorresponding ordmate. Thengn->-0;but"Indoesnot->-~,aswcshouldexpect, buttendstoa value!? equalto;(1178!J8...).Thmitappears thatthelimitmg configuratIOn to whichthecurvesy=Sn(x)approximate contains, besidesthegraphofthefunction 210a(p.3.'il,fig.7),astretchofthey-axis,between theordinates±g,whose ') lengthexceeds the"jump" ofthefunction bynearly 1~1'Infig.9,thentbapproxi- mationcurve ISdrawnforn=9mtheinterval 0...7T,andforn=44theimtial portion isgiven. §51.Products withvariable terms. Givenaproduct oftheform ""lI(!+fn(x»,n=l whosetermsarefunctions ofx,weshalldefine(incomplete analogy with thetheoryofseries)asanintervalofconvergence oftheproduct, aninterval Jateverypointofwhichallthefunctions fn(x)aredefinedandtheproduct itselfisconvergent. Thuse.g.theproducts 00(X2) 00(X2) 00( X) 00(X)111-n2'111+n''111+(-l)n;:;,111+nlog2Tt"'.n=l n=1 n=l 11"----~ areconvergent foreveryrealx,andthesameistrueofanyproduct oftheform II(l+anx),If1:aniseitherabsolutely convergent (v.127,theorem 7)oracon­ ditionally convergent seriesforwhich1:an2converges absolutely (127,theorem 9). ForeveryxinJ,theproduct thenhasaspecificvalueandtherefore definesadeterminate function F(x)inJ.Weagainsay:theproduct represents thefunction F(x)in1,or:F(x)isexpanded inthegivenproduct in./.Themainquestion isasbefore: howfardothefundamental pro­ perties(ofcontinuity, differentiability, etc.)belonging tothetermsfn(x) stillholdforthefunction F(x)represented bytheproduct? Hereagainthe 67:1.W.Gibbs,Nature, Vo\.59(London 1898-91:), p.006.-Cf.alsoT.11. Grollwall, tlberdieGibbssche Erscheinung, Math.Annalen, Vcl.72,p.228,1912. 68Th.,h' I .".37T_57Th emaxIma Int emterva occuratx=n+i'n+-1'n+l'•••,t e fib· h .Th" 27T47Trstemgt egreatest maximum. eminIma occuratx=..n'-n-,.... §51.Products withvanableterms. ;~81 answerwillbeth.1tthisisthecaseinthewidestmeasure, aslongasthe products considered arcumJurmly convergent. Whatthedefinition ofuniform convergence ofaproduct istobe isalmostobvious ifwerefertothecorresponding defimtion forsenes, sinceineithercaseweareessentially concerned With ~equences offunctIOns (cf.190,4.).However, weshallsetdownthedefinitIOn correspondlllg tothe4thform(191,1) forseries: oDefinition 69.Theproduct11(1+fn(x))issaidtobeunifurmly217 cunverKent inanintervalf,1f,glVenE>0,asingle numberN=N(E) depending onlyon£,notonx,canbechosensothat 1(1+fn+!(x))(l-1-fn+2(x)).. ,(1-+-fn+1.(x))-11<E fureverytl>N,everyh2':1andevery 7Uxin;. ItisnotdifficulttoshowthatwithtIllSdefinItIOn asbasisthetheorcms of§17holdsubstantially forinfiniteproducts 71.WeWill,however, leave thedetailstothestudent, whileweproveafewtheorems whichareless far-reaching, butwhichwillamplysufficeforallourapplications, alllI whichhavetheadvantage ofproviding usatthesametimewithcriteria fortheuniformity oftheconvergence ofaproduct. Wefirsthave oTheorem 1.Theproduct11(1+in(x))converges ulllformly inJ218. andrepresents acuntinuous function inthatintenJal,ifthefunctions fn(x) areallcontinuous in;andtheseYlesLIfn(x)Ico/merges ulllformly inf. Proof.IfLIfn(x)Iconvergcs inj,sodoestheproduct11(11-fn(x)), by127,theorem 7;indeed,itconverges absolutely. LetF(x)denotethe function itrepresents. Letuschoose IIIsolargethat Ilm+1(x)1+IIml2(x)1+···-\-I!",+k(x)1<1 foreveryxinJandeveryh~1;thisispossible, byhypothesis. Consider theproduct 00Il(1+In(x)), n~m+l 6.Thesymbol 0inthissectionagamholdsonlywiththesamerestrictions as in§§46-48; cf.p.327,footnote 1. 70Thisdefimtlon mcludes thatofconvergence. IftheI.Jtterbeassumed, wemayspeakofthe"remainder" rn(x)=(1+1n+1(x))(l-I1"+2(x))...and defineuniform convergence asfollows:11(1+In(x))ISsaidtoconverge umformly III/,ifforevery(xn)in/,however chosen, rn(x)--+1. m 00 71Writing11(1+I"(x))=Pm(x)and11(1+I"(x))=F",(x),wemay "=1 "~m+-l quiteeasl1ydeduce e.g.thecontmUlty ofF(x)atXofromthatofthefunctions f"(x)there,bymeansoftherelatIOn F(x)-F(xo)~Pm(x).Fm(x)-Pm(xo) .Fm(xo) =[Pm(x)-P",(xo)]Fm(x)+[Fm(x)-Fm(xo)]Pm(xo). 13· (G51) 382 Chapter XI.Seriesofvariable terms. anddenoteitspartialproducts byP"(x),n>m.LetFm(x)bethefunction represented bythisproduct. Wchave(cf.190,4) .Fm=--Pm+1+(Pm+2--Prn~I)+...+(Pn-Pn-I)+... =Pm+l+Pmn'fm+2+PmH'fm+3+...+Pn-l'f"+...J i.e.Fno(x)isalsoexpressible byaninfiniteseries-asisindeedevident from§30.Now(by192,theorem 3)thisseriesconverges uniformly in.f. Infact,foreveryn>m,wehave Ip"I:S:(1+I/m+1I)'(1+ I/mI21)···(1+Ifnl)<e:Im+1l+1Im+21+...<e<3, andOCJ };Ifn(x)I n=m+1 isuniformly convergent inJ,byhypothesis. Accordingly thesequence ofitspartialsums,i.e.thesequence offunctions Pn(x),tendsuniformly OCJ toFminJ,sothattheproduct11(1+fn(x»isseentoconverge uniformly 1Zm+l in./,andthisproperty isnotaffected whenweprefixthefirstmfactors. By193,Fm(x)isnecessanly continuous in./,sincethetermsofthe serieswhichrepresents itareallcontinuous inthatinterval. Thesame isthentrueofthefunction F(x)=(1+fl(x»..•(1+fm(x»Fm(X), q.e.d. Asimilarproofholdsfor oTheorem 2.Ifthefunctions f"(x)arealldifferentiable in./and ifnotonly);Ifn(x)I,but};Ifn'(x)Iconverges uniformly in./,thenF(x) isalsodifferentiable in.f.Moreover itsdifferential coefficient ateverypoint ofJwhereF(x)'*0isgivenby72 ~(.x2=J;_fn'(x) F(x) n=11+In(x)' Proof. Theproofmaybeputinaformanalogous tothatofthe previous theorem; however, inordertomakeothermethods ofattack familiar, wewillconduct theproofbymeansofthelogarithmic function, asfollows. Letuschoosemsolargethat 1Ifm+1(x)I+IIm+2(x)I+...<2 72Ifg(x)isdifferentiable ataspecialpointxandg(x)'*0there,theratio ~(~)IScalledthelogarithmic differential coefficient ofg(x),becauseitisixlogIg(x)I. Forg(x)=gl(x).ga(x)•..gk(x),wehave,asiswellknown, g'(~_gl'(xl+g2'-iX)+ + gk~(::l g(x)-gl(x)g2(X)•••gdx), provided thatthefunctions gA(x)arealldifferentiable atthepointxinquestion. §51.Products withvariableterms. foreveryxinj,sothat,inparticular, foreveryn>m, 1li"(x)1<2' By127,theorem 8,theseries 00 };log(1-+i"(x)) n=",+1383 (m>0)isthenabsolutely convergent inj.Theseriesobtained fromitbydifferen­ tiatingtermbyterm, ISindeedalsouniformly (andabsolutely) convergent inj.Forsince 1 1Iin(x)I<2foreveryn>m,11+In(x)I>2andtherefore !l-f".k (x)I<2,sothattheuniform convergencc ofthelastseriesfollows fromthatof1,'II,,'(x)I.Accordingly (by196) Fm~(x)_E f,~(~L F;"(x)n=",+11+In(x)' if,asbeforc,weput 00 11(1-+In(x))=Fm' u=m+l I.e. 00 };log(l+in(x))=logFm(x). n=m1-1 Sincefinally F(x)=(1+11(x))...(1+Im(x)).Fm(x), andthelastfactorontherighthasbeenseentobedifferentiable in./, F(x)itselfisdifferentiablc in.f.If,further,F(x)=l=0,thelastrelation leadsatoncetotherequircd result,bytheruleofdifferentiation men­ tionedinthepreceding footnote. Applications. •1.Theproduct (JJt) Fm(x)-Il(1-~. n=m1-1 l isuniformly convergent ineverybounded interval, since,withI"(x)= _x••n- IX"I 1 .E1/"(:1:)1=.E[no=IxI"·.Eno isevidently auniformly convergent seriesinthatinterval. Theproductaccordingly definesafunction Fm(.x)continuous everywhere, which,inparticular, isnever zeroinIxI<m+1.Thisfunction isalsodifferentiable, for1:11,,'(x)I=21x11,'·~n219. 384 ChapterXl.Seriesofvariable terms. isunIformly convergent ineveryboundt"d Interval. HenceforIxI<m+1 !'m'(x)=~+E_2:l:_ Fm(x)x,,~m+lx"-n"' By117, thi~however implies ",hereGm(x)Fm'(x) F,~(x) denotes the1'Trcot'Tr.\:-x functIOnm2xE-"• 11=1X·-u..Gm'(x) -Gm(x)' 4x2 ) (2/~-1)2•sin'TrX--Ill ----.-, x11(1-~) 11=1 mterpreting thisexpres~ion asequal,forx=0,:I:I,•••,±rn,toitslimit(obvi­ ouslyexisten tand-1-0)asxtenustothese"alues. (Thecorresponding conven­ tIonismadeforthemiddletermIntherelation Immediately preceding.) Ifhowever twofunctions F(x)andG(x)havetheirlogarithmic derivatives equalinan1I1terval, In"hlchthetwofunctions nevervarush, Itfollowsthattheycanonlydifferbya (;onstant factor (=1=0).Hence, IIIIxI<m+I, 00( X2 )sin'Trx=c•x'111--.Jn-11=1 ",herecisasuitable constant. Todetermine itsvalue,weneedonlydividethe lastrelation byxandletx->-O.TheleftIl.lndsidethen-->-'Tr,whiletherighthand Side-->-c,becau~e theproduct IScontinuous atx=O.Accordingly c='Trandwe ha"e,firstforIxI<m+I,buthence,asmwasarbitrary, forallx, ~( X2 )sin."tx=1tx.II1 -n". 1l---1 Thisproduct, andthosediscussed below 1112and4.,aswellastheremarkahle product 257,9,andmanyotherfundamental expansions inproducts, areduetoEu/er. 2.Forcos'Trxwenowfind,without further calculatIOn, . 2 2 'TrX•1/(1-4~~) 'L SIll'Tr:1(; n11( COS1tx=-2-';IIl'Tr-~= 2'Trx'11(1-~;Y= k~l1 3.Thesine-product forspecial"aluesofxleadstoimportant numerIcal 1product expansIOns. E.g.forx=2' 1=TT.•11(I_!)='Tr.iI(~n-1)(2n+11). 2 4n"211=I 2 n.2n As¥.,,-±--1 --->-I,wemayclearlyomitthebrackets, amiweaccordmgly write2n 'Tr2·2·4..t·6·6·8·8 ... 2=1--;-:r:3-=-5· 5-:i-:7 .9-.-.• (Wallis' Product)". SmceItfollows fromthiSthat (2)"lim .k_+oo J or 2k 2h-~1.". 2 73Arithmetica infinitorum, Oxford 1656.(Cf.pp.218-9, footnote 1.) Exercises onChapter Xl. weobtainatthesametllnetheremarkable asymptotic relation 1.3.5...(2n-1)= ~n(2n)~(_l)n(-~)r-J~ 2.4.6...2n2n n-yn:n385 fortheratioofthemiddle coefficient inthebinomial expansion ofthe(21!)t~ powertothesum22nofallthecoefhclCnts ofthisexpansion orfortheco­ efficient ofxnintheexpansion of_I . VI-X 04.Thesequence offunctions gn(:r:)=x(x±!)(xn;:l~x+n) =-:"X(1+1)(1+~)...(1+~-), (v.12S -10)cannotbeimmediately replaced byaproduct oftheform1I(1+(n(x), as1I(1+-~)diverges forx1=O.However, thisdivergence ISofsucha kindthat By128,2 and42,3,thisimplies that x·(1+-f)(1+~)...(1+~)------n-"'=--- =gn(x) tends,asn-.00,toaspecific limit,finiteand1=0;-thelatter,ofcourse, onlyifx+0,-1,-2•...,Accordingly 1lI01-(a:) =r(a:)On isadefinite number foreveryx=\00,-I,-2I • •••Thefunction ofxsode­ finediscalledtheG(l1ttIlHt-fllrtctiolt (T-J'ltllction). Itwasintroduced into analysis byEuler(secabove) and,nexttotheelementary function", isone0f themostimportant inanalysis. Further investigation ofitsproperties liesout­ Sidethescopeofthisbook.(Cf.,however, pp.43!l-440 andp.5:m.) Exercises onChapter XI. I.Arbitrary series ofvariable terms. 11'»4.Let(nx)denotethedifference betwcen nxandtheinteger nearest tnX,orthevalue+frlifnxliesexactly inthemiddleoftheinterval belween twoconsecutive integers. Theseries2)in:)isuniformly convergent foralln 21'+1x's.Thefunction represented byit,however, isdiscontinuous forx=---­2q I (1',qintegers), whileitiscontinuous forallotherrational valuesofxand forallirrational valuesofx. 1~~.Ifan-.0, (sinnX)lI5"'a,.x-----~ Inx converges uniformly forallx'sDoesthisremain trueforan=t? 386 Exercises onChapter Xl. 1I'S8.Theproducts a)n(l+(-l)":), c)II(1+sin2:),xb)Hcos~,n converge umformly ineverybounded interval. X2R 1:>7.TheserieswhosepartIal sumshavethevaluess"(x)=1-c,+X"" converges foreveryx.Isthisconvergence uniform ineveryinterval? Draw thecurvesofapproximation. 1I'i8.AseriLs:::fn(x)ofcontinuous positIVe functions certainly convcrg-es uniformly ifitrepr('sents acontmuous function F(x).(Cr.p.344,Rem.3.) 159.Does2}-(1_x~__•converge umformly ineveryinterval? Isthen+nx) function itrepresents continuous? 180.IntheproofofIll,asituation ofthefollowing kindoccurred: Anexpression oftheform F(n)=au(n)+a,en)+...+ak(n)+...+apn(n) isconsidered, inwhich, foreveryfixedk,thetermak(n)lcnJstoalimit CXk asnIllcreases. Atthesametime,thenumber oftermsincreases, P.-00. Mayweinferthat limF(n)=:fCXb n~ClO I.=J provided theseriesontherightconverges? Showthatthisiscertainly per­ missible if,foreverykandeveryn, and converges. -Form'llate thecorresponding theorem forinfimte products. ­ (Cf.Exercise 15,wheresuchterm-by termpassages tothellluitwerenot allowed.' 181.Thetwoseries x8:z:4x5x7x":1:+3-"2+-5-+7-4++-++- ..• x3x2x5x7xC:1:+3-"2+-5+7-4++-++- ... 3arebothconvergent for°~x<1andhavethesamesum"2log2forx=+1. Whatistheirbehaviour whenx-I-o?-Examine thetwosenes, con· vergent forx>1, -121 1 2 1+V-p+1)i'"+-p -g:<++-' .. 1 1111 1+V--2x +""5Z+P--:P-++-··. forx-+1+0. ~[X"X·,,-1x'''J182.Theseries £.J-------n=1n2n-l 2n. isitssum?Isitsconvergence uniformiconverges in°<x~1What Exercises onChapter Xl. 163.Showthat,forx-+1+0 , 001 a)lim(x-I)n~--:nx=1 • b)lim[)'_1_-_1]=C(v.176,1).;;;1n"(x-1) 164.Showthat,forx-+1-0, 00(_1)"-1 x" I a))i;-n--.T+x"- -+~log2, n=l 00"1b)(l-x)..2;'(-I)"-I-x----+. log2 n=1 1 -xe" 2 • 00 nx" 1c)(I-x).)'(_1)"-1 -_+_--. n"";;/1 1_.xe,.4887 nx16:).Thescricswhosepartml sumshavethevalues sn(x)=---­1+n2xt maynotbemtegrated termbytcrmoveranmterval withendpoint O.Draw thecurves ofapproximatIOn. n.lOItY/er scTles. 166.Maywededuce fromthesenes210a, byIntcgration termbyterm: a).2_cos::~~=(x2-x+~).ni, n=l b)~!>in~Jr_n:14 =(~x'-x9+_1_x).Jr' n--;;t n3 3 3 . , c)i'cos~:n3_ =(-~xt+-:-x"--}Xi+9~).nt,etc.I n=l Inwhichintcrvals aretheserelations valid?(Cf.297.) 167.Inthesameway,deduce from210ctherelations a)~sin(~~-I):J:=nx( _) L.J(2n_I)38JlXIn=1 ~cos(2n-1)x_!!-(n)s2 2 sb)L.J--2--=-1)-C- -4tl2-x(ll+nX-x).n=l(n Whatwouldbethercsultsoffurther integratIOns? Inwhichintervals arethcse expansions vahd? 16S.From209,210,andtherelations inthetwopreceding exercises, dcduce thefollowmg- further expansions anddctermine theirexactintervals ofvalidity: cos3xcos5x n a)cosx--a-+-.')---+"·= ±"4' b)cosx_cos_3~+cos5~_+...=.3f_(~_xs) 3353 8 \ 4 • c)sinx-- ..si~~X+Si~~~_+...=n8x(~9_~B),etc. 169.From21:),deduce further expansions bysubstituting :If-zforz orbydifferentiating termbyterm.Isthelattcroperation allowcd? Whatare thenewseriessoobtained? 388 Chapter XII.Seriesotcomplex term~" 170.\Vhatarethesine-series andthecosine-series forea""Whatis thecomplete FOllnty expansion ofesin""Showthatthelatter i~oftheform ~ao+b.sinx--aJcos2x-b3sin3x+a4cos4x+b"sin5z--++... whereayandbyarepositivr-. 171.Ifxandyarepositive and<:If, 1[~.~.~•• 2' 172.Determine thevaluesofthemtegralsifx>y, itx=:Y, ifx<:Y. and (The2 fsinx--dxx u former=1'374!J8...,Tt J_Si~~dX' u thelatter=H~;,I9.••.) 173.Foreveryzandeveryn, .-, Isin2x sinnxIJsinxSl11x+---+",+---::::; -dx2 n-x' o wheretheboundontherighthandsidecannotbedill1in1~hed (cf.thepreceding exercise). (Further exercises onspeCIal Fourier serieswillbegivenintheneXI chapter.) Chapter XII. Seriesofcomplex terms. §52.Complex numbers andsequences. Afterwehavediscussed indetail,asinChapterI,themodesof formation ofalltheconcepts essential forbuilding upthesystemof realnumbers. nonewdifficulties areraisedbytheintroduction offurther typesofnumbers Sincethe(ordinary) complex numbers andtheir algebra areknown tothereader, wemayaccordingly becontent withbrieflymentioning oncortwomainpointshere. 220. 1.Itwasshownin§4thatthesystem ofrealnumbers isin- capable ofanyfurtherextension, andis,mOreOl"er, theonlysystem ofsymbols satl"fying thecondition~ whichwelaiddownforanumber system. Yetthesystemofcomplex numbers ISa.,ystemof"ymbols towhichthenameofnumb('rsystemisapplied. Thisapparent contra· §b2.Complex numbers andsequenct"s. 389 diction iseasilyremoved. Forourdefinition ofthenumber concept wasinacertain senseanarbitrary one,asweemphasized onp.12, footnote 16:Aseriesofproperties whichappealedtousessential inthe caseofrational numbers wasraisedtotherank01characteristic pro­ perties01numbers ingeneral, andthere;,ultJustified ourdoingthis, IIIsofaraswewereableactually toconstruct asystem -inallessen­ tials,asingleone,-whichpossessed alltheseproperties. Ifwede5iretoattribute toothersystems thecharacter ofasystem ofnumbers, wemusttherefore ofneces-,ity diminish thelistofchar­ acteristic properties whichwesetupin4-,1-4-.Thequestion ariseo­ whichoftheseproperties maybedi"pensed withfirstofall;i.e.which ofthemmaybeml"sing fromasystemofsymbols without itsbecoming Impossible tolegard thelatterasanumber system. 2Among theproperties 4ofasystem ofsymbols, thefirstwith whichwemaydispense, Without fearofthesystemlosingthecharacter ofanumber system entirely, arethelawsoforderandmonotony. Tl1<'s(,arebased,by4,1,onthefactthatoftwodlffercnt numbers ofthesystem, theonccanalways becalledlessthalltheother,and thelattergreater thantheformer.Ifwedropthisdistinction andin 4-replace boththesymbols<and>by9=,itappears thatthe mOLlIfied conditions 4-arcsatisfied byanother moregeneral system ofsymbols, namdy thesystem 01ordinary complex numbers, butthat noothersystem substantially different fromthelattercansatisfythem 3.Accordingly, thesystem of(ordinary) complex numbers isa system ofsymbols -which,asisknown, maybeassumed tobeofthe formx+yi,when~xandyarerealnumbers, andiisasymbolwhose manipulation isregulated bythesinglecondition i'J= -1,-for whichthefundamental lawsofarithmetIC 2remalll validwithout ex­ ceptIOn, provided thesymbols<and>aresuilably replaced throughout by9=.Inshort:Except forthelast-named restrictIon, wemaywork formally withcomplex numbers exactly aswithrealnumbers. 4.Inaknown manner (cLp.8),complex numhers maybe brought into(1,1)correspondence withthepointsofaplaneandmay thusberepresented bythese:withthecomplex number x--f-yiwe associate thepoint(x,y)ofanxy-plane. Everycalculation maythen beinterpreted geometrically. Insteadofrepresenting thenumberx-+-yi bythepoint(x,y),itisoftenmoreconvenient torepresent itbya directed line(vector) coincident inmagnitude anddirection withthe linefrom(0,0)to(x,y). 5.Complex numbers willbedenoted inthesequelbyasingle letter:z,C,a,b,...;andunlessIhecontrary isexpressly mentioned orfollows without ambiguity fromthecontext, suchletterswillin­ variably denotecomp7ex numbers. 390 Chapter XII.Seriesofcomplex terms. 6.Bytheabsolute value(ormodulus)IzIofthecomplex numbel x+yi,ismeantthenon-negative realvalueVx,J-+y2;byitsamplitude (amz,z=+=0),wemeantheanglerpforwhichbothcosrp=~1and sinrp=I:I'\Vhenwecalculate withabsolute values,therules3,Il, 1-4holdunchanged, while5.losesallmeaning. Sincewemaya(;cordingly operate, broadly speaking, inprecisely thesamewayswithcomplex aswithrealnumbers, byrarthegreater partofourprevious investigations maybecarriedoutinanentirely analogous manner intheredlmofcomplex numbers, ortransferred tothelatter,asthecasemaybe.TheonlyconsideratIons whichwill havetobeomitted orsuitably modified arethoseinwhichthenumbers themselves (notmerely theirabsolute values) areconnected bythe symbol<or>. Inordertoavoidrepetitions, whichthisparallel coursewould otherwise involve, wehaveprefixed thesign0toalldefinitions and theorems, fromChapterIIonwards, whichremainvalidwordforword whenarbitrary realnumbers arereplaced bycomplex numbers, (this validity extending equally totheproofs, withafewsmallalterations whichwillbeexplained immediately). Weneedonlyglancerapidlyover thewholeofourpreceding developments andindIcate ateachplace whatmodification isrequired whenwetransfer themtotherealmof complex numbers. Afewwordswillalsobesaidonthesubjectof thesomewhat different geometrical representation. Definition 23remains unaltered Asequence ofnumbers willnow berepresented byasequence ofpoints(eachcounted onceormore thanonce)intheplane.Ifitisbounded (24,1),noneofitspoints lieoutside acizcleof(suitably chosen) radiusKwithoriginatO. Definition 2~,thatofanullsequence, andthetheorems 26, 27,and28relating tosuchnullsequences remain entirely unaltered. Thesequences (zn)wlth (n=1,2,3,...) anddivergence of remain unaltered,areexamples ofnullsequences whosetermsarenotallreal.Thestudent shouldformanexactideaoftheposition ofthecorresponding setsofpoints andprovethatthesequences areactually nullsequences. Thedefinitions in§7ofroots,ofpowersinthegeneral ~ense,and oflogarithms wereessentially basedonthelawsoforderforreal numbers. Theycannot, therefore, betransferred totherealmof complex numbers inthatform(d.§55below). Thefundamental notions oftheconvergence asequence ofnumbers (39and40,1)still §52.Complex numbers andsequences. 391 although therepresentation ofZn-+,nowbecomes thefollowing 1:If acircleofarbitrary (positive) radius €isdescribed aboutthepoint,as centre,wecanalwaysassigna(positive) number nosuchthatallterms ofthesequence (zn)withindexn>noliewithinthegivencircle.The remark39,6(1sthalf)therefore holdswordforword,provided wein­ terpretthe..-neighbourhood ofacomplexnumber,asbeingthecirclementioned above. Insettingupthedefinitions 40,2,3, thesymbols<and> playedanessential part;theycannot, therefore, beretained unaltered. Andalthough itwouldnotbedifficult totransfer theirmaincontent tothecomplex realm,wewilldropthementirely, andaccordingly inthecomplex realmwcshallcalleverynon-convergent sequence divergent 2, Theorems 41,1to12,andtheimportant groupoftheorems 43, withtheexception oftheorem 3,remain wordforwordthesame, together withalltheproofs. Themostimportant ofthesetheorems weretheCauchy-Toeplitz limit-theorems 43,4and5,andsincewehaveinthemeantime gained complete familiarity withinfiniteseries,weshallformulate themonce moreinthisplace,withtheextension indicated in44,10,andfor complex numbers. Theorem 1.Thecoefficients ofthematrix 221. (A)aoo'a01'a02'...,aon'... a10,all'(/12'...,a1n, a20,a21,a22,...,a2n, areassumed tosatisfythetwoconditions: (a)thetermsfneachcolumn formanullsequence, i.e./orevery fixedn:?0, akn-O ask-oo. 1Forcomplex numbers andsequences, wepreferably useinthesequel theletters B,C,Z,.... 'Wemightsay,inthecaseIBnl-'+oo, that(Bn)isdefinitely divergent withthelimit00,ortendsordiverges (orevenconverges) to00. Thatwouldbequiteaconsistent definition, suchasisindeedconstantly made inthetheoryoffunctions. However, itevidently involves asmallinconsist­ encyrelative totheuseofthetermsintherealdomain, thate.g.thesequence ofnumbers(-1)nnshouldbecalleddefinitely orindefinitely convergent, accordmg asitisconsidered inthecomplex orintherealdomain. Andeven though, withalittleattention, thismaynotgiveusanytrouble, weprefer toavoidthedefinition here. 392 Chapter XII.Seriesofcomplex terms. (b)thereexistsaconstant J(suchthatthesumoftheabsolute valuesofanymemberoftermsinanyone rowremains lessthanK, i.e.,foreveryfixedk:20,andanyn: IakOI+IakII+...+IaknI<K. Undertheseconditions, when (ZIPZl'•••)isanynullsequence, the numbers Zk'=akOZo+aklZI+...-.f;ak..z..,,=0 alsoformanullsequence 8. Theorem 2.Thecoelliciellts ale..ofthematrix(A),besidessatis­ fyingthetwuconditions (a)and(b),areassumed tosatisfythefurther condition 3 (c)'"};{lk"c=Ale--+1ask-~00. n-O Inthiscase,if%..--C,wehavealso z,:=akO%0+akl%1+...==.i;aknZ..-t;. ..=0 (Forapplicatlons ofthistheorem, seemoreespecially 233,aswell as§§60,62and63.) Unfortunately, welosethefirstofthetwomaincriteriaof§9, whichwas the moreusefulofthetwo.Moreover, theproofofthe secondmaincriterion cannotbetransferred tothecaseofcomplex numbers, asitmakesuseoftheorems oforderthroughout. Inspite ofthis,weshallatonceseethatthesecondmaincriterion ttself­ inallitsforms-remains valtdforcomplex numbers. Theproof maybeconducted intwodifferent ways:eitherwereducethenew (complex) theorem totheold(real)one,orweconstruct freshfounda­ tionsfortheproofofthenewtheorem, byextending thedevelop­ mentsof§10tocomplex numbers. Bothwaysareequallysimple andmaybeindicated briefly: 1.Thereduction ofcomplex sequences torealsequences ismost easilyaccomplished bysplitting upthetermsintotheirrealand imaginary parts.Ifwewritez..=xn+iY..andC=~+i'Yj,wehave thefollowing theorem, whichcompletdy reduces thequestion ofthe convergence ordivergence ofcomplex sequences tothecorresponding realproblem: 222. Theorem 1,Thesequencetz..)=(x..+iy..)converges tol;=~Ti'YJ if.andonlyif,therealpartsx..converge to~antItheimaginary parts y..converge to'Yj. 3Inconsequence of(b),Ale=f,aknisabsolutely convergent andthere­ fore.asthezn'sarebounded, by41,Theorem 2,thesenes1:ak"z..=zk'ISalso absolutely convergent. n §52.CompleI numbers andsequences. 393 Proof. a)Ifx"-~andy"-1J,(x"-~)and(y"-1J)arenull sequences, By26,1,thesameistrueofi(y"-1J)and,by28,1,of (x,,-~)+i(Y,,-1J), i.e.of(z,,-C). b)Ifz"-c,Iz"-CIisanullsequence; since 4 Ix"-~I~Iz"-CIandIy"-1JI:::::;:Iz"-Cl, (x"-~)and(y"-1J)arealsonullsequences, by26,2,i.e.wehave both x,,_~ andy.,_fj. Thetheorem isestablished. Thetheorem atwhichweareaiming follows immediately: Theorem 2.Fortheconvergence otacomplex sequence (Z.,),the conditions otthesecondmaincriterion47areagainnecessary and sutttcient, -namely, that,toreverychoiceote>0,we~houldbeable toassignnosothat loreveryn>noandeveryn'>no' Proof. a)If(z,,)converges, sodo(x.,)and(y.,)bythepreceJing theorem Asthesearerealsequences, wemayapply47,and, givene>0,wemaychoose nlandn2sothat Ix"'-x"I<~-foreveryn>n1andeveryn'>nl' and Iy,,'-y"I<;foreveryn>1t2andeveryn'>1t2, Taking nogreater thannlandn2,wchaveaccordingly, foreveryn>no andeveryn'>no' IZ,,'-Z.,I=I(x...-X.,)+i(y",-y,,)I~Ix..'-x..I+Iy..'-Y..I ee<"2-+2=e. Theconditions ofourtheorem aretherefore necessary. b)If,conversely, (z..)fulfilstheconditions ofthetheorem, ­ i,e.given8>0,wecandetermine nosothatIz..,-z"I<e,provided onlythatnandn'areboth>no'-wehavealso,forthesamen andn'(byourlastfootnote) Ixn'-x.,I<8andIy,,'-y.,1<8. 6Wehaveingeneral sinceIffi(z)I~IzIand13(z)I<,./ Z~} IZI}- )'1<x'+,,'Ii.e.I)'I<vZI+,,'=,.i. 394: Chapter XII.Seriesofcomplex terms. By47,thisimplies that(x,,)and(y,,)areconvergent, sothat(z,,)must alsoconverge, bythepreceding theorem; theconditions ofourtheorem aretherefore alsosufficient. 2.Directtreatment ofcomplex sequences. Inthetreatment ofreal sequences, nestsofintervals constituted ourmostfrequent resource. Inthecomplex domain, nestsofsquares willrenderusthesame services: 223. Definition. LetQo'Q1'Q2'.,.denotesquares, whosesideswill forsimplicity beassumed parallel tothecoordinate-axes. Ifeachsquare isentirely contained inthepreceding andifthelengthslo'l1'...of thesidesformanullsequence, weshallsaythatthesquares form anest. Fornestsofsquares, wehavethe Theorem. Thereexistsoneandonlyonepointbelonging toallthe squaresofagivennestofsquares. (Principle oftheinnermost point.) Proof. LetthelefthandbottomcornerofQ"bedenoted by a"+ia~andtherighthanduppercornerbyb"+ib~.Apoint z=x+iybelongs tothesquareQ"if,andonlyif6, a,,'::::;::x<b"and a~<y<b~. Now,inconsequence ofourhypotheses, theintervals In=an".b"on thex-axis,andsimilarly theintervals I~=a~i...b~ionthey-axis, formanestofintervals. Thereistherefore exactlyonepoint ~on thex-axisandexactlyonepointi'YJonthey.axisbelonging toallthe intervals ofthecorresponding nest.Butthismeansthatthereisalso exactlyonepointC=~+ir;,belonging toallthesquaresQ". Wearenowinaposition totransferdefinition 52andtheorem54 tothecomplex domain: 224. Definition. It(z,,)isanarbitrary sequence, t;issaidtobea lill/iting pointorpointofaccumulation ofthesequence if,givenan arbitrary e>0,therelation Iz"-t;1<B issatisfied foraninfinity ofvaluesofn(inparticular, foratleast onen>anygivenno)' 225. Theorem. Everybounded sequence possesses atleastonelimiting point.(Bolzano- Weiersfrass Theorem.) Proof. SupposeIz"1<Kanddrawthesquare Qowhosesides lieontheparallels totheaxesthrough±Kand±iK.Allthez,,'s AThisstatement atthesametimeexpresses, inpurearithmetical lang· uage,therelations ofmagnitude framed ingeometrical forminthetheorem anddefinition ~~3. §52.Complex numbers andsequences. 395 arecontained init,i.e.certainly aninfinity ofz,,'s.Qoisdivided by theCOOldinateaxesintofourequalsquares Oneatleastofthefour mustcontain aninfilllty ofz,,'s.(Infdct,iftherewereonlya finitenumber ineach,therewould al~obeonlyafinitenumber inQu'whichisnotthecase) LetQldenote the fir~tquarter'\ whichhasthisproperty. Thisweagainproceed todiVideintofourequal squares, denollng byQ'Jthefirstquarter whichcontains aninfinityof pointsz",andsoon.Thesequence Qo'Q1'Q2'•..formsanestat squares, slllceeachQ"lieswithintheprecedmg andthelengths ofthe sldesformanullsequence, namely(2K·;;;).Lett;denote the innermost pointofthisnest7;?;isapointofaccumulation of(z,,). ForIfeisgiven>0andmischosen sothatthesideofQmisless than~~,thewholeofthesquareQmlieswithinthee-neighbourhood of1;,and,withit,aninfinite number ofpointsz"alsolieinthis neighbourhood. Therefore l;isapointofaccumulation of(z,,),and theexistence ofsuchapointisestablished. Thevalidity ofthesecondmaincriterion forthecomplex domain, -i.e.ofthetheorem 222,2, formulated above-maynowbe e~tablished oncemore,butwithout anyappeal tothe"real"theorems, onthesamelinesasin47. Proof. a)Ifz"-.1;,i.e.(z"-t;)isanullsequence, wecan determine nosothat Iz"-t;I<~-andIz;-1;I<-~- provided onlythatnandn'aresimultaneously >no[seeparta)of theproofof47].Forthesen'sandn"s,wetherefore alsohave Iz"-zn'/<Iz;-t;I+Izn-t;I<8. Thecondition isaccordingly necessary. b)If,conversely, thee-condition isfulfilled,(z,,)iscertainly bounded. Infact,ifm>noandn>m, Izn-zm1<8, i.e.everyz"withn>mliesinthecircleofradiuseround Taking KtobelargerthanallthemnumbersIzll,Iz21, :z..._ll,IZml+e, wehaveIz"I<K foreveryn...-, 6Weregardthefourquarters asnumbered intheorderinwhichthe fourquadrants ofthexy-planearehabitually taken. •Theprocess ofobtaining thispointcorresponds exactly tothemethod. 0/successive bisection sooftenapplied intherealdomain. 396 Chapter XII.Seriesofcomplex tE'rms~' Byourprecedmg theorem, itfollows limitmg pointC.Supposing thereeXIsts C'+1;,choosethat(zn)lasatleastone asecondli7gpoil41 whichisposItive. By224,thedefinition oflimiting point,wecan choose noaslargeaswepleaseandyethaveann>noforwhich Iz"-?;1<eandalsoann'>noforwhichIz,,'_?;'I<e.Thus aboveanynumber no'however large,thereexistapairofindicesn andn'forwhich 8 IZ,,'-znI>e, This contrad~cts ourhypothesIs. Accordingly?; mustbetheunique limiting point,andoutside theCIrcleofradiUS 13round ?;thereis onlyafinitenumber ofPOllltsz,,'IfnoissUItably chosen, wethere­ forehaveIz"-1;I<eforeveryn>no'andconsequently z"-.1;. Thecondition ofthetheorem istherefore sufficient also9. §53.Seriesofcomplex terms. AsaseriesZa"ofcomplex termsmustobviously beinterpreted asthesequence ofitspartialsums,thebasisfortheextensIon of ourtheoryofinfiniteserieshasalreadybeenprovided bytheabove. Corresponding to222,1,wehavefirstthe 226. Theorem. AseriesL:a"ofcomplex termsisconvergent if,and onlyzt,theseriesZm(a,,)attherealpartsatitstermsandtheseries 2,'S(at.)ottheirimaginary partsconverge separately. Fztrther, itthese twoserieshavethesumss'andS"respectively, thesumat.2'anis S=S'+is". Inaccordance with222,2 thesecondprincipal criterion (SI)for theconvergence ofinfiniteseriesremain'> unaltered inallitsforms, and,atthesametime,thetheorems S3deduced fromit,onthealgebra ofconvergent series,alsoretaintheirfullvalidity. Since,inthesameway,theoremS5alsoremains unchanged, weshall,asbefore,distinguish between absolute andnon-absolute con vergence ofseriesofcomplex terms(Def.S6). • Z,,'-ZII=(C'-C)+(ZII'-C')+(C-ZII), hence IZII'-ZnI;:;;;IC'-Cl-IZII'-C'I-IZII-Cl>3E-E-11=11. 9Hencewemayalsosay:(ZII)converges if,andonlyif,itisbounded andpossesses onlyonepointofaccumulation. Thisisthenatthesametime' tbelimitofthesequence. §53.Seriesofcomplc% terms. 397 Hereagainwehavethe Theorem. TheseriesXa"ofcomplex termsisabsolutely con·227. vergentif,andonlyif,boththeseriesXgt(a,,)andX3(a,,)areab­ solutely convergent. Theproofresultssimplyfromthefactthateverycomplex number 2=X+iysatisfies theinequalities (cf.p.393,footnote 4) :;:}~lzl~lxl+IYI. Inconsequence ofthissimpletheorem, itisatonceclearthat, withseriesofcomplex termsaswithrealseries,theorderoftheterms isimmaterial iftheseriesconverges absolutely (Theorem SS,1). If,however, Xanisnotabsolutely convergent, eitherxm(a,.)or X3(a")mustbeconditionally convergent. Byasuitable rearrangement oftheterms,theconvergence oftheseriesXanmaytherefore bedes­ troyedinanycase,asintheproofoftheorem S9,2,thatis:Inthe caseofseriesofcomplex termsalso,theconvergence, whenitisnot absolute, depends essentially ontheorderofsuccession oftheterms. (Regarding theextension toseriesofcomplex termsofRiemann's rearrangement theorem§44,cf.theremarks onthefollowing page.) Thenexttheorems, S9,3and4,asalsothemainrearrangement theorem90,whichrelatetoabsolutely convergent series,stillremain valid,without modification oraddition, forseriesofcomplex terms. Sincethedetermination oftheabsolute convergence ofaseries isaquestion relating toseriesofpositive terms,thewholetheoryof seriesofpositive termsisagainenlisted forthestudyofseriesof complex terms:Everything thatwasprovedforabsolutely convergent seriesofrealtermsmaybeutilized forabsolutely convergent series ofcomplex terms Ifweomitpowerseriesfromconsideration forthepresent, we observe, onlooking overthelatersections ofPartII(§§18-27), that thedevelopments ofChapter Xarethefirstforwhichthereisany question oftransference toseriesofcomplex terms. Abel'spartialsummation IS2,beingofapurelyformalnature, anditscorollaryIS3,ofcourseholdalsoforcomplex numbers, and sodoestheconvergence-test IS4whichwasbaseddirectlyonthem. Thespecialformsofthistestmayalsoallberetained, provided we keeptotheconvention agreedonin220,5,inaccordance withwhich allsequences assumed tobemonotone arereal.InthecaseofduBois­ Reymond's andDedekilld's tests,eventhisprecaution becomes unnecessary: theyholdwordforwordandwithout anyrestriction forarbitrary series oftheform1:anb",withcomplex anandbn. Riemann's rearrangement theorem (§44)is,onthecontrary, essen- 398 Chapter XII.Seriesofcomplex terms. tiallya"real"theorem. Infact,ifaseries~anofcomplex termsis notabsolutely convergent, soisoneatleastofthetwoseries~m(an) and~~(an)'by227.Byasuitable rearrangement, wecantherefore, inaccordance withRiemann's theorem, produce inoneofthesetwo seriesaprescribed typeofconvergence ordivergence. Buttheother oneofthetwoserieswillberearranged inprecisely thesamemanner, andthereisnoimmediate meansofforeseeing whattheeffectofthe rearrangement onthis5eriesoron2'anitselfwillbe.-Ithasrecently beenshown,however, thatif~:anisnotabsolutely convergent, itmay betransformed byasuitable rearrangement intoaseries,againcon­ vergent, whosesummaybeprescribed tohaveeitheranyvaluein thewholecomplex planeoranyvalueonaparticular straight linein thisplane,according tothecircumstances ofthecase10. Thetheorems 188and189ofMertens andAbelonmulti· plication ofseries(§45)againremainvalidwordforword,together withtheproofs.Forthesecondofthesetheorems wemust,itistrue, relyonthesecondproof(Cesaro's) alone,aswehaveprovisionally skipped theconsideration ofpowerseries(cf.later232). Atthispointweareinpossession ofthewholemachinery required forthemastery ofseriesofcomplex termsandwecanat onceproceed tothemostimportant ofitsapplications. Beforedoingso,however, weshallfirstdeduce thefollowing extremely far-reaching criterion. OX> 228. Weierstrass' criterion 11.Aseries2)an0/complexterms,forwhich,,=0 an±!.-=1 _~_An a,. nni. withAnbounded, -whereaiscomplexandarbitrary, and12'\>1,- 10Wethushavethefollowing veryeleganttheorem, whichinacertainsense completes thesolution oftherearrangement problem: The"rangeofsummation" ofaseries1:a"ofcomplex terms-i.e.thesetofvalueswhichmaybeobtained assumsofconvergent rearrangements of1:a"-iseitheradefinite point,ora definite straight line,ortheentireplane.Othercasescannotoccur.Aproofis givenbyP.Levy(Nouv.Annales (4),Vol.6,p.506,WO';),butanunexceptIOn­ ablestatement oftheproofisnotfoundearlierthaninE.Steinitz (Bedmgt kon­ vergente Reihenundkonvexe Systeme,J.f.d.reineu.angew.Math.,Vol.14:J. 1913;Vol.144,1914;Vol.146,1915). Forthe(morerestricted) resultthateveryconditionally convergent series Ea..=scanberearranged togiveanother convergent seriesEa,,'=s'withs'*'s, W.Threlfall hasgivenafairlyshortproof(Bedingt konvergente Reihen, Math. Zschr.,Vol.24,p.212,1926). 11J.f.d.reineu.angew.Math.,Vol.51,p.29,1856; WerkeI,p.186. 12Anequality ofthiskindmayofcoursealwaysbeassumed; weneedonly . A '(1 IXant.l) dfi. "Wh' .I'hd" Writen=n~- - -----asa emtlOn. atIScssentla Int econItlOn 71an ishere,aspreviously (cf.footnote to166),thatwhen IXand,\aresuitably chosen theA,,'sshouldbebounded. -Itissubstantially thesamethingtoassumethat 4,,/an+1-1+rt./n+BTj/nAWith,\>1andB"bounded. §53.Seriesofcomplex terms. 399 isabsolutely convergentit,andonlyit,m(a)>1.Forffi(a)<0the seriesisinvariably divergent. 1/0<m(0:)<1,boththeseries 1:l(an-an+1)1andi'(-1)"a" n=O n=O areconvergent 18. Proof. 1.Leta=fJ+ir andletusfirstassumefJ=ffi(a)>1. Inthatcase,ifIAnI<K,say,wewrite,asispermissible, Ian+1_1~11_fJ+i'YI+~. a"- n n}.' anditfollowsatoncethat,iffJ'isanynumber suchthat1<fJ'<fJ, IanHI~1-.£.. an1- n foreverysufficiently largen.ByRaabe's test,theseries ~Ianiis therefore convergent. 2.Nowsupposem(a)=fJ<1.Inthatcase,since I~..:t--'-I;;;:::1 --~-~a"- nn}. forsufficiently largevaluesofn,itfollows fromGauss's test172that 2,'IanIisdivergent. 3a.If,ontheotherhand,m(ex)=f3<0,ourlastinequality showsthat then Therefore ~anmustnowdiverge. 3b.1£m(a)=fJ=O, i.e. Ian+1I=1 _~_An an nn.l itISeasytoverifythatwethenhave '~1=1- A~a" n}. whereA'>1andisthesmaller ofthetwonumbers 2andA,and theA~'sareagainbounded. Accordingly, ifcdenotes asuitable constant, Ia"+l1;;;:::1---.:.->0an- n}. 13AsregardstheseriesEanitself,itwasshownbyWeierstrass, I.e.,thatthisis alsodivergent whenever !Jl(oc)~1.Theproofissomewhat troublesome. - A furthermoreexaetinvestigation oftheseriesEanitselfintheease0;5Ht(oc)~I isgivenbyA.Prin/!sheim (Arehiv d.Math.undPhys.(3),Vo!.4,pp.I-lP,in particular pp.13-17. 1902),J.A.Gmeiner, Monatshefte f.Math.u.Phys.,Vol. 19,pp.149-103. 1908. 400 Chapter XII.Seriesofcomplex terms. foreveryn~m,say.Itfollows bymultiplication that I.!!!!.-I=I!!-"'+1/..'1-~I>Y/(1--;)>Jf(1-"c"')=Cm>O.am 4,,, Qn-l J'=m " t'=m HenceIanI>Cm'1amI,foreveryn>m,andancannottendto0, sothat.L:allagaindIverges (cf.170,1). 4.If,finally, ffi(a)={J>0,wehavetoshowthatboththe series areconvergent. Nowasin1.wehave,foreverysufficiently largen, Ian+l,<1-£, an nwith0<{J'<(J, sothatIanIdiminishes monotoncly fromsomestageon,andthere foretendstoadefinite limit~O.Accordingly, a)theseries.L:(IanI- Ian+1I)isconvergent, by131,andhas, moreover, allitstermspositive forsufficiently largen's.Now Ian-aq1I _11-~::lI<i:+:~I. lanl-Ia,~- 1-1a~~ll=p'-' all n sincethefraction ontherighthandsidetendstothepositive limit';.I whenn-+oo, thatontheleftis,foreverysufficiently largen,less thanasuitable constantA.By70,2,thismeans that2,'Ian-a"+1I converges withI(IanI- Ian+1I).-Wecanshowmoreprecisely, how­ ever,that b)an-O.Foritagainfollows, bymultiplication, from that(n~m) I~I<(1-£)(1-J~__)...(1-1_).am m m+1 n-l Therighthandside(by126,2) tendsto0asn-+00,hence (cf.170,1)wemusthavean_O.Nowtheseries co (ao-a1)+(aJ-as)+(a4-all)+...==2(a~"-a2H1) k=O isasubseriesof.L:(an-a,,+l)andtherefore converges absolutely, bya)jalso,sinceIanI+Iall+11-0withan'wemayomitthe brackets, byS3,supplement totheorem 2.Thisproves thecon­ vergence ofI(-1)"a... Thistheorem enables ustodeduce easilythefollowing further theorem, whichwillbeofusetousshortly: §54.Powerseries.Analytic functions. Theorem. 11,asinthepreceding theorem,401 229. an+1_1 Cl:An{ce:arbitrary, A.>1, -a--;:-- -n-tIT (A,,)bounded, theseries:Ea"z"isabsolutely convergent lorIzI<1,divergent lorevery IzI>1,andtorthepointsofthecircumferenceIzI=1,theserieswill a)converge absolutely, ifm(ce:)>1, b)convergeconditionally, il0<m(ce:)':::;:1,exceptpossibly 14forthe singlepointz=+1• c)diverge,ifm(ce:).:::;:O. Proof. Sincelan::;:+l!_I:I, thestatements relative toIzI:e:1areimmediately verified. For IzI=1,thestatement a)isanimmedIate consequence ofthecon­ vergence of~IanIensured bythepreceding theorem. Similarly c) isanimmediate consequence .ofthefactestablished above,thatinthis caseIallIremains greaterthanacertainpositive number forevery sufficiently largen. Finally, if0<m(ce:).:::;:1andz=+=+1,theconvergence of2'a"z" followsfromDedekind's testIS4,3.Forweprovedinthepreceding theorem that:EIan-an-r1Iconverges anda"--0;thatthepartial sumsof2'z"arebouncled, forevery(fixed)z++1onthecircum· ferenceIz1=1,followssimplyfromthefactthatforeveryn §54.Powerseries.Analytic functions. Theterm"powerseries"isagalDusedheretodenoteaserie~ oftheform2'anz",or,moregenerally, oftheform:Eall(z-zo)'" wherenowboththecoeflicients anandthequantities zandZomay becomplex. Thetheoryoftheseseriesdeveloped in§§18to21remains valid without anyessential mocllfication. Intransferring theconsiderations of thosesections, wemaytherefore bequitebrief. Sincethetheorems 93,1and2remainentirelyunaltered inthe newdomain, thesameistrueofthefundamental theorem93itself, onthebehaviour ofpowerseriesintherealdomain. Onlythegeo­ metricalinterpretation issomewhat different: Thepowerseries ~'allzn 14Ifw{'takeintoaccount PrmgshBlm's resultmentioned Intheprecedinll footnote, weIllaystatehere,moredefinitely' except/orz=+1. 4:02 Chapter XII.Seriesofcomplex terms. 230.converges -indeedabsolutely -foreveryzinterior tothecircleof radius 'Yroundtheorigin0,whileitdiverges forallpointsoutside thatcircle.Thiscircleiscalledthecircleofconvergence ofthepower series-andthename'Yadiusapplied tothenumber 'Ythusbecomes, forthefirsttime,completely intelligible. Itsmagnitude isgivenasbefore bytheCauchy-H adama'Yd theorem 94. Regarding convergence onthecircumference oftheci'Ycleofcon· vt::rgence, wecannomoregiveageneral verdictthanwecouldre· garding thebehaviour at theendpointsoftheinte'Yval ofconvergence inthecaseofrealpowerseries.(Theexamples whichfollowimmedla­ tclywillshowthatthisbehavIOur maybeofthemostdiversenature.) Theremaining theorems of§18alsoretaintheirvalidityunaltered. Examples. I,Iz"i '=1.Intheinterior oftheunitcircle,theseriesisconvergent, withthesuml~' Ontheboundary, i.e.forIzI=1,itiseverywhere dl­-z vergent,as,"doesnot-+0there. z"2..2-.;,=1.ThisserIes 16remains (absolutely) convergent atalln theboundary pointsIzI=1. ,"8..2-ir=1.Theseriesiscertainly notconvergent forallthen boundary points,forz=1givesthedivergent series.2~. However, itisalson notdivergent forallthesepoints,since'= -1givesaconvergent series.In fact,theorem229ofthepreceding- sectionshows,moreprecisely, thatthe seriesmustconverge conditIOnally atallpointsofthecircumference IzI=1 different from+1;forwehavehere a"n-l1 1 a"_l=-n=-n' Thesameresultmayalsobededuced dIrectly fromDiflchlet's testIS4,2,since Iz"hasbounded partialsumsfor,=l=+1andIzI=1(cf.thelastformula of thepreceding section) and~tendsmonotonely toO.As.21..:-"-1=\-._~n n"'::'n' theconvergence can,however, onlybeconditionallB• z·"4..2~;,=1.Thisseriesdiverges atthefourboundary points±1 and±i,andconverges conditionally ateveryotherpointoftheboundary. UIfIanz"hasrealcoefficients (asinmostofthesubsequent examples) thispowerseriesofcoursehasthesameradiusastherealpowerseriesIanx". 18Thesefactsregarding convergence mayalsobededuced fromIS:,),5, bysplitting uptheseriesintoitsrealandimaginary parts.Conversely, how­ ever,theabovemodeofreasoning provides anewproofoftheconvergence ofthesetworealseries. §54.Powerseries. Analytic functions. 403 z"5.For.2)-,1'=+00. For2,'n!z",1'=0;thusthisseriesconvergesn! nowhere butatz=O. '" ,Ok '" ZOk+l 6.Theseriesk~(-ll(2k)1 and~(-l)k(2k+l)T areeverywhere convergent. 7.Apowersenesofthegeneral form ~:all(z-zo)"converges absolutely atallinterior pomtsofthecircleofradiusrroundZoIRnddiverges outside thiscircle,whererdenotes theradiusof~a"z". Beforeproceeding toexamine theproperties ofpowerseriesin moredetail,wemayinsertoneortworemarks on Functions ofacomplex variable. Iftoeverypoint Zwithinacirclesr(ormoregenerally, a domain 17ill)avaluewismadetocorrespond inanyparticular manner, wcsaythatafunction w=f(z)ofthecomplex variablezisgivenin thiscircle(ordomain). Thecorrespondence maybebrought aboutin agreatnumber ofways(cf.thecorresponding remarkontheconcept ofarealfunction, §19,Def.1);inallthatfollows, however, thefunc­ tIOnalvaluewillalmostalwaysbecapable ofexpressIOn byanexplicit formula intermsofz,orelsewillbethesumofaconvergent series whosetermsareexplicitly given.Numerous examples willoccurvery shortly; forthemoment wemaythinkofthevaluew,forinstance, whichateachpointzwithinthecircleofconvergence ofagiven powerseriesrepresents thesumoftheseriesatthatpoint. Theconcepts ofthelImit,thecontinuity, andthedifferentiability ofa functIOn arethosewhichchieflyinterest usinthisconnection, andtheir definitions, insubstance, followprecisely thesamelinesasinthereal domain: 1.Definition oflimit.Ifthefunctionw=f(z)isdefined 18for231. everyzinaneighbourhood ofthefixedpoint ~,wesaythat iimf(z) =Wz...., or f(z)-Wfor 17Astrictdefinition oftheword"domain" isnotneededhere.Inthe sequel,weshallalwaysbeconcerned withtheIIlterior ofplaneareasbounded byafimtenumber ofstraight linesorarcsofcircles, inparticul,lr withcircles andhalf-planes 18f(z)neednotbedefinedatthepointCItself,butonlyforallz'swhich satisfythecondition 0<Iz-CI<e.The15oftheabovedefinition mustthen ofcoursebeassumed<(l. 404 Cbapter XII.Seriesofcomplex terms. if,givenanarbitraryc>0,wecanassign t5~t5(e)> 0sothat Ifez)-ill1<e foreveryzsatisfying theconditIon 0<Iz-Cl<<5;or-whichcomes toexactly thesamethll1g19-ifforeverysequence (z..)converging to1,;,whosetermslieinthegivenneighbourhood ofI,;anddonot coincide with1,;,thecorresponding functional valuesw"=fez,,)con· vergetoill. Ifweconsider thevaluesoff(z),notatallthepointsofaneigh· bourhood of1;,butonlyatthosewhichlie,forinstance, onaparti­ culararcofacurveendingat1;,orinananglew1thitsvertexat1,;, or,moregenerally, whichbelong toasetofpointsM,forwhich I,; isapointofaccumulation, -wesaythatlimfez)=illorfez)-ill asz-I,;alongthatarc,orwithinthatangle,orinthatsetM,1fthe aboveconditions arefulfilled, atleastforallpomtszofthesetMwhich comeintoconsideration intheprocess. 2.Definition ofcontinuity. Ifthefunction w=fez)isdefined inaneighbourhood ofI;andatI;itself,wesaythatfez)iscontinuous atthepoint1;,if limfez) z-+C existsandisequaltothevalueofthefunction at1,;,i.e.iff(z)·-+f(1;). Wemayalsodefinethecontinuity offez)atI;whenzisrestncted toan arcofacurvecontaining thepoint1;,orananglewithitsvertexatC, orany()~hersetofpointsMthatcontains ,andofwhich,isalimiting point;thedefinitions areobvious from1. 3.Definition ofdifferentiability. IfthefunctIon w=fez)isde· fined 1Daneighbourhood of1;andat1;itself,f(z)issaidtobediffer. entiable at1,;,1fthelimit lim((z)-f(l;) z-+cz-l; existsinaccordance with1.Itsvalueiscalledthedifferential coeffi· cientoffez)atI,;andisdenoted byf'(1;).(Hereagainthemodeof variation ofzmaybesubjected torestnctions.) Wemustbecontent withthesefewdefinitions concerning the general functions ofacomplex variable. Thestudyofthesefunctions indetailconstitutes theobjectoftheso·calledtheoryoffunctions, one ofthemostextensive domains ofmodern mathematics, intowhichwe ofcoursecannotenterfurtherinthisplace20• 19Sameproofai>intherealdomain. 110Arapidviewofthemostimportant fundamental factsofthetheory offunctions maybeobtained fromtwoshorttractsbytheauthor: Funktionen· §54.Powerseries.AI,lllytic functions. 405 Theaboveexplanations areabundantly sufficient toenable liSto tramfer themostimportant ofthcdevclopmcnts of§§20and21to powerserieswithcomplex terms. Infact,thosedevelopments remain valIdwithout exception for ourpresent case,Ifwesuitably change thewords"interval ofconver­ gence" to"CIrcleofconvergence" throughout. Theorem 5(99)isthe onlyonetowhichwecanformnoanalogue, smcetheconcept of integral hasnotbeenintroduced forfunctions ofacomplex argument. AllthisissoSImplethatthereaderwillhavenotrouble, onlooking through thesetwosections again,tointerpret themasiftheyhadbeen intended fromthefirsttorelatetopowerserieswithcomplex terms. Atthemost,afewremarks maybenecessary inconnection with Abet'slImittheorem 100andtheorem 107onthereversIOn of apowerseries. Inthecaseofthelatter,theconvergence ofthesenes y+fJ2y2+.."andhenceofIheseriesy+b2y'~+.."whIchsatIs­ fiedtheconditions ofthetheorem, wereonlyprovedforrealvaluesofy. ThIsISclearlysufficient, however, aswchavetherebyprovedthatthispower seneshasapositive radiusofconvergence, whichisallthatisreql11reLl. Asregards Abel'shmittheorem, wemayeven-corresponding tothegreaterdegreeoffreedom ofthevariable pointz-provemore thanbefore,andforthiSrea~onwewillgointothematteroncemore: Letussuppose :Eanzntobeagivenpowerscries,noteverywhere convergcnt, butwIthapositive radiusofconvergence. Wefirstobserve that,exactlyasbefore,wemay,lssume thisradius=1without intro duc1l1gallYsubstantial restnctlOn Onthecircumference ofthecircle ofconvergence, IzI=1,weassume thatatleastonepointZoeXists atwhichtheseriescontmues toconverge. Hereagainwemayassume thatZoisthespecialpoint+1.Infact,ifZo=f=+1,weneedonlyput anzo"=an'; theseries:Ean'z"alsohastheradius1andconverges atthepoint+1. Theproofongmally gIven,whereeverything maynowbe interpreted as"complex", thenestablishes the Theorem. Ifthepowerseries.2:anznhastheradius 1andremains232. convergent atthepoznt+1oftheunztcircle.,andit.2:an=s,then wealsohave lim(.2:anzn)=S z-.+1 ifzapproaches thepoint+1alongtheposz"tz"verealaxz"sfromthe origin 21O. theorie,I.Teil,Grundlagen derallgemeinen Theorie, 4thed.,LeipZIg 1930;11. Teil,Anwendungen undWeiterfuhrung derallgemeinen Theone, 4thed.,Leipzig 1931(Sammlung Goschen, Nos.668and703). 21Wearetherefore dealing withalimitofthekIndmentioned abovein 231,1.l' (a51) 406 Chapter XIl.Seriesofcomplex terms. :l33. Wecannoweasilyprovemorethanthis: Extension ofAbel'stheorem. Withtheconditions ofthepreceding theorem, the1'elation Em(Ianz")=S %-++1 '"L+1 ~0= Fig.10.1'emain\ trueifthemodeofapproach ofZto+1is1'estricted only bythecondition thatzshould1'emamwithintheunitcircleandin theanglebetween twoarbi­ trary(fixed)rayswhichpene­ trateintotheinteriorofthe unitcircle,starting fromthe point+1(seeFig.10). Theproofwillbecon­ ductedquiteindependently of previous considerations, sothat weshallthusobtainathird proofofAbel'stheorem. LetzO'Z1'...,z",..•be anysequence ofpointsof limit+1inthedescribed portion oftheunitcircle Wehavetoshowthat f(Zk)-S if,asbefore, wewriteIanz"=f(z).InToeplitz' theorem221,2, chooseforakthevalue n a=(1-z).Z "kn k k andapplythetheorem tothesequence ofpartialsums s"=au-1-a1+...+an' which,byhypothesis, converges tos.Itfollowsimmediately that <Xl <Xl <Xl.2(1-zk)·zt,sn =(1-Zk)'-I;snzk" =-I;anzt =f(Zk) 1'=0 ,,=0 ,,=0 alsotendstosaskincreases. Thisprovesthestatement, provided wecanshowthatthe,hosennumbers aknsatisfytheconditions (a), (b)and(c)of221.Now(a)isclearlyfulfilled, aszk-1, andthe sumofthekthrowisnowAI,=(1-Zk)1;zk"=1,sothat(c)is n=U fulfilled. Finally(b)requires theexistence ofaconstantKsuchthat 11_Z1.1;Iz"1=11-zr<]{ 1'=01-1zI forallpointsz=z.9=+1intheangle(oranysector-shaped portioll ofitwithitsverte>..at+1).Itonlyremains, therefore. toestabhsh §54.Powerseries.Analytic functIOns. 407 theexistence ofsuchaconstant. Thisreduces (v.Fig.10)toproving thefollowing statement:IIz=1-e(cosP+isinp)withIPI<Po<-;­ and0<e:::;;:eo<2cosPo'aconstant A=A(Po'eo)exists.depending onlyonCPoandeo'suchthat 11-zl<A1-lzl= forevery Z01thetypedescribed. Intheproofofthisstatement. itis sufficient toassume eo=cosCPo'andinthatcasewemayatonce showthatA=_2_isaconstant ofthedesired kind.Infact.thecos'Po statement thenruns: (!<_2_ 1-f1-2(!cos'P+(!'=cos'Po or - 2ecoscP+e'< -(!cosPo+i(!'cos·%, for0<e<cosPoandIpI<Cf'o' Byreplacing cPbyCPoand(}.byecosPoonthelefthandside,thelatterisincreased; therefore Itcer­ tainlysuffices toshowthat 1-ecos'Po< -ecos'Po+4,e2cos2'Po, -whichisobviously true.-Thisextension ofAbel'stheorem to"com- plexmodesofapproach" or"approach withinanangle"isdueto O.Stolz 22. Thiscompletes theextension tothecaseofcomplex numbers ofallthe theorems of§§20and21-withthesingleexception ofthetheorem oninte­ gration,whichwehavenotdefinedinthepresentconnection. Inparticular, itisthereby established thatapowerseriesintheinteriorofitscircleof convergence definesafunction ofacomplex variable, whichiscontinuous anddifferentiable -thelatter"termbyterm"andasoftenaswcplease -inthatdomain, andaccordingly possesses thetwoproperties which aboveallothersarerequired, inthecaseofafunction, forallpurposes ofpractical application. Forthisreason,andonaccount oftheirgreat importance infurther developments ofthetheory, aspecialnamehas beenreserved forfunctions representable intheneighbourhood ofapoint 2.Zeitschrift f.Math.u.Phys.,Vo!.20,p.369,1875.Inrecentyearsthe question oftheconverse ofAbel'stheorem hasbeentheobjectofnumerous investi­ gations, -i.e.thequestion, underwhat(minimum of)assumptions relating to thecoefficrents an'theexistence ofthehmitofj(z)asz->1(within theangle) entailstheconvergence of1:an'Anexhaustive surveyofthepresentstateofresearch inthisrespectisgiveninpapersbyG.H.HardyandY.E.Llttlewood, Abel'stheorem anditsconverse, Proc.Lond.Math.Soc.(2),I.Vo!.18,pp.205--235, 1920;II. Vol.22,pp.254-269, 1923;Ill.Vo!.25,pp.219-23li, 1921i.-Cf. alsotheorems 278and287. 408 ChapterXII.Seriesofcomplex terms. ZobyapowerseriesEan(z-zo)n.Theyaresaid tobeanalyticorregular atzooBy99,suchafunction isthenanalyticateveryotherinteriorpoint ofthecircleofconvergence; itistherefore saidsimplytobeanalytic or regularinthiscircle23.Inparticular, aserieseverywhere convergent re­ presents afunction regularinthewholeplane,whichistherefore shortly calledanintegralfunction. Allthetheorems whichwehaveprovedaboutfunctions expressed bypowerseriesaretheorems aboutanalyticfunctions. Onlythetwofol­ lowing,whichareofspecialimportance inthesequel,needbeexpressly formulated again. 234. 1.Iftwofunctions areanalyticinoneandthesamecircle,thensoare (by§21)theirsum,theirdifference, andtheirproduct. Forthequotient thecorresponding statement isprimarily true(by 105,1)onlyifthefunction inthedenominator isnotzeroatthecentre ofthecircle,andprovided, ifnecessary, thatthiscircleisreplaced bya smallerone. 2.Ift'wofunctions, analytic inoneandthesamecircle,coincideina neighbourhood, however small,ofitscentre(orindeedatallpointsofaset havingthiscentreaspointofaccumulation), thetwofunctions arecompletely identical inthecircle(Identity theorem forpowerseries97). Besidesstatingthesetwotheorems, whicharenewonlyinform, weshallprovethefollowing important theorem, whichgivesu:;some information ontheconnection between themoduliofthecoefficients of apowerseriesandthemodulus ofthefunction itrepresents: 00 23i'i. Theorem. Iff(z)=Ean(z-zo)nconverges forIz-ZoI<r,then n=O IavIS~ (p=0,1,2,...),e if0<e<randM=M(e)isanumberwhichIf(z)Ineverexceedsalong thecircumference Iz-ZoI=e.(Cauchy's inequality.) Proof 24.Wefirstchooseacomplex number 'Yj,ofmodulus=1, forwhichhowever 'Yjq*'1foranyintegral 25exponent q~O.Nowwe consider thefunction g(z)=a.(z-zoY< 21Afunction isaccordingly saidtobe"analytic" or"regular" inacircle.Il' whenItcanberepresented byapowerserieswhichconverges InthiSCircle. ..Thefollowing veryelegantproofisduetoWelerstrass (WerkeIf,p.224) anddatesasfarbackas1841.Cauchy(Memoire lithogr., Turint831)provedthe formula indirectly bymeanqofhisexpression forf(z)Intheformofanintegral. Theexistence ofaconstant MthatIj(z)Ineverexceed.,onIz-ZoIO~eISpractically obvious, ofcourse,sinceM,1:IanIenclearlyhasthisproperty. ThisMisobviously alsosuchthatIanIen;SM.Buttheabovetheorem statesthateveryMthatIj(z)I neverexceeds hastheproperty thatIanIenISalways:"; M. 26Suchnumbers TJofcourseexist,forIfTJ=cos(a:1T)+I~InCot1T),then TJq=cos(qa:1T)+iSill(qa:1T);thisisneverIifa:ischosenirrational. §54.Powerseries.Analytic functions. 409 foraspecific integral valueoftheexponent k:§0andanarbitrary constant coefficient a.Ifwedenote bygo'gl'g'J'...thevaluesof thisfunction for%=%0+e''YJ", 11=0,1,2,...,wehaveforn2:1 k1-'1kn gO+gl+,,·+gn-l=a·e·I_'1/'" hence Igo+gt+ •.•+gn-11<!.k·l_a_l· n1=ne1-'1" Theexpression ontherighthandsidecontlins onlyconstants, besides thedenominator n;itthereforc follows thatthearithmetic mean go+gt+...+gn-l_ 0 n asnincreasc!>. Inthecasek=0,weshouldbeconcerned withthe identically constant functIOn g(z)=a,forwhich go+gt+...+gn-l---- n----a, sincetheratioisequaltoaforeveryn,inthiscase.Ifweconsider ther.tthermoregeneral funcllon where1andmarefixedintegers:20,andnowformthearith­ meticnlean go+gl+...+g,,-l n (where, asbefore,g"=g(zo-+-er;»,,.=0,1,...),thisclearly -bu' bythetwocasesjusttreated.If,further,itisknownthatthefunctiong(z), foreveryzofthecircumference Iz-ZoI=e,isnevergreater thana certamconstantK,wehavealso andtherefore also Withthesepreliminary remarks, theproofofthetheorem isnow quitesimple: Letpbeaspecific integer2:O.As.17l cln!Q"con verges, givene>0,wecandetermine q>Psothat 410 Chapter XII.Seriesofcomplex terms. Afortiori, wethenhaveforallvaluesof%suchthatIz-ZoI=(!, I1;an(z-zotI<8 n=1l+1 andtherefore, forthesamevaluesofz, qI.2an(z-%otI<M+e, n=O ifMhasthemeaning giveninthetext.Accordingly, onthecircum· ferenceIz-ZoI=(}, I-a_o-+...+ap_t+a+a(z-z)+...+a(z-z)q-pI(8-Zo)p 8-Z0P1'+1 0 q 0 <M+6. =eP Thefunction between themodulus signsisofthekindjustconsidered. TheinequalityIboI<Kthereobtained nowbecomes laI<M+e P=e'P' and,asewasarbitraryand>0,wehave,infact,(cf.footnote to41,1) Mlapl:::;:1>'e q.e.d. §55.Theelementary analytic functions. I.Rational functions. expressible asapowerseries forh. . 1 .1.T eratIOnal functIOn w=1---IS-8 everycentre Zo=l=+1: 1 1 1 1 '"1 1-z=1-Zo-(z-zo)=1 -zo'1 _z-Zon~(1_zo)n+l'(z-%0)n; I-zo andthisseriesconverges forI% -ZoI<11-%0Ii.e.forevery % nearer to%0than+1;inotherwords,thecircleofconvergence of theseriesisthecirclewithcentre %0passing through thepoint+1. Thefunction 1-.!--isthusanalytic ateverypointdifferent from+1--z Withreference tothisexample, wemaybrieflydrawattention tothe following phenomenon, whichbecomes offundamental importance inthetheory offunctions: Ifthegeometric senes:EZR,whosecircleofconvergence isthe unitcircle,isexpanded byTayloy's theorem aboutanewcentre ZIwithinthe unitcircle,wecouldassertwithcertainty, bythattheorem, thatthenewseries converges atleastinthecircleofcentrez,whichtouches theunitcircleon theinside.'Vcnowseethatthecircleofconvergence ofthenewseriesmay verypossibly extendbeyond theboundary oftheold.Thiswillalways.bethe case,infact,when ZIisnotrealandpositive.IfZIisrealandnegative, the newcirclewillindeedinclude theoldoneentirely. (Ct.foomote to99,p176,) §55.Theelementary analytic functions. -I.National functions. 411 2.Sincearational integral function ao+a1z+a~z'l+...+amZ'" mayberegarded asapowerseries, convergent everywhere, such functions areanalytic inthewholeplane.Hencetherational functions ofgeneral type ao+a,Z++amz... 110+b,Z++bkif" areanalytic atallpointsoftheplaneatwhichthedenominator is not0,-i.e.everywhere, withtheexception ofafimtenumber of points. Theirexpansion inpowerseriesatapointzo'atwhichthe denominator is=+=0,isobtained asfollows:Ifzisreplaced by Zo+(z-zo)bothinthenumerator anddenominator ofsuchafunction, thesebeingthenrearranged inpowers of(z-zo)'thefunction takes theform act+a/(if-Zo)+ + a~(.c-.co)- b~'+b/(if--za)+ +b;:(z-.co)k, where,onaccount otourassumption, bo'+O.Wemaynowcarry outthedivision inaccordance with103,4andexpand thequotient intherequlred powerseries26oftheform££""(.3'-.3'0)". 11.Theexponential function. Theseries Z.II .I"I+Ti+2'+···+"I+··· isapowerseriesconverging everywhere, andtherefore definesafunc­ tionregular Inthewholeplane,i.e.anintegral function. Toevery pointzofthecomplex planetherecorresponds adefinite number UJ, thesumoftheaboveseries. Thisfunction, whichforrealvaluesofzhasthevalueelllasde­ finedin83,maybeusedtodefinepowers ofthebasee(andthen furtherthoseofanypositive base)forallcomplex exponents: IIIAnalternative method consists infirstsplitting upthefunction into partialfractions. Leaving outofaccount anypartwhichrepresents arational integral function, wearethenconcerned withthesumofafiDltenumber of fractions oftheform A A (1)!l(if_a)!l= (_a)!l'1-;', eachofwhichwemay,by1,expandseparately inapowerseriesoftheform Ie"(if-ifa)",provided .co=1=a.-Thismethod enables ustosee,moreover, that theradinsoftheresulting expansion willbeequaltothedistance ofifOfromthp nearest pointatwhichthedenominator ofthegivenfunction vanishes. 412 Chapter XII.Series 01complex terms. 236. Definition. Forallrealorcomplexexponents, themeaning tobe attributed tothepowereZisdefined,without ambiguity, bythe relation Z_2 _n e3=1+11+;!+...+;l!+..., AndifPisanypositive number,pzshalldenotethevaluedetermined, without ambiguity, bytheformula pz=ezIOgp, wherelogpisthe(real)naturallogarithm ofpasdefined 27in36. (Foranon-positive baseb,thepowerbzcannolongerbeuniquely defined; cf.,however, 244.) Astherewasnomeamng- attached persetotheideaofpowers withcomplex exponents, wemayinterpret 1heminanymanner weplease. Reasons ofsUItability andconvenIence canalonedetermine thechOIce ofaparticular interpretation. Thatthedefinition justgivenisatlllr· oughly suitable onc,resultsfromformula 91,example 3(leaving outofaccount theobvious requirement thatthenewdefinition must coincide withtheoldoneforrealvaluesoftheexponent 28);thisformula wasprovedbymeansofamulllplicatlOll ofseries,thevalidltyofwhIch holdsequally forrealandcomplex variable«, andtheformula must accordingly alsoholdforanycomplex exponent; itis 237. 238.whence also Thisimportant fundamental lawforthealgebra ofpowers therefore certainly remains true.Atthesametimeitprovides uswiththekey tothef1,II"therstudyofthefunction eZ. 1.Calculation ofea,Forrealy's,wehave . '";(~y)n 00 ky2k .00 ky2k+l eUl=~o---n!=k~(-1)(2k)I+~d:(-:-1)(2k+l)T =cosy+isiny, 2.Itmaybenotedhowfarremoved thisdefinition isfromtheelementary definition "xkistheproduct ofkfactorsallequaltox".-Atfirstsight,thereis noknowing whatvaluebelongs e.g.to2ijyetthisvalueisinanycnseuni­ quelydetermined bytheabovedefinition. 28By234,2, therecanexistnoothet'function thanthefunction eZjust defined whichisregulnr intheneighbourhood oftheoriginandcoincides on thereal axi~z=.'1:withthefunction eZdefined by33.Forthisreasonwe mayindeedsaythateverydefinition ofeZdiffering fromtheabovewouhl necessarily beunsuitable, §55.Theelementary analytic functions. -H.Theexponential function. 413 Henceitfollows that,forz=x-+-iy, e"=eX+ill=eXeill=er(co,y+isiny). BymeansofthIs1ormula'.lOthevalueofeZmayeasIlybedetermined forallcomplex z's. Thisformula enables us,besides, toobtaininaconvement and complete manner anIdeaofthevalueswhichthefunction eZassumes a~thevanous pointsofthecomplex plane(inshort,ofitsstock01 vatues). Wenotethefollowing facts. 2.WehaveIeZ1=e91(z)=e"'.Infact Ieillj=Icosy-+-iSinYI~veos'.ly-+-sin2y=1, henceIeZI=le'"1·1eilll=e"',because e'">0andthesecond factor =1.SlIlularly, ameZ=~(z)=y, abofromtheformula23S,1Justused. 3.eZhastheperiods 2k7li,thatistosay,forallvaluesofz, (k~O, integral). oForifweincrease zby2niitsimagmary partyincreases by2n, \\hileitsrealpartremains unaltered, andby1.and§24,2,thisleaves thevalueofthefunction unchanged. Everyvaluewhich eZisab:e toassume accordingly occurs inthe strip~7l<~(z)=y<n,orinany stripwhichmaybeobtained fromit byaparallel translation. Everysuch stripIScalledaperiod-strip; Fig.11 represents thefirst·namedofthesestrips. 4.eZhasnootherperiod, ~in­ deed,moreprecisely: ifbetween two specialnumbers ZlandZlIwehavetherelation .------------ ~Jlr-------- thisnecessarily implies that Z2=Zl-+-2kni.Fig.11. ForwefirstinferthateZ'-Z'=1,thenwenotethatif ell=eHill=e:l:(cosy-+-isiny)=1, 29Eule,.:Intr.inAnalysin iDt.Vot.I,§138.1748..,,- (a51) 414 Chapter XII.Seriesotcomplex terms. wemustby2.haveeX=1,hencex=O.Further, wealsohave cosy+isiny=1 i.e. cosy=1,siny=0. hencey=2k:n.Thus.asasserted, Z=Z2-Z1=2kni. 5.eZassumes everyvaluew=1=0onceandonlyonceintheperiod strip;or:theequation eZ=wl'forgivenw1=1=0,hasoneamIonly onesolution inthatstrip. Ifw1=R1(cosW1+isinlJ>1)withRI>0,thenumber ZI=logRI+ifJ>1 iscertainly asolution ofeZ=wl'as eZ'=e1ogR,ei'!>,=R(cos([J+isinfJ>)-W1 1 1-l' By3.,thenumbers (k=0,±1,±2,...) arealsosolutions ofthesameequation, andby4.noothersolutions canexist.Nowkmayalwaysbechosen, inoneandonlyoneway, sothat -n<~(Z1+2kni)<+n, 6.Thevalue0isneverassumed byeZ;for,by237, eZ'e-Z=1,q.e.d. sothateZcanneverbeO. 7.Thederivative (ez),ofeZisagain=eO,asfollowsatoncebydiffer­ entiating term-by-term thepowerseriesthatdefinese". 8.From238,1,wealsodeducethespecialvalues 111 -nz e2m=1,e"j=-1,e-2=i,e-2-=-i. Ill.Thefunctions coszandsinz. Inthecaseofthetrigonometrical functions, wecanagainusethe expansions inpowerseriesconvergent everywhere todefinethefunctions 239.forcomplex valuesofthevariable. Definition. Thesumofthepowerseries,convergent everywhere, Z2z4 Z2k1-2!+:n-+...+(-l)k(~rkyJ+..., isdenotedbycosz,thatofthepowerseries,alsoconvergent everywhere, ZZ3Z5 Z2k+1 f!-3!+In-+...+(-1)k(2k+T)l+..., bysinz,-foreverycomplexz. §55.Theelementary analytic functions. -Ill.Thefunctions cos1andsinI.415 Forrealz=x,thiscertainly givesustheformerfunctions cosx andsinx.Wehaveonlytoverify,asbefore, whether thesedefini­ tionsaresuitableones,inthesensethatthefunctions defined, -which area1>alytic inthewholeplane,i.e.integral functions, -possess the samefundamental properties astherealfunctions 30cosxandsinx. Thatthisisagainthecase,tothefullestextent,isshownbyth~ following statement oftheirmainproperties: 1.Foreverycomplex z,wehavetheformulae cosZ+'isinz=eh,cosz-isinz=e-i", whencefurther eh+e-b cos,~= 2 (Eltler's formula,e). Theprooffollowsimmediately byreplacing thefunctions onboth sidesbythepowerserieswhichdefinethem. 2.Theaddition theorems remain validforcomplex valuesofz: cos(z]+z~)=cosZ]cosZ'l-sinZlsinz~, sin(Z1+Z2)=cosZ1sinZ2+sinZ1cosZ2' Thisfollows from1.,sinceby237 andthelatterinvolves cos(zJ+Z2)+isin(z]+Z2) =(cosz]+isinZl)(cosZ2+isinz2) =(cosZ1cosZ2-sinZ1sinZ2)+i(cosZ1sinz~+sinz,cos%2)' Substituting -Z1and-Z2forz]andZ2'andtakmgintoaccount the factthatcosZisaneven,sinzanoddfunctIOn, weobtainasimilar formula, whichdiffersfromthelastonlyinthatiappearstobechanged to-ioneItherside.Addition andsubtraction ofthetworelations giveustherequired addition formulae. 3.Thefactthattheaddition theorems forourtwointegral func· tionsareformally thesameasthoseforthefunctions cosxandsinx oftherealvariablex,notonlysufficiently justifies ourdesignating thesefunctions bycosZandsinz,butshows,atthesametime,that theentireformalmachinery oftheso-called goniometry, sinceitis evolved fromtheaddition theorems, remains unaltered. Inparticular, 10Hereagainaremark analogous tothatonp.412,footnote 28,may bemade.24.0. 416 Chapter XII.SeriesotcompJ<>x terms. wehavetheformulae cos'JZ+sin'Jz=1, sin2z=2sinzcosz,cos2z=cos'JZ -sin~z, etc. validwithout change foreverycomplex z. 4.Theperiod-pi operties ofthefunctions arealsoretained Inthe complex domain. Foritfollows fromtheaddition theorems that cos(z+2n)=cosz·cos2n-sinz·sin2n=cosz, sin(z+2n)=cosz·sin2n+sinz·cos2n=sinz. 5.Thefunctions coszandsinzpossessnootherzerosinthecom· plexdomain besidesthosealready knownintherealdomam:l1•Infact, cosz=0necessanly involvcs, by1.,eiz= -e-izor e2iz=-1=en' i.e. e2iz-ni =1. By238,4, thiscanonlyoccurwhen 2iz-ni=2kni orz=(2k+1);-. Similarly, sinz=O implieseiz=e-- iz,ore2iz=1,I.C.2iz=2kni. orz=kn,q.e.d. 6.Therelation cosZ1=cosZ2issatIsfied it,andonlyif, Z'J=±Z1+2k:ll,-i.e.underthcsamecondition asinthereal domain. Similarly smZl=~inz'.!if,andonly1/,Z9=Zl+2k nor ZlI=n-Z1+2kn.Itfollows infactfrom cosZ-cosZ= -2!';!TI.ll+~2SIl1~~.l2_=0 1 'J •2 2 ' by5.,thateither Zl~-!gor~;~!must=kn;slllularly itfollowsfrom .. .ll+Z2•.ll-.l20smz-smZ=2cos---sm-----= 1 'J 2 2 • by5.,thateither 81;-Z2=knor!1__;~=(2k+1)-;-. 7.Thefunctions coszandsinzassumeeverycomplex valuew intheperiod-strip, i.e.inthestrip-:re<9{(z)<+n.theequations cosz=wandsinz=whaveindeedexactlytwosolutions inthatstrip, jfw-+±1,butonlyone,ifw=±1. Z2 11Orinotherwords:Thesumofthepowerseries1 - -+_...=0if,21 nandonlyif,Zhasoneofthevalues(2k+1)-2-'k=0,±1,±2,...;and ,similarly for thesineseries. §55.theelementary analydc lunctions. -iv.'Ihe[unctions cotzandtanz.41? o FIg.12.le:::::=- ~~ L_~~-c=­~. t:....=..== ~=--~Proof.Inordertohavecosz=-w,wcmusthaveeiz+e-Cz=2UJ oreiz=w-I-v'w~= 1.(Here v'-;Tco~(p--+isin(p)isdefined asone ofthetwonumbers, forinstance r}(cos.~-I-isin-~),whosesquareis thequantity undertheradical<ign.) Sinceinanycase32w+Vw2-1'*0, therecertainly existsacomplex num­ berz'suchthat-n<S(z')<-I-n, fOlwhicheZ'=w-t-Vw2-=,1, by238,5.Writing - iz'=Z,we have--n<m(z)<-I-nandeH =w+v'W2-~-1,orcosZ=w.This equation therefore certalIlly hasat leastonesolutIOn inthepenod-stnp. By6.,however, asecondsolutIOn, dlllerent fromit,(viz.-z),exi~ts intheperiod.stripif,andonlyif, z9=0and9=n,i.e.w9=±1. Wereasoninprecisely thesamemanner withregardtothe equation smz=.w.Inthiscase,wecanalsoeasilyconvlllce our· selvesthatthere ISalwaysoneandonlyoncsolutIon oftheequation intheportion oftheperiod·striplefttmshaded inFig12,ifwein­ cludethepartsoftherimindicated inblack,butomitthepartsrc' presented bythedottedlines(seeVIbelow). 8.Forthederivatives, wehaveasintherealcase, (cusz)'~-sinz,(sinz)'.=cosz. IV.Thefunctions cotzandtanz. 1.Sincecoszandsinzarcanalyticinthewholeplane,thefunctions cotz=~osZandtanz=si":._": sinz cosz willalsoberegularinthewholeplane,withtheexception ofthepoints k7Tfortheformerand(2k+1)~forthelatter,whicharethezerosof sinzandcoszrespectively. Theirexpansions inpowerseriesmaybe obtained bycarrying outthedivisionofthecosineandsineseries.Since thisoperation isofapurelyformalnature,theresultmustbethesame asitwasintherealdomain. Accordingly, by§24,4,wheretheresult ofthisdivision wasobtained byaspecialartifice,wehave en 22kB 241zcotz=E(-l)k 2kz2k, • k-0 (2k)! ~_~(_l)k-l22k(22k-1)B2k 2k-ltan N-k""1 (2k)Iz • 33Infact,sincewll-1=Fw2•v'w3-1=F±w. 418 Chapter XII.Seriesotcomplex terms. Onaccount of94and136,wearenowalsoinaposlDon tode termine theexactradiusofconvergence oftheseseries. Theabsolute valueofthecoefficient ofZ2kinthefirstseries,by136,is _12U.B2k 2.29k 001(-1)k-(2k)1 =(2n)2k'n~nU' Its(2k;throotis 1ifs2kdenotes thesum.2n2k'Thelatterliesbetween 1and2for ]l2everyk=1,2,...(foritIS-6-whenk=1,andislessthanthisfor everyotherk,but>1);therefore 2{1-_1)k-1~!32k__~rl (2k)In' andtheradiusofthecot-series =:n,by94.Similarly thatofthe tan-series isfoundtobei'. 2.cotzandtanzhavetheperiod:n.Forcoszandsinzboth change insignalonewhen %ISincreased by:n.Hereagamwemay show,moreprecisely, that cot%1=cotZ2andtan%1=tan%2 involve cosZlcosz,sin(Z2-z.)cot%-cot%.=-.- --.--=-.---.- 1 2sInZ.sInZ2sInZl.SInZ,InfactIitfollows from(k=0,±1,...). thatinthecaseofthefirstequation sin(Z2-%1)=0,i.e.Z2-%1=k:n. Similarly inthecaseofthesecond. 3.Inthe"period-strip", i.e.inthestrip-~-<lR(z)~+i-' cotzandtanzassume everycomplex valuew9=+iiustonce;the valuesw=±iareneverassumed. Toseethis,writee2iz=C.The equation cot%=Wthenbecomes .'+1 ,.=w+~. ~,_1=wor..w_• Foreachw+±i,Cisadefinite complex number+0and(by Il,5)thereaccordingly existsaz'suchthat-:n<S(z')~:n, forwhicheZ'=1;.For%= -i~,wethenhave -!!..<ffi(z).::;;:!!-.andcotz=fJJ,2 - 2 i.e.zisasolution ofthelatterequation intheprescribed strip. By2.therecanbenoothersolution inthisstrip.Theimpossibility §55.Theelementary analytic functions. -V."lhelogarithmic series.419 ofasolution forcotz=±iresultsfromthefactthattheseequations bothinvolve cot'JZ+1=0 whichcannotbesatisfied byanyvalueofz,ascos'Jz+sin'Jz=1. Fortanztheprocedure isquitesimilar. 4.Theexpansion inpartial fractions deduced in§24,5fO'1'the cotangent intherealdomain remains validinthesameformforevery complex zdifferent from0,±1,±2,.•.(andsimilarly fortheex- pansions oftanz,_._1_etc.).Indeed thecomplete reasoning givensmz theremaybeinterpreted inthe"cClmplex" sense,without altering a singleword33.Inparticular, foreveryzsatisfying theabovecon· dition, .nzcotnz =1+z.i;[_1_+-1-1=-1+2Z21;--2~.n=1z-n z+n_ n=1Z-n Now itfollows, ifwesubstitute zfor2inz,that z6z+1 002z'2-z--1=1+.2Z2+4n2n9;e- n=1 henceweobtaintheexpansion 1[1 11] 002 ----;1-z-----;--2~=~z"+4n9n9'- e n=1 validforeverycomplex z9=2k7Ti(k~0,integer). Thisistheex­ tension tothecomplex variablezoftheremarkable expansion inpartial fractions obtained onp.378,anditexhibits thetrueconnection between thisexpansion andthatofcotz,whichpreviously seemed ratherfortuitolls. V.Thelogarithmic series. In§25,wesawthattheseries ~(_1~-1 ..Y=~ X n=1 n represents foreveryIxI<1theinverse function oftheexponentIal function ell-1;i.e.substituting foryin y9yB Y+21+S1+··· 33Itwasprecisely forthispurpose thatatthetimeweframedsomeofour estimates inaformsomewhat dIfferent fromthatreqUIred fortherealdomain (c.g.thoseonpp.206-207 towhichfootnote 26refers). 420 ChapterXII.Seriesofcomplex terms. theaboveseriesandrearranging (asiscertainly allowed) inpowersof x,wereducethenewseriessimplytox.Thisfact-becauseitispurely formalincharacter -necessarIly remains whencomplex quantities are considered. Hence,foreveryIzI<1, eW-1=zoreW=1+z, ifwdenotesthesumoftheseries (L)en(l)n-lw=E-=-- zn. n-ln Wenowadoptforthecomplex domainthe 242. DefinitiOn. Anumberaissaidtoheanaturallogarithm ofc,zn symbols, a=-logC, ifea=c. Inaccordance withIl,5,wemaythenassertthateverycomplex number c=F°possesses one,andonlyone,logarithm whoseimaginary partliesbetween -7Texclusive and+7Tinclusive (tothenumber 0, however, byIl,6,nologarithm canbeassigned atall).Thisuniquely definedvaluewillbemoreespecially referred toastheprincipal value ofthenaturallogarithm ofc.Besidesthisvalue,thereisaninfinityof otherlogarithms ofc,sineewithea=cwehavealsoea+2km=c;thus ifaistheprincipal valueofthelogarithm ofc,thenumbers a+2k7Ti (k~O,integer) mustalsobecalledlogarithms ofc.Thesevaluesofthelogarithm (for k=F0)arecalleditssubsidiary values 34.By238,4therecanbenofurther logarithms ofc.Wehave,foreachofitsvalues, ill(logc)=logIcl, ~(logc)=amc, ifinthefirstoftheserelations logIcIdenotesthe(single-valued) real logarithm ofthepositivenumberIcl,andthesecondisinterpreted as meaning that,takenasawhole,allthevaluesoftheonesideareequalto allthevaluesoftheother. Withthesedefinitions, wemayassertinanycasethattheaboveseries (L)provides alogarithm of(1+z).Butwemayatonceprovemore, namelythe 243. Theorem. Thelogarithmic series(L)gives,ateachpointoftheunit circle(including itsrim,withtheexception ofthepoint-1),theprincipal valueoflog(1+z). ..Ifcisrealandpositive, theprincIpal valueoflog'ccoincide~ withthe(real) naturalloganthm asformerly dcfined(36,Ocf.). §55.Theelementary analytic functions -VI.Themversesmeseries. 421 Proof. Thattheseriesconverges foreachz=1=-1forwhich IzI:::::1wasshownin230,3.(Wehaveonlytoput-zforzthere.) Forthisz,am(1+z)hasprecisely thatvalue.pforwhich Hencewehave,fortheimaginary partofthesumwoftheseries(L), ~'!(w)=~\(log(1+-z»=t/J+2ki, withintegralk.Nowwisacontinuous function ofzinIzI<1,and assumes thevalue1forz=o.Henee ~i(w)tooisacontmuous function inIzI<1.Therefore, intheequation (.\),hmusthavethesamevalue forallthesez.Butforz=c0\\ehaveclearlytotakek=0;hencethis isitsvalueinthewholeofIzI<1.Finally\\elcarnfromtheapplication ofAbet'slimittheorem thatthesumofoursenes ISstillequaltotheprin­ cipalvalueoflog(1+z)atthepomtsz*-1forwhichIzI=1. VI.Theinverse sineseries. WesawinIll,7thattheequation sinw=z,foragivencomplex zCF:1::1,hasexactly twosolutions, --forz'=l:1exactly one,-in thestrip-n<~.H(n)<+-n.Thetwosolutions (byIll,6)aresym- metrical, eitherwithrespectto+-~or-~;accordingly, wemayassert moreprecisely thattheequation sinw=-z,foranarbitr.lry given Z(in­ clUSiveof±1),hasoneandonlyO1lesolution inthestrip -i<!l1{w)<+i' ifthelowerportions ofitsrim,fromtherealaxisdownwards, areomitted (cf.Fig.12,whcrethepartsoftherimnotcounted Withthestripare drawnindottcdlines,andtheothersaremarked byacontinuous black line).Thisvalueofthesolution oftheequation sinw=z,winchisthus uniquely defined foreverycomplex z,IScalledtheprincipal valueofthe function w=sin--Iz. Alltheremaining valuesarecontained, byIll,(i,111thetwoformulae sin-1z+2I~n, 'TT--sin-1z+2I~n, andmaybecalledsubsidiary 'valuesofthefunction. 422 Chapter XII.Seriesofcomplex terms. ForrealvaluesofxsuchthatIxI~1,theseries123, Ix81·3xby=x+--+--+ ...2 3 2·45 represents theinverse seriesofthesinepowerseries ,,8".y-3T+5T-+ .... Exactly thesameconsiderations asinV.forthecaseofthelog· arithmic seriesnowshowthat,forcomplex valuesofzsuchthatIzI<1, theseries 1.181·3.I"w=z+23""+24T+··· istheinverse seriesofthesinepowerseriesw-~~+-....It therefore givesatanyrateoneofthevaluesofsin-1z.Thatthis actually istheprincipal value,maybeseenfromthefdctthat,for Izl<1, z=j=±1, . . 1Izl8l·gIzl"Im(sm-1z)I<Ism-1zI~IzI+2-3-+2~4.-5-+... =sin-1IzI<sin-11=;J - acondition whichtheprincipal valuealonefulfils. VII.Theinverse tangent series. Theequation tanw=z,asweknowfromIV,3,hasforeverygiven z=1=±ioncandonlyonesolution inthestrip-;<ffi(w)~+;. Thisiscalledtheprincipal valueofthefunction w=tan-1z theothervaluesofwhich(byIV,2)arethenobtained fromtheformula tan-1z+kn.Theequations tanZ=±ihavenosolutions whatever. Almost wordforwordthesameconsiderations asaboveagain showthat,forIzI<1,theseries .18,,8(A) w=z-3""+5"-+... givesoneofthesolutions oftanw=z.Toshowthatthisisactually theprincipal valueoftan-1z,wehavetoshowthattherealpartof thesumoftheseriesliesbetween -;(exclusive) and+;(inclusive). Thisremains trueforeveryz9=±ionIz1=1,aswellasforIz1<1, andisproved asfollows: Thesumwoftheseries(A),asmaybeseenbysubstituting the log-series, is w=2\log(1+iz)-21 jlog(1-iz) §55.Theelementary analytic functions. -VIII.Thebinomial series.423 foreveryIzI~1,z=+=±i,whereprincipal valuesaretakenforboth logarithms. Accordingly, !R(w)=~SIOg(l+iz)- ~Slog(l-iz); by243,bothtermsofthedifference liebetween --;and+:. hence!R(w)liesbetween -;and+;.thetwoextreme values beingexcluded ineithercase.Thustheseries(A)certainly represents theprincipal valueoftan-1z,providedIzI<1andz9=±i,q.e.d. VIII.Thebinomial series. Tocomplete ourpresent treatment ofthespecialpowerseriesin­ vestigated intherealdomain, wehaveonlytoconsider thebino­ mialseries (1+xt=n~(:)x· IIIthecasewherethequantities occurring there-i.e.theexponent t% aswellasthevariable x-assume complex values.Westartwiththe Definition. Thenameofprincipal valueatthepower ba,where244. aandbdenoteanycomplex numbers, withb=+=0astheonlycondi- tion,isgiventothenumberuniquely defined bytheformula ba=ealogb whenlogbisgivenitsprincipal value.-Bychoosing othervaluesof logb,weobtainfurther\aluesofthepower,whichmaybecalledits subsidiary values. Allthesevaluesarecontained intheformula ba=eallollb+2bdl, eachvaluebeingrepresented exactlyonce,iflogbISgivenitsprin­ cipalvalueandktakesallintegral values ~O. Remarks andExamples. 1.ApowerblJaccordingly hasaninfinite number ofvaluesingeneral, but poss(,~SE's one/mdonlyoneprincipal valuc. 2.Thesymbol •ItforInstance, dcnotes theinfinity ofnumbers (allreal numbers, moreover) i(n'+2knl)-~-2k" el(logH9kni)=8 2 =8 ~ •(k=O,±1,±2,..) If ofwhich8-2"istheprincipal valueofthepower. I. 3.Theonlycaseinwhichapowerbawillnothaveaninfinite number ofvalues ISthatinwhich (11=0,±1,±2,...) 424 Chapter XII.Seriesofcomplex terll1S. gIvesonlyajlnltenumber ofvalues; thiswilloccurif,andonlyif,k.a assume~, fork=0,±1,±2,...,onlyafmitl'nLllllber ofessentwlly different valul·s. lien' Iwo nlllllh('r~ ,1Tl' de~crihed (JlI'tforthemoment) .ISessen/wily different If,andonlyIf,theydonotdiffermerely bya(re.d)IlIteger. "ow thIsISthecas..If,andoulyIf,aisarealratIOnal number. asmaybeseen atonce;andthenLlmoer of"e"cntlally different" valLles whIchmayinthis casebeassumed byk·aisgivenbythesmallest positive denollllllator WIth whichamaybewntten infractional form, I 4.Ttfollows thatbm=~/b,wheremisapositive mteger, hasexactly mdIfferent values, oneofwhIchisqllltedefmitely distmgllIshedastheprin­ CIpalvalne, .i.T,lenLl'l1bl'r ofdifferent value~ofhawdlreducetoone,by3,and4"if, andonlyIf,aISarational number ofdenOllllna!or I,1.e.,IrealInteger. Forall realmtegral exponents (butforthesealone),thepowerthu,rernams nowdSbefore asmgle-valued symbol. 6.Ifbispositive andareal,thevallleformerly defined (v.33)asthe power baISnowtheprwnpal valueofth"power. 7.Sinlllarly, thevaluesdefined It12:UjforeZandpZ,(1':>0),arenow, morepr(,clsely, theprtnclpal valuesofthesepowers Inthemselvls, the~eSylll­ bol~wouldrepresent, forcomplex vallles ofz,an1I1[lnlty ofvalUeS, IIIac­ cordance Withour la~tdefJl1Ition. Nevertheles" weshallAeep<nfuturetothe conventlOll thate",andr:enerally pzforallYpo.'ltLVep.shallrepresent thevalue defined by2:16. 1etheprincipal va/urvllly 8.Thefollowmg theorems willshowthatItISconslst('nt todefll1eb"also forb=°whenm(a):>O.Thevaluealtnbuted tothepo"erInth,lt ca~1'IS0 (uniquely). Aftermakmg theseprellmlllary preparatIOns, wcproceed toprove thefollowmg far-reachlllg 241>. Theorem35•Foranycomplex exponent IXandanycomplex ztn IzI<1,thebinomial series n~~(:)z"==1+(7)z+(~)z~+...+(:)z"+... converges andhastorsumtheprincipal valueottltepower (1+zt. Proof Theconvergence follows wordforwordasinthecase ofrealzo'sandu's(vpp2()!'l-210),sothatWl'hall'onlytoprovethe statement astothesumofthesenes.Nowforrealx'ssuchthatIxI<1, andreala's,wcmay sub~tttute 00(_l)n-l" IXZ----x=IXlog(1+x) n=1 n 35-II,t!1.fdrelnt' 11anj!pw\lnth,\.I,p.311]826. §55.Theelementary analytic functions. -VIII.Thebinomial series.425 y~foryIntheexponential series ell=1+y+27+...andsoobtain, afterlearranging inpowers ofx(allowed by104-),thepowerseries fore"lngtl+:rI=(l+x)", i.e.thebinomial series.27(:)x". Letus proceed InthIsmanner, purelyformally Inthefirstinstance, assuming IX complex amiwntmg zforx;i.e.wesubstttute "',(_1)n-l B W=et·.L; ~----z m n=l 11 andrearrange inpowers ofz.vVenecessarily obtam-without refer­ enceasyettoanyquestton ofconvergence -theseries n~:(:)ZB, whosesumwouldtherdore beproved tobee"lng(1+z) =(1+z)"(where thelJfll1upal valueIStakenfortheIoganthm alldhenceforthepower also),ifwecouldshowthattherearrangement carried OLltwasper­ mIssIble. Nowby104-thiS l~cert,lll1ly so;IIItlcttlwexponential (1)"-1senesconverges everywhere andthesenesa.2,'--=----Znremainsn convergent forIzI<1when etandallthetermsofthesenesare replaced bytheirabsolute values. '1."hlSprovesthetheorem inItsfull extent. Ifwesplitup(1+z)exintoItsrealandimaginary parts,weobtain aformula duetoAbet,whichiscomphcatecl mappearance, butwhIch forthatveryreason showshowfar-rea,hingaresultiscontall1ed in thepreceding theorem, amifromwlllchwealsoobtainameans for evaluating thepower (1+zt.Writing z='Y(cos'p+ism(fJ)and a=fJ+ir.0<,.<1,'T',fJ,yallreal,andwnting 1+z=R(cosep+isint/J), wehave R=)/1+2'Ycosif'+,.2, r[J=principal value 36oftan-1 -,,-smj~-.1+rcosQ WiththesevaluesofRand (/).wethusobtain (1+z)"=e(P+y.)[lngRH'1>J =RfI.e-YrJ)•[cos(fJep+rlogR)+isin(fJ(/)+rlogR)]. ForthecaseIzI<1,theorem24Sandtheremark justmade completely answer thequestion astothesumofthebInomial senes. Wehavenowonlytoconsider thepointsofthecircumference Iz1=1. FromAbel'stheorem, together withthecontinuity oftheprincipal value oflog(1+z)foreveryz9='---1inIzI<1andthecontinUlty ufthe exponential function, weatoncededuce the . n n 86~hasaccordingly tobechosen between+2and--2' 426 Chapter XII.Seriesofcomplex terms. 246. 247.Theorem. AteverypointoftherimIzI=1oftheunitcircle, atwhichthebinomial seriescontinues toconverge, exceptpossibly for z=-1,itssumremains,wwaspreviously theprincipal valueof (1+zt. Thedetermination whether, andforwhatvaluesofIXandz,the binomial seriescontinues toconverge ontherimoftheunitcircle presents nodifficulties afterthepreparations madeinthisrespect(and chieflyforthispurpose) in§53.Thetheorem wehaveisthefollowing, whichsumsuptheentirequestion oncemore: Theorem. Thebinomial seriesi:(CC)z"reduces, forrealinteg- n=On ralvaluesofIX>0,toafinitesum,andhasthenthe(ipsofacto unique)value(1+zt;inparticular forIX=0ithasthevalue1(also whenz=-1).Ifadoesnothaveoneofthesevalues,theseriescon­ vergesabsolutely forIzI<1anddiverges forIzI>1,whileitexhibits thefollowing behaviour onthecircumferenceIzI=1: a)ifm(a)>O.itconverges absolutely atallpointsonthecircum­ ference; b)ifm(IX)<-1,itdivergesatallthesepoints; c)if-1<m(IX)<0,itdiverges atz=-1andconverges con­ ditionally ateveryotherpointofthecircumference. Thesumoftheserieswhenitconverges isinvariably theprincipal valueof(1+zt;inparticular, itsvalueis0inthecasez=-1. Proof. Writing(-It(:)=a,,+l' wehave ""+1_(:)_n-(a+l)_1 cc+I.a:----(CC-)-n---n-' n-I hencetheorem229maybeapplied, andthevalidity ofa),b)andc) followsimmediately. Onlythecaseofthepointz= -1,i.e.theCOIl­ verg~nce oftheseries requires specialinvestigation. Now (a)(CC\ a(a-I) ()( CC)1-1+2)=1-a+-(.-2-= I-a1-2, (a)(CC)(cc\ (a) cc(a-1)(a-2)1-1+2--3)=(1-IX)1-2-"3--1:2-- =(1-a)(1- ~H1- ~l. §55.Theelementary analytic functions. -IV.Thebinomial series. 427 andingeneral, asmayatoncebeverified byinduction: 1-(~)-+-(;)---1-.••-+-(-1)"(:)=(1-ex)(1- ~-)...(1-:); thepartialsumsofourseriesarcequaltothepaltIalproducts, with thesameindexn,ofthcproductiT(1----,=-).Thebehaviour ofthis n=l n product isimmediately evident. Infact 1.Ifm(ex)={J>0,choose {J'suchthat0<{J'<{J;forevery 5ufficlClltly largen,sayn;::::'m, hence By126,2,ItfuJuws atoncethatthepartial lJroLl~cts, a/Idhencethe partialsumsofourseries,tendtoO.l'heserzestherejore converges 37 tothesumO. 2.If,however, R(ex)=-{J<0,wehave 11-~1>1-+-~, whence itagainfollows bymultIplIcatIOn that I(1-I)(1-i)'..(1-~)I> (1+V(1+g)...(1+~), andhencethatthelefthandSidetendsto00.Thesenestheretore diverges inthiscase. 3.If,finally,m(Ot)=0,Ot=-iy,say,withy::;0,thenthpartial slimofourseriesis (1+ir)(1+!J-) ...(1+J:). ThefactthatthisvaluetendstonolImitasn--..-+-00maybeproved mostspeedily inthepre~cntconnection asfollows: Onaccount oftheab· soluteconvergence oftheserics2Jc::r, wehave,by§29,theorem 10, (1-+-ii)(1+If)...(1-+-i;:)__e'1'(1+++"'+~). Lettingn--..-+-00,therighthandsideevidently tendstonolimit;on thecontrary, thepointswhichitrepresents forsuccessive valuesofn circulate incessantly roundthecircumference oftheunitcircleina constant sense,theintervalbetween successive pointsbecoming smaller 37Themereconvergence ofS(-1)"(:)follows already from228and weseethattheconvergence isabsolute whenlR(0:)>O.Itisthefactofthe sumbeing0whichrequires theartifice employed above foritsdetectIon. ·128 Chapter XII.Seriesofcomplex terms. andsmaller ateachturn.[nviewofthea!>ymptotic relationship, thcsameisthereforc trueofthelefthand"ide.Henceourseries 2(-1)"(:) alsodiverge!> whenm(a)=O. Thustheorem 247 IS established Inallitsparts,thebehaviour ofthebmOlmal seriesisde­ termined foreveryvalueofzandufa,anditssumforallpoints ofconvcrgence isgivenbymeansofa"clo~ed expression". §56.Seriesofvariable terms. Uniform convergence. JVeler.'4tl'a~.'j· theorem ondoubleseries. Thefundamental remarks onsenesofvanable terms ifn(z) n=O aresubstantially thesameforthecomplex asfortherealdomain (v.§46);butIIlstead ofthecommon mterval ofdefinitIOn wemust nowassumc Clcommon regionofdefimtion, whichforslll1pliclty ­ thisisalsoquitesuffiCIent formostpurposes -weshallsuppose to beacircle(cf.p.403, footnote 17).\Veaccordingly assume that 1.AcircleIZ-ZoI<rexists,inwhichthefltnctwnsfn(z)are alldefined. 2.Foreveryindividual zintheclrcleIz-ZoI<r,theseries ifn(z) n=O isconvergent. ThescnesXfn(z)thcllhas,foreveryzintheCircle, Cldefil11te sum,whosevaluetherefore defines afUllction ofz(inthesenseof thedefiniuon onp.403). Weaccordlllgly write .ffn(z)=F(z). n=O Thesameproblems asthosediscussed in§§46and47forthe caseofrealvanables allse 1Ilconnection withthefunctions represented bycomplex senesofvariable terms. Intherealdomain, however, itisofthegreatest importance, bothforthetheoryanditsappli­ cations, tomakeuseoftheconcept offunction inItsmostgeneral form,whileinthecomplex domain thIShasnotbeenfou:ldprofitable. Theusualrestriction, which lSsufficiently wideforallordinary pur­ poses,istoconsid~r analytic functions only.Wetherefore assume further that 3.Thefunctionsfn(z)areallanalytic inthecircleIz-Zo1<r, i.e.expressible bypowerserieswithZoascentreandradiusnot les.~ thanafixednumber r. §56.Seriesofvariable terms. 429 Wethenspeakforbrevity ofs,'ri,:sofanalytic functions 38; thechief prob~em concerning suchaseriesisthefollowing: Isthe function F(z)whichitrepresents analytic inthecircleIz-ZoI<", ornot?Precisely asintherealdomain, itmaybeshownbyexamples thatwithout furtherassumptions thisneednotbethecase.Onthe otherhand,thedesiredbehaviour ofF(z)maybeensured bystipul­ ating(cf.§47,firstparagraph) thattheseriesconverges uniformly. Thedefinition forthisisalmostwordforwordarepetition of191: Definition (2ndform39).AseriesIf..(z),all0/whosetermsare248. definedinthecircleIZ-ZoI<rorinthecircleIZ-ZoI<1',and whichconverges inIhiscircle,issaidtoconverge uniformly inthis circleif,forevery E>0,itispossible tochooseasinglenumbe, N>0(independent, therefore, ofz)suchthat If"+l(Z)+f"H(Z)+ ..·1=Irn(z)t<I! foreveryn>Nandevery Zinthecircleconsidered. Remarks. 1.Uniformity ofconvergence ishereconsidered relative toallthepointsof anopenorclosedcircle'". ofcourseothertypesofrl'glonormdeedarcs ofcurvesoranyotherset ~JIofpOints, notmerely finiteInnl/mber, maybe takenasabasisforthedeflnltlOn. ThedefInItIOn remains thesameInsub­ ~tance. -Inapplications, weshallusually beconcerned withthecaseIn whichtheterm~f..(z)firedefined, andthesene~ ~.:f.(z)converges, atevery pointinterior toacircleIz-ZoI<r(oradomain (I)),bllttheconvergcnce i~uniform onlyinasmallerCircleIz-Zo1<e,"hereIt<Y,(orinasmalleysub­ domam (,1)"whIch,together WIthItsboundary, belongs tothemterior ofllj) 2Ifthepowersenes ~·a..(z-zo)" hastheradlllsr,andO<e<Y, theseries ISUnIformly convergent Inthe(closed) circleI;;-ZoI:;:;I!'Proofwordfor wordasonp.33;~. 3.IfrIStheexactradIUSofconvergence of~a..(z-zo)',theconver­ genceisnotn...ce~sanly uniform inthecircleIz-Zo1<r.Example thegeo­ metncsenes,proofonp333. 4.Exactly asbefolc,wemayverifythatourdefinition iscompletely cquivalenttothe101l0wlOg: a8Hereagainwemayremark (cf.190,4) thatthereisnosubstantial difference between thetreatment ofseyiesofvariable tennsandthatofsequences offunctIOns AseYles ~f.(z)isequivalent tothesequence ofitspnrti,,1sums So(z),SI(z),.." -andasequence offunctions s.(z)ISequivalent 10thesenes '~o(z)+(SI(z)-So(z))+.".Forslluphcity, weshallhereafter forlllulate all ddmitlOns andtheorems forseriesalone;thestudent willeaSilybeableto enunCIate themforsequences. a.ThISdefinItIOn corresponds totheformer 2Ddform.The10\form191 mayherebeomItted, asitdidnotappear essentiul fortheappllcallon of theconcept ofUniform convergencc, hutonlyforitsIntroducllon. 4]Theset01pointsofacirclc(or,forshort,theCIlcleitself)issaidto beclosedoropenaccording usthepointsofthecircumference areregarded asincluded inthesetornot. 430 Chapter XII.Seriesofcomplex terms. 249.srdform.2:f"(z)issaidtobeuniformly convergent inI:-zoI~(!(orIn ,heset9JI).If.foreverychoiceofpOints 11"belongIng tothIscircle(orset).the corresponding remamdel's r"(IIn)t1lw(lY~ formanullsequence. The4thand5t•formsofthedefinition (p.335)alsoremainentirely un alteredandwemaydispense withaspecialstatement ofthemhere. Ontheotherhand,itisimpos!>lble togiveasimpressive ageometrical representation ofuniform andnon-uniform convergence ofaseriesasinthe realdomain. Wearenowinaposition toformulate andprovethetheorem announced. JYt,,'iersi1'ass' theorem ondoubleseries 41.Wesuppose givena series 1;fk(z) k=O each01whosetermsfk(z)isanalytic atleastlorIZ-ZoI<r,sothat theexpansions 42 fo(z)=ao(0)+al(0)(z-zo)++a,,(O)(z-zo)"+... fl(z)=ao(l)+atCl)(z-zo)+ +allCl)(z-zor'+,.. allexistandconverge atleastlorIz-ZoI<r.Further. weassume thattheseries.J:fk(z)converges unilormly inthecircleIz-ZoI<e, loreveryf!<r,sothattheseriesconverges, inparticular, everywhere withinthecircleIz-ZoI<r,andrepresents adelinite lunction F(z) there.Itmaythenbeshownthat. 1.Thecoel/icients inaverticalcolumnlormaconvergent series: ~a(k)=A(I.d 0 1 2 )£.J" "txen,= , , , .... k=O 2.i;An(z-zotconverges lorIz-ZoI<7. n=O 3,ForIz-ZoI<r,thelunction F(z)=1;fk(z) k=O ISagama1!-alytic, with F(z)=1;An(z-ZO),I. n=O 41Werke, Vol.1,p.70.Theproofdatesfromtheyear1841• ..Theupperindex,inthecoefficient a,,(1t),indicates theplaceoccupied inthegivenseriesbythecorresponding functIOn, whilethelowerindex relate~ tothepositIOn, intheexpansion ofthisfunction, ofthetermtowhichthf,0 _6.6.=_=__"'-L_'_ §56.Seriesofvariable terms. 4.ForIz-ZoI<randforevery(fixed)v=1,2,..••431 (a)<Xl F(v)(z)=Ef,/v)(z) k-O i.e.thesuccessive derivedfunctions ofF(z)maybeobtained byterm-by-term differentiation ofthegivenseries,andeachofthenewseriesconverges uniformly ineverycircleIz-ZoI<e,withe<r. Remarks. 1.Ifwedirectourattention primarily toexpansions inpowerseries,the theorem simplystatesthatwiththeassumptions detatled above,anltlfinitenumber ofpowerseries"may"beaddedtermbyterm.Ifontheotherhandwelookrather attheanalytic character ofthevarious functions, wehavethefollowing Theorem. Ifeachofthefunctionsf"(z)isregularforIz-ZoI<randthe series1:f"(z)converges uniformly inIz-ZoI~e,foreverye<r,thenthisseries represents ananalytic functIOn F(z),regularinthecircleIz-ZoI<r.Thesucces­ sivederivedfunctions F(v)(z)ofF(z),foreveryv~I,arerepresented, inthatcircle, bytheserie<1:f,,(v)(z),obtained from1:f"(z)bydifferentiating termbyterm,vtimes insuccession. Eachoftheseseriesconverges uniformly ineverycircleIz-ZoI~e, withe<r. 2.Theassumption that1:/"(z)converges InIz-ZoISeforeverye<T issatisfied, forinstance, byeverypowerseries1:c"(z-Zo)"WIthradIUsofcon­ ~"vergence r.Itisalsosatisfied e.g.bythesenesEC~zkforr=1;cf.§'38,C. 3.Thefirstofourfourstatements showsthatthepresent theorem cannot beprovedsimply asanapphcation ofMarkofJ's transformation ofseries; forthe latterassumes theconvergence ofthecolumns, -herethISisdeduced fromthe otherhypotheses. Proof. 1.Letanindexm,apositivee<randane:>0bechosen tobekeptfixedthroughout. Byhypothesis, wecandetermine aI~osuch that,throughoutIz-ZoI:S:::e, Is",-s"I<e:'=e:•em• foreveryksuchthatk'>k>ko,ifwewrite s"=s"(z)=fa(z)+...+f"(z). Nowthefunction Sk'(z)-Sk(z)isadcfinite powersencs,whosemtb coefficient is a~:+1)+a~:+2)+...+a~:'). ByCauchy's inequality 235,wetherefore havc Iark+1)+ark+2)+...+aW)I:S:::-I!;~=e. m m 111 -e'"' Hencetheseries <Xl a~)-+a~,:)+...+a~:)+...=::Ea~:p k-O isconvergent, by81.LetAll,beitssum.Asmcouldbcchosenarbitrarily, thefirstofourstatements isthusestablished. 432 ChapterXII.Seriesofcomplex terms. ..,2.NowletM'hethemaximum 43ofISic.1-1(z)Ialongthecircum· ferenccI;;;--ZoI=e.\Vchavethenforeveryk:>flOonthesamecir­ cumference ISk(z)I<ISk.+l(z)I+ISic(z)-Sk.+I(z)I<M'+E'=M. Again,usingCauchy's inequality, weobtain,foreveryn--,0,I,2,. \an(O)+an(l)+...+an(klI~~,e whatever thevalueofk.Hence IAnl~~,e andJ;An(z-zo)ntherefore converges forIz-ZoI<e.Sincetheonly n~lJ restriction onewasthatitshouldbe<r,theseriesmustevenconverge forIz-ZoI<r.(Infact,ifzisanydeterminate pointsatisfying the inequa~ltyIZ-ZoI<T,itisalwayspossible toassumeetobechosen sothatIz-ZoI<e<T.)Letusforthemoment denotebyPI(z)the function represented bytheseriesEAn(z-zo)n;itisthus,byitsdefini­ tion,ananalytic function inIz--ZoI<r. 3.WehavenowtoshowthatFI(z) =-0F(z),sothatF(z)isitself ananalytic function regularinIz-ZoI<r.Forthispurpose, wechoose, asinthefirstpartofourproof,apositivee'<r,apositiveeinr/<P.<r, andanE>0,fixed.Wecandetermine kosothat,forallzinIz--ZoI<:::::e, e-e'Is,:-SicI<E'=E'-e- foreveryksuchthatk'>k>ko'ByCauchy's inequality, itfollowsas beforethat,fork'>k>koandjoreveryn?;0, Ia(k+l)+a(k+2)+...+a(k')I<~'n n n en' Makingk'-++00,weinferthat,foreveryk>koandeveryn;:.-;;0, IA-(a(0)+a(1)+...+a(k»I~£.nnn n--en Nowtheexpression between themodulus signsisthenthcoefficient in k theexpansion ofFI(z)-Ej..(z)inpowersof(z-zo).Hencewehave, v~O forIz-ZoI<e: IFI(z)-Ejv(z)I~E'•[1+Iz-ZoI+Iz-2z,,-l~+..'J. v-o e e Therighthandsideis,forIz-ZoI<:::::e', ~E'•[1+~+(~')2+..J=e'.e!-e'=e. asISk.+1(z)Iisacontinuous function ofamz='llalongthecircumference in que~tlOn and(epbeingreal)attainsadefimte m'lXlmUm onthiscircumference. §56.Seriesofvariable terms. 433 Thus,when 8>0ande'<e<rhavebeenchosen arbitrarily, wc candetermine kosothat k IFI(z)-.2f"(z)I<8 ..=0 F1(z)=if,,(z),1.e.=F(z). ,,=0 Thenumberse'andeweresubjected tonorestriction otherthan o<e'<e<r;hence(asabove)itfollows thattheequation holds foreveryzmtenor tothecircle 1z-ZoI<r. 4.\Vewrite fo'(z)-=a1(0)+2a~CO)(z~-zo)+3a3(0)(z-zo)'J+ . fI'(z)~a1(1)+2a2(1)(z--zo)+:3a3(l)(z-ZO)2+ .foreveryk>koandeveryIz-ZoI<e'·Thisimplies, however, thatfor thesevaluesofz A1+2A'.l (z-zo)+3A a(Z-ZO)2+ ..., wherethesumofthecoefficients inanyone column converges tothe valuewritten immediately belowthem.Justasin3.{wehaveonly t?beginourevaluations witheo'=(e~e'r.e)wededuce thatfor Iz-Zo:~e'<e<randeveryk>ko, IF'(z)-v~!:(z)I~e'[1+2~:+3~':+...J-=e'.(e~~"-e')2=e. 00 Henceforthosevaluesofz,F'(z)=E.h'(z).Indeed, bythesame k-0 reasoning asbefore,thisseriesconverges uniformly inIz-ZoI<r,for everye<r.Ifwewritedownthecorresponding systemofseriesforthe vthderived functions, weobtain,inthesamemanner: 00 F(")(z)=ENv)(z)(v=1,2,...,fixed) kO foreveryIz-ZoI<r;i.e.theseriesEI",(v)(z)obtained bydiffer­ entiating termbyterm,vtimesinsuccession, converges inthewholecircle Iz-~ZoI<r(andconverges uniformly ineverycircleIz-ZoI:S:e<r) andgivesthevthderived function ofF(z)there. Remarks. 1.Afewexamples ofparticular importance willbediscussed indetailin thenextsectionbutone. 2.Thefactofassuming theconvergence uniform inacircular domain IS immaterial forthemostessential partofthetheorem:IfGISadomain ofarbitrary shape" andIfeverypointZoofthedomain isthecentreofaCircleIz-ZoI~e (forsomee)whichbelongs entirely tothedomain, issuchthateachtermoftheseriesIf"(z)isanalytIc there.andisacircleofuniform convergence ofthegivenseries, thenthissenesalsorepresents afunction F(z)analytic inthedomain inquestion, whosederivedfunctions maybeobtained bydifferentiatIOn temlbyterm.-Examples ofthISwillalsobegivenIn§08. UCLp.403,footnote 17. '34 Chapter Xll.Seriesofcomplex term.. §57.Products withcomplex terms. Thedevelopments ofChapter VIIwereconducted insuchaway thatalldefinitions andtheorems relating toproducts with"arbitrary" termsholdwithoutalteration whenweadmitcomplex valuesforthe factors. Inparticular thedefinition ofconvergence 12:Jandthetheo­ rems1,2and5connected withit,aswellastheproofsofthelatter, remainentirelyunchanged. Thereisalsonothingtomodifyin127,the definilion ofabsolute convergence, andtherelatedtheorems 6and7. Ontheotherhand,somedoubtmightariseastotheliteraltrans­ ference oftheorem 8tothecomplex domain. Hereagain,however, everything maybeinterpreted as"complex", provided weagreeto takelog(1+a,,)tomeantheprincipal valteeofthelogarithm, forevery sufficiently largen.Thereasoning requires care,andweshalltherefore carryouttheproofinfull: 2C'iO. Theorem. TheproductlI(l+an)converges if,andonlyif.the series,startingwithasuitableindexm, .2log(1+a,,), ,,=m+1 ,,;hosetermsaretheprincipal valuesoflog(1+an),converges. IfLmis thesumofthisseries,wehave,moreover, if; lI(l+an)=(1+a1)(1+a2)•••(1+am).eLm. n-I Proof. a)Theconditions aresufficient. Forifthesenes i;log(1+an)'withtheprincipal valuesofthelogarithms, iscon· ,,~m+1 vergent, itspartialsumssn'(n>m),tendtoadefinite limItL,and consequently, sincetheexponential function iscontinous atevery point, e6"=(1+am+l)(1+amH)..,(1+an)-eL i.e.itcertainly tendstoavalue+O.Hencetheproduct iscon· vergentinaccordance withthedefinition 12riandhasthevalue stated. b)Theconditions arenecessary. For,iftheproduct converges, givenapositive e.whichwemayassume<1,wecandetermine no sothat (a) 1(1+0,,+1)(1+a"+2)'..(1+a"+k)-11<; foreveryn~noandeveryk21.Wethenhave,inparticular, IanI<;<-~-foreveryn:>no'andthemequalityIanI<-~.isthus certainly fulfilled foreveryngreaterthanacertainindexm.Wemay now!;hnwfnrthprthatfnrthp.!;ampvalup.!;oftt.and11(usinp-the §57.Products withcomplex terms. 435 (b)principal valuesofthelogarithms) 45 I"IkI"!-:.llog(1+a.)<e: andtherefore theseries1:log(1+an)isconvergent. Infact,asIa.1<~2 1J=m;-1 foreveryv>no,wealsohave46,forthesevaluesofv, (c) Ilog(l+a v)I<e:, andlikewise, by(a), Ilog[(1+an+I)••.(1+an+7,)]I<e: foreveryn~noandeveryk~1.Accordingly, forsomesuitable integer 47 q,wecertainly have Ilog(1+an+I)+log(1+a1l1-2)+...+log(1+an+k)+2q7TiI<e:, anditonlyremains toshowthatqmayineveryeasebetaken=O.Now ifwetakeanyparticular n~no,thisiscertainly truefork=1,by(c). Itfollowst!1atitistruefork=2.Forintheexpression log(1+an+1)+log(1-I-ant2)+-2q7Ti themodulus ofeitherofthetwofirstterms<e:,by(c),andby(d)the modulus ofthewholeexpression hastobe<e:;ase:<1,qcannot,there­ fore,beanintegerdifferent from0.Forcorresponding reasons, italso followsthatforh=3theintegerqmustbe0,andthisistheneasilyseen byinduction tobetrueforeveryk.Thisestablishes thetheorem. Thepartoftheorem 127,8relating toabsolute convergence may alsobeimmediately transferred tothecomplex domain, -viz. 00 00 theseriesElog(1+an)andtheproduct11(1+an) n=m-l-l '11-mi1 aresimultaneously absolutely or/lon-absolutely convergent, in Similarly thetheorems !J-llof§§29and30remain valid. remains trueforcomplex an'sofmodulus<~thatin lag(l+an)=an+&nan2everycase. Infact,it ••Thelogarithms arealwaystakentohavetheirprincipal 'Valuesinwhat follows. 1 46Infact,forIzI<2' Ilog(1+z)I;?:IzI+L~I"+...;?:IzI+Iz"+...=1~'llzf-s:2IzI. 47Fortheprincipal valueofthelogarIthm ofaproduct isnotnecessarily thesumoftheprIncipal valuesofthelogarithms ofthefactors,butmaydifferfrom thissumbyamultiple of2'1Ti.Thuse.g.logi="'i,but log(i.i.i.i)=log1=0, ifwetakeprincipal valuesthroughout. 436 Chapter )Ul,Series0/complex terms. (I-z)(I+z)(1+z")(I+z4)...(1+z~n)=I_z2n+1 ,Ithequantities {}narebounded, -sllJcewhenIzI<~ log(1+z)=z+[- -~+-i-~~+-...J.z~, whiletheexpression insquarebrackets clearly ha~itsmodulus<1 forthosez's. Finally, theremarks onthegeneral connection between series andproducts alsoholdwithoutalteration, sincetheywerepurelyformal IIIcharacter. 251. Examples. 1.II(1+~)isdivergent. ForJ:Ian12==.2~Iiisconvergent, sothat by§29,theorem 10,thepartialproducts ~) ~) ( ~) l(1+.!.+...+.!.) PlC=(1+T(1+'2.., 1+11""'"(J 2 n; therighthandexpression represents, forsucceS~lve valuesofn,pointsonthe circumference oftheunitcircle,whichcirculate incessantly round tillScircum. ference atshorter andshorter intervals. Pntherefore tendstonolimiting value.(Cf.pp.427-8.) 2.j\'in(n+1.)+(1._+l)= _1.Infact,htb.IJ. tenpartlaproduct isatoncen=on(n+ I)+(I-,) 1+(n+I)~ .seentobe ,which -+-1.1-(n+I)l loo,(9ft)1 f8.ForIzI<1, 1+z-=1-z.Inacttheabsolute) convergence n=O ofthisproduct isobvious by127,7and it~nthpartialproducl lIIultiplied by (I-z)is whichtendsto1. TheconsideratIOn complex variable,ofproducts whosetermsarefunctions ofa if(1+fn(z)), 11=1 --likethatofseriesofvariable termsinthepreceding section, -­ willberestricted tuthesimplest, butalsothemostImportant case, inwhichthefunctions In(z)areallanalytic inoneandthesamecircleIz-ZoI<,(i.e.possessanexpansion inpowerseriescallicrgentin thatcircle)andinwhichtheproduct alsoconverges everywhere in thecircle.Theproduct thenrepresents adefinite function F(z)in thecircle,whichissaid,conversely, tobeexpanded inthegiven broduct. .Wenextenquire underwhatconvenient conditions thefunction F(z)represented bytheproduct isalsoanalytic inthecircleIz-ZoI<,.Forthegreatmajority ofapplications, thefollowing theorem issufficient: §57.Products withcomplex terms. 437 Theorem. Iftliefunctionsfl(z),f2(z),...,in(z),...arfallanalytie 2~2. atleastinthe(fixed)circleIz-ZoI<r;If,further, theseries converges uniformly inthesmallercircleIz-ZoI~e,fore1)eryPOSItIve (!<r;thentheproductII(1+fn(z»converges everywhere inIz-ZoI<r andYl'presents afunctionF(z)u'hichisitselfanalytic inthatcircle. Theprooffollows thesamelineofargument asthatofthe continuity theorem 218,1almostwordforword.Toestablish thecon­ vergence andanalytic character oftheproduct ataparticular pointZI inthecircleIz-ZoI<r,wechooseae<randprovcthetwofacts firstforeveryzofthecircleIz-ZoI<e.Theseries};It~(z)Iconverges uniformly inthewholeofIz-ZoI;;:::e,sothattheproductII(l+fn(z» certainly converges there(indeedabsolutely). Choosemsolargethat foreveryn>mandeveryIz-ZoI~e;thenforallthesen'sandz's, Itfollowsprecisely asonp.382thattheseries Pmt-l+(Pm+2-Pm+l)+...+(Pn-Pn-t)+... converges un~rormly inIz-ZoI~e.Asallthetermsofthisseriesare malytic inIz-ZoI<r,theseriesitself,by249,therefore represents afunction Fm(z)analytic 111Iz-.0'0I<e.Hence '"F(z)=II(1+f"(z»=(1+fl(z»...(1+fm(z».Fm(Z) 11-1 isalsoananalyticfunction, regularinthatcircle. Fromtheaboveconsiderations, wemaydeducetwofurthertheorems, whichprovideananalogue toWeierstrass' theorem ondoubleseries: Theorem 1.Withtheassumptions ofthepreceding theorem, theex-253. pansioninpowerseriesofF(z)maybeobtained byexpanding theproduct termbyterm.Moreprecisely, weknowthatthe(finite)product k Pk(z)=II(1+j.,(z» ....=1 maybeexpanded inapowerseriesofcentreZowhirhconverges for Iz-ZoI<r,sincethisisthecasewitheachofthefunctions fl'f2'.. 15 (051) 438 Chapter XII.Seriesofcomplex terms, Lettheexpansion be P,,(z)=A~")+Afk'(z-zo)+AJkI(z-Zo,2+."+A~")(Z-2'0)"+,.", Thenforeach(fixed)n=0,1,2,","'thelimit limA~A)=An k....+«> exists,and F(z)=H(l+f,,(z)=iA,,(z-zo)"· k=I n=O Proof. By§46,theorem 2,theuniform convergence, ID 1%-ZoI~e,ofthesenes Pm+1+(PmH-Pm+!)+.... usedinthepreceding proof,implies theuniform convergence inthe samecircleoftheseries 48 PI(z)+[P2(Z)-PI(Z)]+...+[Pk(Z)-P/'-I(z)]+.,', Applying Weierstrass' theorem ondoublesenestothisseries,we obtainprecisely thetheorem stated. Finallyweproveatheorem aboutthederived function ofF(z), quitesimilar to218,2: Theorem 2.ForeveryzinIz-ZoI<rforwhz'chF(z)=1=0, wehave 1<"(z)_1:1',:(z) F(z)-..=11+/~d;::) I i.e.theseriesontherighthandsideconverges foralithesevalues ofzandgivestheratioonthelefthandside,thelogarithmic dif· ferential coefficient ofF(z), Proof. Wesawthattheexpansion F(z)=PI(z)+(P2(z)-PI(z»+... wasuniformly convergent inIz-ZoI<e<r.By249, F'(z)=P;(z)+(P;(z)-P;(z)+,.., whichimplies that P~(z)-F'(z) ateverypointinthecircle.Ifataparticular pointF(z)+0,we havePn(z)9=0foreachn,andhenceby41,11, 1',.'(z)F'(z) P,.(z)--+F(z)-" csFortheremainders ofthelattersc.riesonlydifferfromthoseof theformer inthattheycontain thecommon factorP".(z),whichisacon· tinuous function foreveryzinthecircleIz-ZoI~(bandhenceisbound<.'u inthisclosedcircle. Since,however,~57.Products withcomplex terms. Po'(I),.f/(z)--L}---Pn(I)-V=11+fv(Z), thisisprecisely whatourtheorem asserts.439 00 Aa=L}a,.a"a"1.,<).,<1.. ' , •Examples. 1.If~'allisanyabsolutely convergent seriesofcon~tant terms,theproduct254. 00 J[(1+anz) n=1 reprc~cnts afunction regular inthewholeplane,by252. By253,itsex· pansion, i,!powerseries,whichisconvergent everywherc, is 1+A,I+A.z·+A3Z3+..•+AkZk+... with 00 Al=.2a,.a.,1.,<1., 1, 00 ...,Ak=2,'a•.a....a.,'1.,<...<1., I, Heretheindlccs ).1')..,••.,)./,independently takefortheirvaluesallthenatural numbers, subject onlytothecondition .I.,<.I..<...<J.k'ThecXlstence of thesumsAllA.,...issecured bytheorem253itselfiitisalsoea~ytoverify thattheyaremdependent oftheorderoftheterms.-Itwasbyarplying thIStheorem thatEuler 49andlaterC.G.J.Jacobl bOwereledtoanabundance ofmostremarkable formulae. 2.Wehave 00(%2)sin7r:z=7r:z,][ 1--.,, ..=1n" wheretheprc>duct ontherig-hthandsillcconverges inthewholeplaneThe proofiswordforwordthesamensthatgivenin219,1forarealvariable. 3.Taking z=iintheabovesineproduct, weobtam 00(1 ) e-;r;_en"i/l.1+11""2=sinnz=-2J- or jj(1+-;-)=_e~_-;;e-n • n==1 n :..JlC (Cf.however theextremely easyevaluation ofII(1-~.) 4.Thesequence offunctions ()z(z+I)(z+2)",(z+1I)gnz= ,nlnz converges foreveryzinthewholeplane.-Infactin128,6). n=1,~,•.•• gn(z)=Z(1+~)(1+~)...(1+:).n-z; by127,theorem 10, (1+~)(1+~_)...(1+:)'"eZ(1+++" .+~)i '0Introductio innnalysin inr.Vo!.1,Chap.15.1748. 5QFundamenta nova,K6nigsberg 1829. 440 Chapter XII.Series 01complex terms. 1 g,.(z)also,by128,2,thenl1mber~ i'n=(1+-}+...+~)-log110tend,asn-++00, toEuler's constant C,sothattherighthandexpression, -whichis ez(1ogn+Yn) =nZeYnz,_ whendivided bynZ,tendstoadefinite limitas110-++CO.ThISprovesthe statement. -Further, thelunit,K(z)say,becomes 0onlyforz=0,-1, 2,••••Excluding thesevalues, wehave,forallothervaluesofz, 1 n'nz1Hm-= 11111-------'------=- -=l'(Z). tI-++ QC)gn(z)n-++ QC)Z(z+1)(z+2)...(z+110)I<(z) Thisfunction ofacomplex variable z(restncted onlytobe4~0,-1,-2,...) ISth('so-called (;allnl/(l-j'lIlIction r(z)whichwehavealready dcfined on p.385rorrealvaluesofthe,irgument. \Veproceed to~howthatR(z)isanalytic IIIthewholeplane (I.e.an wtcgral function). ForthIS,Itsuffices toshowthattheseries R(z)=g.(z)+(gg(z)-g,(z»+...+(g..(Z)-gn_.(Z)+... converges umformly IIIeverycircle Iz1<e.Now gn(z)-gn-1(z)=g"-1(z)[(1+:)(1--t-)Z-1]; alsol\constant Aexists 51suchthatI~I'(z)I:s:Aforevery 'V~1,2,3,...and everyl::rI~e,andfurther, wemaywrite(scep.283andp.442,footnote 54) (1_~)Z=1_~+!..i:>.. 110 110 1109 "hereI{}71(z)IremalOS lessthansomeconstant Bforevery 110=2,:3,•••and 61Let1zI:::;eand110>m>2e.Then g.(z)=z(1+n···(1+,;).(1+m~1)...(1+:).110-. ( 1 1 ) 'Im+l 1/,. =Z(1+T)'"(1+~). /\m-~i+"'-1-n-logn .elm-:t-Il"+"'+",-, (Z)Z"".IzI1 wherelog 1+--;=-;+-;;;<. As-;<2(cf.p.435) wehave1",.I<izj·<c9, n' (!J..---;- andthelastfactorinthepreceding expression therefore remains<e6=Aa' foreveryIz/<eandeveryn>m. Similarly thela<;tfactorbutone(spep.2!.J5), alsoremains lessthanafixednumber Ag•AstheH'mulning- factorisalso always le'>sthanafixednumber A.foreveryIzI<e,itfollows that Ig,.(Z)I:::;A••Ag.Aaforallth('sevaluesofzandevery 110>m.Ontheother hand,thefirstInfunctions Igj(Z)I,Ig.(z)I,"',Igm(z)Ialsoremain bounded foreveryIzI:Sf!;theexistence ofthenumber Aasasserted inthetext i~ thusestablished. Ifzisrestricted tolieinacirclesr,intheinterior andontheboundar)' ot\\hichzof0,-1,-2I •••andIzI~f!,thenforevery 110>m 1 -z(m~l+··,+;'-IOgn) -(:::;'I'-"'-~--c-----:-----,----:-. e •e . Z(1++)...(1+;J Fromthisweinferinexactly thesamewaythataconstant A'existssuch that/-),-:,I<A'insr,{orevery 110=1,2,•••• §1)8.Special classes 01seriesofanalytic functions. -A.Dirichlet's series.441 everyIzI:5(!.Thusforallthesez'sandn's, Ig(z)-gn-I(z)I~A,1__~+~~~-I-~'l'}n(z)'I<_C_n - n2nil n8=n'J, whereCisasuitable constant Dy197,itfollows thatthesenesforK(z) converges uniformly inthecircleIzI~(!,-indeedthe serie~ofab,olute values ~:Ien(z)-g..-I(z)Idoesso,-and,by249,J((z)isanalytic 111the wholeplane §58.Special classesofseriesofanalytic functions. A.Diricllll't's series. ADirichlet serzesisaseriesoftheform62 iat:. ,,=1n Here:theterms-asexpollentJal functions -areanalytic inthe wholeplane.Thechiefquestion w1l1therefore betodetermine whether amIwheretheqeriesconverges and,inparticular, whether andwhere itconverges uniformly. Wehave Theorem 1.ToeveryDirichlet seriestherecorresponds areal23i5. numberA-knownastheabsdssn ofconl'el'(lence oftheseries- suchthattheseriesconverges whenm(z)>}.anddiverge~when ~)t(z)<}.. Thenumber AmayaLsobe-00or+00;intheformercase theseriesconverges everywhere, inthelatternowhere. Further, if 1++00and;:>1,theseriesisuniformly convergent inevery circleofthehalf-planem(z)~A'andaccordmgly theseries,bylVeier- strass'theorem249,r"presents aJunctlOll analyticandregular IIIevery SUCil circleandht11ceinthehalf-plane 63'11(z)>A. Theprooffollows alineofargument similartothatusedinthe caseofpowerseries(cf.93)Wetlrstshowthatiftheseriesmn· vergesatapointzo'itconverges ateveryothelpointzforwinch gt(z)>lJt(zo)'Ashowever X"'an_X"'a"1 £.J-;;;--LJ~.~z-;;' itsuffices, by1St,iJa,toshowthattheseries )11_1__----!-I ==01;__1~-..1(1+_~_)Z-Zo -11 ..-;;;1nZ-ZO(n+1)z-zon=1(n+ 1)91(z-zo) n .,Moregenerally. a~('riesiscalledaDlrlchlet berles\\henitisofthe form '\"1·:lIn.oroftheform.:Ea"e-I'n',wherethep,.'sarepOSItIVe 1llft1llJeYl ~Po' andtheAn'sanyrealnumbers Increasing monotonely to+00. bJTheexistence ofthehall-plane 01convergence wasproved by].L.IV.V. JCllsen(Tidskrift forl\Iathematlk (5),Vol.2,p.63.1884);theuniformity ofthe convergence andthereby theanalytic character' ofthefunction represented werepointed outbyE.Cahen(Annales Ec.Norm.Slip.(3),Vol.11.p.7.1.1894) 4t2 Chapter XII.Seriesofcomplex terms. i<;convergent. Writing (forafixedexponent (z-%0)) (l+~r-z·=l+~' thenumbers &n~(.3'-.3'0),asisatonceseen54;theyaretherefore cer­ tainlybounded,I&nI<A,say.Thenthtermoftheaboveseriesistherefore A<n1+91(z-zo), andtheseriesisaccordingly convergent whenm(z-zo)>O. Asacorollary, wehavethestatement: IfaDirichlet seriesh divergent atapointz=Zl'itisdivergent ateveryotherpointwhose realpartislessthanthatofZ1'Supposing thatagivenDirichlet seriesdoesnotconverge everywhere ornowhere, theexistence ofthe limiting abscissa Aisinferred (asin93)asfollows: Letz'bea pointofdivergence andz"apointofconvergence oftheseries,and choose Xo<m(z')andYo>91(z"), -bothreal.Forz=Xothe serieswilldiverge, forz=Yoitwillconverge. Nowapplythemethod ofsuccessive bisection, wordforwordasin93,totheinterval 10=Xo...Yoontherealaxis.ThevalueAsoobtained willbethe required abscissa. Nowsuppose A'>A(forA= -00,A'maytherefore beanyreal number); ifzisrestricted tolieinadomain Ginwhich 91(z)2A' andIzI<R,-sothatingeneral Gwilltaketheshapeofaseg­ mentofacircle,-ourseriesi"uniformly convergent inthatdomain. Toshowthis,letuschooseapointZoforwhichA<m(zo)<A';as before,wewrite 1 31Moregenerally, wemayatonceobserve thatifIzI~-2andIwI~R, andifwewrite,takingtheprincipal value, (I+z)W=1+zw+{}'Z2, thefactor(),whichdepends onzandw,remains lessthanafixedconstant forallthevaluesallowed forzandw.-Pro0f: 1zZ2(l+z)w=ewlog(lH)=eW(Z+'lz'), with1/=--+----+....2 3 4 ForeveryIzI<~Iwetherefore have11/I<1jhencein (• W"Z2(1+.,Z)2eWz+'lz)=1+wz(l +17z)+--21·-- + . =1+wz+[w17+w2 (1_-t_7)Z)_2.+~3j1+.?Lz)3.+J.Zll2! 3! theexpres!>ion insquare brackets, whichwasdenoted by(),satisfies thein­ equality 1001<e2R• Thisisatonceobvious ifwereplace allthequantities inthebrackets bytheir converges uniformly 1areuniformlynZ-ZO§58.Special classesofsenesofanalytic functions. -A.Dirichlet's series. 4~3 2~isaconvergent seriesofconstant terms;by198,3aitthere- n20 foresuffices toshowthat nJ:1nZ~z.-(n-t-:)Z-ZOI inthedomain inquestion andthatthefactors bounded inG.Now,writing A'-m(Zo)=t5(>0), InZ~2.-(n+:)Z-z.l<n1 "·1(1+~r-z·-11· Usingtheevaluation givenintheprecedmg footnote(orelsedirectly, by (1)Z-ZO (Z-Z.)lOIl(1+~) expanding 1+n=e ninpowers of(z-zo))wenow seethataconstant Acertainly existssuchthatthedifference within themodulus signsontherighthandsideoftheaboveinequality is inabsolute value A<-11 foreveryzinourdomain andeveryn=1,2,3,....Thewhole expression ontherightisthus A<1+;l'n Ontheotherhand,since'--1-1<_1_,thefactors1arenZ-zo -nfY nZ-2'o uniformly bounded inG.By198,3a,thisprovesthattheDirichlet seriesisuniformly convergent inthedomain stated,andhence,in particular, thateveryDirichlet seriesrepresents afunction whichis analytic IIItheIlltenor oftheregIOnofconvergence ofthesenes(the half-plane ~n(z)>A). From itfollows atoncethatIfaDirichlet seriesconverges absolutely ata pointzo'itdoessoatanypointzforwhichm(z)>m(zo)'andifit doesnotconverge absolutely atzo'thenitcannotdosoatanypoint ,forwhichm(z)<m(zo)'Justasbeforeweobtain Theorem 2.Thereexistsadelinite realnumber 1(whichmay alsobe+00or-(0)suchthattheDirichlet seriesconverges ab· solutely 101'm(z)>l,butnot101'm(z)<l. OfcoursewehaveA<l;overandabovethis,therelati"e posi· tionsofthetwostraight lineslH(z)=;.andm(z)=1issubjecttothe following 4,14 Chapter XII.Seriesofcomplex terms. Theorem 3.Wehaveineverycasel-l<1. Proof.IfZ~isconvergent andffi(z)>m(zo)+1.then nZO z::15absolutely convergent, forI::I=I:;01-;;'1~(~:::7~1 WIth m(z-zo)>1.Thisprovesthestatement atonce. Remarks andExamples. 256. 1.IfaDirichlet senesisnotmerelyeverywhere ornowhere convergent the situation WIllmgeneral beasfollows. thehalf-plane ~J/(z)<Aofdivergence of thescriesISfollowed byastripA<m(z)<Iofcondllional converl(etlce ofthesertes; thebreadthofthiSstnpISInanycaseatmost1,andintheremammg half-planem(z)>l,thescnesconverges absolutely. 2.Itmaybeshownbyeasyex,nnples thatthedifferencei-Amayassume anyvaluebetween 0and1(bolhmcluslve), andthatthebehavIOur ontheboundmg hnesiR(z)=AandHI(z)=lmayvaryindifferent cases. 3.ThetwosenesE2n-~nZandE~:provide simpleexamples ofDirichlet serieswhichconverge everywhele andnowhere. 4.1:~zhastheabscissa ofconvergence A=1;thusitrepresents ananalyticn functIOn, regularmthehalf-plane HI(z)>1.ItISknownasRlemann\ {-function (v.197,2,3)andISusedIntheanalytical theoryofnumbers, onaccount ofItS connection \\Iththedlstnbution ofprunenumbers (scebelow,Rem.9)5•• 5.JustastheradlU"ofapowerscnescanbededuced directly fromItsco­ effiCients (theorem 94),sowcmaymferfromthecoefficients ofaglvcnD1Tlchlet scricswhatpOSitIOns thetwoIlnutmg straight hnesoccupy. \Vehavethefollowmg Theorem. Theabscissaofconvergence AoftheDmchlet seriesEa~isinvariably givenbytheformula n A=lim1log!a"+l+aU+I+...+avIx-?+oc x wherexincreases continuously and rertJJ=u,[e"]=v. Substill/ting anforaninthISformula, weobtainl,thelimiting abscissa0/absolute converl(ence 5•• Il.Aconciseaccountofthemostimportant resultsInthetheoryofDmchlet's senesmaybefoundinG.H.HardyandM.RICSZ,TheoryofDiriclzlet's senes, Cambndge 19111. 66Adetailed investigation ofthiSremarkable function (aswellasofarbitrary Dlrichlet series)ISgivenbyE.Landau, Handbuch derLehrevanderVerteilung deI Primzahlen, Leip.lIg 1909,2Vols.,InE.Landau, Vorlesungen uberZahlentheorie, LeipZIg 1927,3Vols.,andInE.C.Tltchmarsh, TheZeta-Function ofRiemann, Cam­ bridge1930, 6.Asregardstheproof,wemustrcfcrtoanotebytheauthor: "Oberdie Abszisse derGrcnzgeraden emcr Dtrlchlct~chen RClhe"intheSltzungsberichte der Berliner Mathematischen Gesellschaft (Vo!.X,p.2,1910). §58.Special classesofseriesofanalytic functions.-A.Dirichlet's series.445 7.Byrepeated term-by-tenn dIfferentiatIOn ofaDITll:hlet senesF(z)-~Eai, weobtaintheDlrichlet ,cnes n (-1)"i:~J)og n)~ n.1n"(fixedv). AsanImmediate consequence ofWei"strass' theorem ondoubleseries,theseneces­ sanlycannothavealargerabscissa ofconvergence thantheonginal series,and, oWingtotheadditiotJal factors 10gVn,theycanobvIOusly nothaveasmallerone either.Theyrepresent, IntheInteriorofthehalf-plane ofconvergence, thedenved functions F(v)(z). 8.By2i)i'),thefunction represented byaDlrnhlet senescanbeexpanded Inapowerseriesaboutanypointinterior tothehalf-plane ofconvergence as centre. TheexpansIon Itselfisprovided byIVe.erstrass' theorem ondouble 001 series. If,for in~t,tllce, itisrequired toexpand thefunction I;(z)=,J;­ k=lll" abollt Zo=+2ascentre,wehaveforII=2,3,... ~=~._ ~_=~.r("-2)lngk =~i:(_I)n(log11)"(z-2)" (llfixed), kZk"kz-2kQk2..=0 n! andthiscontinues toholdfork= 1provided weinterpret (log1)°ashaving thevalue1.Henceforn~0 (_l)n00(logk)n A"=-ni-.2)kQ- (nfixed), k=l whichgivesthedesllcdexpansion 257. andtheproduct001.2)- n=ln"theser,es00(-1)"[-r"(logk)"JnQ(00100"k)l;(z)~2)--,- 2)--Q-(z-2)n :=-,-.2}-iJ- (z-2)+- .... n=On.k=1k 6k=lk 9.For!Jt(z)>1, (whereptakesforItsvaluesallthepTlmenumbers 2,3,5,7,..InsuccessIOn) haveeverywhere thesamevalue,andaccordingly bothrepresent theR,emann 1;-func­ tIOnI;(z).(Euler,1737;v.Introd.inanalysin, p.22.')) Proof.1ctzbeadefinite pointsuchthat ~t(z)=1+J>1.Byour remark 4and]27,7,theseriesandproduct certainly converl!e absolutely at thispoint.Wehaveonlytoprovethattheyhavethesamevalue. Now 1 1 1 1---=1+-+-;;-+-+ ...;1-r" p"rZpSz multiplying theseexpansions together, forallprimenllmbersp;;;;N,-whl're Ndenotes aninteger keptfixedforthemoment, -the(fu1lte) product so obtained is wheretheaccentonthe1.'indicates thatonlysome,andnotall,oftheterms oftheserieswritten downaretaken.Herewehavemadeuseoftheelemen­ taryproposition thateverynatural number ~2canbeexpressed inoneand onlyonewayasaproduct ofpowers ofdistinctprimes(provided onlypositive 15- (G51) 4-46 Chapter XII.Seriesotcomplex terms. integral e:xponents areallowed andtheorderofsuccession ofthefactorsIs leftoutofaccount). Accordingly IJl_1-z-1~,:;::;1;l\J. 'P-;;,NI-p n=lnn=N+l n Ontherighthandsidewehavetheremainder ofaconvergent series,which tendsto0whenN-++00.Thisproves theequality ofthevaluesofthe ~n/lnlte product andofthein/lnlte series,aswasrequired. 10.By21')7,wehaveform(z)>1 -!-=II(1--P-Z)=Il(1-~)=1;ft(n) C(z) p ppZn=lnZ where I-'(I)=1,I-'(2)= -1,I-'(3)= -I,I-'(4)=0,I-'(5)= -I,,.(6)=+I,••• andgenerally I-'(n)=0,+1,or- 1according as11ISdiyislble bythesquareof apnmenumber, orisaproduct ofanevennumber ofpnmes, alldIfferent, orof anoddnumber ofprimes, all,bffernll. Theproduct-expansion ofthe~-functlOn alsoshowsthatforlJl(z)>1,wealwayshll\e~(z)et-n.Thecunous coefficients I-'(n)areknownas1I1obius' coefficients. Thereisnosuperficial regulanty 10the modeofsucccssion ofthevalues0,+1,- 1amongthenumbers p.(n). 11.SinceC(z)=.2..!..converges absolutely form(z)>1,wemayform nZ thesquare(C(Z))9bymultiplying theseriesbyItselftermbytermandre­ arranging inorderofincreaslllg denominators (asisallowed by91).Wethus obtain 00TC9(z)~~.2 ~, n=ln where Tndenotes thenumber01dWlsors 01n.-Tlnseexamples maysuffice toexplain theImportance oftheC-function inproblems inthetheory of numbers. B.Faculty series. i·~ n=lnZ(F)Afacultysenes(ofthefirstkind)isaSerIesoftheform .i; n!an n=tz(z+l)...(z+n)' whichofcoursehasameaning onlyifz=1=0,-1,-2,....The questions ofconvergence, elucidated inthefirstinstance by]ensen, arecompletely solvedbythefollo\\ing 208. Theorem ofr,(/,,,[nu57.Thefacultyseries(F)converges -WIth theexclusion ofthepoints0,-1,-2,...-wherever the"asso­ ciated"Dirichlet senes converges, andconversely thelatterconverges wherever theseries(F)con­ verges.Theconvergence isuniforminaCIrcleforeitherseries,whenit issolortheother,provided thecirclecontains noneofthepoints 0,-1,-2,. . .eitherinitsinterior 01'onitsboundary. ~VberdieGrundlagen derTheorie derFakultlitenreihen. MUnch. Ber Vol. ~ti,pp.151-218. 1906. §58.Special classesofdcriesof4nlllytic functions. -H.Faculty series.447 Proof. 1.Wefirstshowthattheconvergence oftheDirichlet seriesatanyparticular point9=0,-1,-2,••.involves thatof thefacultyseriesatthesamepoint.As n!an an1 Z(z+~(z+n) =,~z·gn(z)' Ifg"(z)hasthesamesignifIcance asin2;>4,example 4,itissufficient, byIS4,3a,toshowthattheseries 1;11_ -_1_1-i;~+ 1(z)-=-~n(,2)I ,,=1gn(Z) gn+l(Z) -,,=1 !gn(Z)·gn+l(Z) I ISconvergent. Now 1()tendstoafinitelimItasnIncreases, namelyguz tdthevaluer(z);hlnce,inparticular, thisfactorremains bounded for <illvaluesofn(zbeingfixed).Henceitsuffices toestablish thecon­ vergence oftheseries l'ign(z)-g,,+1(z)I· n=1 Butthishasbeendonealready in254,example 4. 2.Thefactthattheconvergence ofthefaculty seriesatany pointinvohesthatoftheDirich1et seriesfollows inpreCisely thesame manner, asagain,byIS4,3a,everything turnsontheconvergence of 2'!gn(z)-g"+1(z)l· 3.NowletSfbeacircleinwhIchtheDirichlet seriesconverges absolutely andwhichcontains noneofthepoints0,-1,-2,..., eItherasinterior orboundary pOints. \Vehavetoshowthatthe facultyseriesalsoconverges uniformly inthatcircle. By198,3a, thisagainreduces toproving that ;;IgnH(z)-gn~' n=1gn(z).gn+1(z) ISuniformly convergent inSfandthatthefunctions 1Ign(z)remain uniformly bounded inSf.ThelInifnrm convergence of 1;Ign+l(z)-g"(z)I n=1 wasalready established in254,4.Alsoitwa"shownonp.440, footnote 51,thatthereexistsaconstant A'suchthat Ign1 (Z)I<A' foreveryzinSfandeveryn.Thisisallthatisrequired. (Cf.§·Hi. theorem 3.) 4.Theconverse, thattheDirichlet seriesconverges uniformly in everycircleinwhichthefacultyseriesdoesso,follows atonceI:>y 198,3afromtheuniform convergence oftheseriesIIgn+1(z)-gll(z)I andtheuniform boundedness ofthefunctions en(z)inthecircle,both ofwhichwereestablished in24')4,4. 448 Chapter XII.Seriesorcomplex terms. Examples. 1.Thefaculty series1:1 n! ..=12n-+1z(z+ 1)...(z+n) converges ateverypointoftheplane =1=0,-1,....FortheDirichlet series 001 ..~2n+'."z isevidently convergent everywhere. As 1 1 1 1 ..1-=----=~--,--::-:-xxx+Ix(x+I)' AS.!..=__~_2J_~ Ak.!..= kI , xx(x+1)(x+2)' ...,xX(.l:+1)..•(x+k) thegivenfaculty seriesresults simply, byElIler's transCormatlOn 144,from theseries 00(_I)n1 1 12)-+=---+1+ +2-+..··..=0z" zez 2.Itisalsoeasilyseen(cf.pp.265-6) thatfor!R(z)>0 101 11 (n-I)l ;g=z(z+1)+z(z+l)(z+2)+...+z(z+1)...(z+n)+...t i.e. 1 001 nl ZO=..~n'z(z+lF-·-=--(z+n)' Toshowthis,wehaveonlytosubtract thetermsoftherighthandsidesue cessively fromthelefthandside.Afterthenthsubtraction wehave nl 1 n!nZ=--. ,ZB(Z+1)(z+2)...(z+n) z.nllz(z+1)...(z+n) andthis,by254,example 4,tendsto0whenn-co,provided !R(z)>0 (Stirling Methodus dlfferentialis, London 1730,p.6seqq.) C.L(11}tbert's series. ALambert seriesisaseriesoftheform58 Ifweagaininquire whatisthepreciseregionofconvergence ofthe series,itmustfirstbenotedthatforeveryzforwhichz"-1can beequaltozero,aninfinite number ofthetermsoftheseriesbe­ comemeaningless. Forthisreason,thecircumference oftheunitcircle willbeentirely excluded fromconsideration 59whilewediscuss the &8Amoreextensive treatment ofthistypeofseriesistobefound IDapaper bytheauthor: OberLambertsche Reihen. Journ.f.d.reineu.angew.Mathem., Vol.14-2,pp.28:J-315. 1913. 5.Thisdoesnotimplythatthissenesmaynotconverge atsomepoints Zl ofthiscircumference, forwhich Zl"=l=+1foreveryn;;;:;1.Thismayactually happen; butwewdlnotconsider thecasehere. §58Specialclassesofseriesofanalytic functions. -C.Lambert's series.449 question ofconvergence oftheseseries,andthepointsinsideand outside thecirclewillbeexamllled separately. Wehavethefollowing theorem, whichcompletely solvesthequestion ofconvergence in thisrespect: Theorem. IfZanconverges, theLambert seriesconverges foreveryz2:>9. whosemodulus is9=1.If2,'anisnotconvergent, theLambert series converges atprecisely thesamepointsasthe"associated" power series2,'anz"-providedIzI9=1asbefore. Further, theconvergence isuniform ineverycirclesrwhichlies completely (circumference included) withinoneoftheregionsofconvergence ottheseriesandcontains nopomtofmodulus 1. Proof. 1.Suppose Zandivergent. TheradIUsrof2,'allz"is inthatcase neces~arily<1andwehavetoshow fir~tthattheLambert seriesandtheassonated powerseriesconverge anddIverge together foreveryIzI<1,andthattheLambert seriesdIverges forIzI>1. Now andz~ ,,1.2anI------n==.2anz.-1 n • -z -z Accordmgly, itsuffices, by184,3a,toestablIsh theconvergence of thetwoseries X1(1-zn+1)-(1-z'l)I=ZIz"-zn+11=11-zI·2,'IZnI and '--'11 1II I",Iz~I£..JI_zo+t-I_zo =1-z ·£..Jj(I-z~)(I-z~+t)1 forIzI<1.Thefirstofthesefactsisobvious, however, whilethe 1secondfollowsfromtheremark thatforIz1<1,wehave/1-znI>-2- forallsufficiently largen's. Ontheotherhand,iftheLambert seriesconverged atapoint ZO' whereIZoI>1,thepowerseries L;-~z"I-zoo wouldconverge forz=zo'andby93,theorem 1,wouldhavealso toconverge forz=+1.Hencetheseries '"an'" Zo0 '"~T=----zn-~an1_zn=~an " 0 wouldalsohavetoconverge, whichiscontrary tohypothesis. Finally, thefactthattheLambert seriesconverges uniformly in IzI<(}<rmayatoncebeinferred fromthecorrespondmg factin thecaseofthepowerseries2'ianz"I,by§46,2,-invirtueofthe inequality 450 Chapter Xll.Seriesofcomplex termS. ThecasewhereXandiverges ISthuscompletely dealtwith. 2.Nowsuppose 2,'anconvergent, soth.lt2,'allz"hasaradIUs l'~1.TheLambert ~enesiscertall1ly convergent foreveryIzI<1and indeeduniformly soforallvaluesofzsuchthatIzI.::;:u<1. ForIzI~e'>1,wehave z" (~)n .J)an1_z"=-.J)an-.J)an(1)"; 1---z andasI!I.::;:.!,<1,thisreduces thelaterassertions tothepre·z-e ceding ones,andthetheorem istherefore establl',hed inallItsparts. Bytheabove,averysimpleconnection eXIsts,inthecasewhere ~:aniscamergent, between thesumoftheseriesatapointz outside theunitcircleandthesamesumatthepoint..!.-inside H.z Accordingly itWIllsuffice ifweconsider onlyth.1tregion of convergellce oftheseneswhIchliesinsidetheunitCircle. ThISIS eItherthecircleIzI<"ortheumtcIrcleIzI<1Itself,.tccording as theradius l'oftheseries2.'anznis<1or~1.Let1'1denotethe radiusofthiSperfectly defilllte regionofconvergence. ThetermsofaLambert seriesareanalytic functIOns regular III IzI<1'1'andfore\erypositivee<1'\,theseriesisul1lformly con­ vergent inIzI<e:hencewem.tyapplyWeierstrass' theorem ondouble seriestoobtaintheexpansion inpowersenesofthefunctIOn re­ presented byaLambert :,cries InIzI<"1'Wehelve a_z_=az+aZ2+az:l+aZ4+az1'>+aZll+az7+...11-z 1 1 I 1 1 I 1 Z2 all1_Z2= a2Z2+a'JZ4+a2Zll+... Z3a - aaz3 +a.,zll -1_•••al-z3- ~ -, z<a.1_z<= a.z' ---j_••• andwemayaddallthesesencstogether termbyterm.Inthekth row,agivenpowerznwilloccurif,andonlyIf,nisamultiple ofk, orkadivisor ofn.Therefore An'thecoefficient ofz"intheresult­ ingseries, willbeequaltothesumofthosecoefficients a"wh05e suffix 'Visadivisor ofn(includmg 1orn).Thiswewritesym· bolically 60 andwethenhave,forIzI<"1' 00 Zn ao 2}anl-z;o=2-'Atlz". n=1 ,,=1 O.Inwords: thesumofallaa'sforwhich disadivisor ofn. §58.Special classesofseriesofanalytic functions -C.Lambert's series.451 Examples. 260. 1.an=1.HereAnisequaltothenumberofdivisorsofn,which(asin257, example 11)wedenotebyTn;then rLJ .....n 'Y:J 1:1:::..~n~1:T,.03'" (IzI<1) n=l ~n=l =-Z-+-2z'-+-2Z3-+-3o3'·-+-2Z5-+-4Zl-+-203'7-t-4o3'·+.... Inthiscurious powerseries,theterms03'''whoseexponents areprimellumbers are distingUished bythecoefficient 2.Itwasduetothemisleadingly closeconnection between thIsspecialLambert seriesandtheproblem ofprimesthatthisseries(as arulecalledsimplytheLambert series) 61playedaconsiderable partintheearlier attempts todealwiththIsproblem. Butnothing ofimportance wasobtained in thi,manner forsometime.OnlyqUiterecentlyN.TV/ener6•succeeded bythismeans inproving thefamous primenumber theorem. 2.an'"n.HereAnisequaltothesumofallthediVisors ofn,whichwe WilldenotebyT,.'.ThusforIzI<1 rr., ...n rf)1:11i~~~n=1:Tn'zn=,Z+3z·+4Z3+7o3'·+6o3'·-+-12Z6+...• 1l1 .... 11=I :I.TherelatIOn An~1:adISuniquely reversible, i.e.forgivenA,.'s,the din coefficients ancanbedetermined InoneandonlyonewaysoastosatisfytherelatIOn. WethenhaveInfact _(11). an-1:I-'l[Ad• dill where I-'(k)denotes theMub11l' coefficients defined in257,example 10,whose valuesare0,+Iand-I.Inconsequence ofthisfact,notonlycanaLambert senesalwaysbeexpanded IIIapowerseries,butcon\ersely everypowersenes maybeexpressed asaLambert serieq,provided Itvamshes forz~~0,I.e.Ao--O. ButItshouldbeobserved thatarelatIOnoftheform ~n 1:Anz"~Ean1--~~"-", neednotremaIntrueforIzI>1,evenwhenbothsenesconverge there. 4.Forin'tanee, ifAI=1andeveryotherAn=0, an=IL(n), andweha\ethecurious identity '" ~n Zc=1.:IL(n)--- - (IzI<1). 7l~1 1-z,. 5.Similarly, wefindtherepresentation, valIdforI03'I<I, ..... t:Lj ....n (1--=-Z)-3=};tp(1/)i-:::..03''', 11-1 where,? (n)denotes thenumber ofintegers lessthannandprimeton,-anumber mtroduced byEll/er. clJ ...n ,.." 6.Writing1:anf::-::-,.=f(03')and1:anzn=g(z),andgrouping theterms 111..... 111 bydiagonals Inthedoubleexpansion oftheLambert seriesonp.450(\\hich IS allowed), weobtain m f(z)=g(z)-+-g(Z3)+...=1:g(zm). m·1 61Lambert, JJ1.Anlage zurArchltektonik, Vo!.2,p.507.Riga1771.I'TVieller,N.,anewmethod inTaubenan theorems, J.Math.Masqachusetts, Vol.7,pp.lUI-IS4, 1!J:!l>,anuTllubcnan theorems, Ann.ofMath.(2),Vo!.33, pp.1-100, 11l32. 452 Chapter XII.Seriesofcomplex terms. <Ia:I<I),-::;11};(-I)'H'''1 Elz" n1--z1t (-I)"1z"};n--.1d) f)c) e)1(_1)10-1 7.1<oran=(_1)10-1, =n,=(_1)10-1 n,=-n-'=-;;--, =a:n,.•••we obtain InthlSway,successively, thefollOWing remarkable Identities. validfor 111I<I,111whichthesummatlons aretakenfromn~1to()J: oyn zn a)P(-1)10-1-- '"" 1-zn="1+-z'" z" zn b)};"1 _~n =};(1_z1l)l' zn ztl -zn=);(1+zn)2' 1 =};Iog1::.....zn' etc. 8.Inthetwoidentities d)ande)wehaveonthenghthandsideasenes ofloganthms (forwhIchofcoursewetakethepnncipal values); thu~simplecon­ nections canheestablished between certamLambert senesandinfimte products. E.g.fromthetwoIdentities inquestIOn: 1 ~n/1(1-zn)-~eW,\\1thw~-£--...­nl_~nJ (_1)10-1zn11(1f-z")=eW,withw~};- -n1__zn' (v,~0,I,2,...), 1=O.(Cf.Ex.114.)9.Asanmterestmg numencal example wemaymentIOn thefollowing: Takmg Un--0,IIII,andforevery n>I,Un~"10_1+Un_2,weobtain FlbollflCCl'S sequence (cf.6,7) 0,I, I, 2,3,0,8,13,21,34,5,';, vVethenhave i1=I+1+1+1+1+...=V5[L(3-=-V'5)_L(7_=::!y'5)] ' 11-1u2k 3 8 2155 2 2 x..£I _xn'TheprooflSbased whereL(x)denotes thesumofthe{,ambert ~enes 63 onthefact,which ISeaSIlyestabhshed, that a:V-pI' UV=------a:-fJ wherea:andfJaretherootsofthequadratic equatIOn x·-x- Exercises onChapter XII84• 174.Suppose zn->-,andbn-)-b=l=O.Underwhatconditions mayw~infer thatb..Z"--+b'? 175.Suppose zn--+Xl(i.e.IznI--++co).UnderwhatcondItions may wethenmferthat a)(1+z~rn --+e", b)zn'(ZI/Zn-1)--+logz? .aLandall, E.:Bull.delaSoc.math.deFrance, Vo!'27,p.29R.1899. ••Intheseexercises, wherever thecontrary doesnotfollowclearlyfromthl= J:0ntcxt, allnumbers aretoberegarded ascomplex. Exercises onChapter XII. 453 176.Theprincipal valueofz'remains, forallvaluesofz,lessinabsolute valuethansomefixedbound. Whatisthehehaviour <O.Zn=1:(-1)"(Z),v0 v according asffi(z)>0or177.If 1either Z"--70or --70,zn of(zn)whenffi(z)~O? 178.Leta,b,c,dbefourconstants forwhichad-bC=F0andletZobe arbitrary. Inve~tlgate thesequence ofnumbers (z",Z10Z2,•••)givenbythere­ currence formula thesenesE----.!-~­n'+'Ylogn'aZn+b Zn~1=c"zn+-d(n=0,I,2,...). Whatarethenecessary and&UffiClent conditIOns that(z,,)or(:)shouldconverge?-n AndIfneither ofthetwoconverges, underwhatconditIOns canzpbecome ="z. agamforsomeindexp?Whenareallthezn'sIdentically equal? 179.Letabegiven=F0andZochosenarbltranly, andwnteforeachn~0 Z"+1=~(zn+~} (zn)converges If,andonlyIf,Zodoesnotlieontheperpendicular tothestraight lmejommg thetwovaluesofvathrough itsmiddlepoint.IfthiScondItIOn is fulfilled, (z,,)convcrgl'S tothevalueofVanearesttozooWhatISthebehavIOur of (.~n)when ZoIte~OIltheperpendicular InquestIOn? 180.TheseriesE~!+".-doesnotcon\"erge foranyredIy;n1'Y - ontheotherhand,doesconverge foreveryrealyi=O. 180a.Therefinement ofWeierstrass's theorem 228thatwasmentIOned in footnote ]3,p.:{(J(l,maybeproved asfollows InconnectIOn Withtheforegomg example: Fromthea~sumptlOns, Itfollows, firstly,thatwemaywnte (n+.1)':''!.n-t!~1+~ (A'~Mm(A,2)>I),n-xannA. wheretheBn'sarebounded; hence,secondly, thatwemaywnte a"=:-x(1+nS~)), (C)-1with Cconstant amItheCn'sbounded. Thefactorsbn=1+n~'~1satisfytheas- sumptions ofthetest184,3.IfEanweretoconverge, thenEanbn=};C-wouldn-x alsohavetoconverge, contrary tothepreceding example andtheorem 255. 181.Forafixedvalueofzandasuitable determination oftheloganthm, does [1' 1 ]z·+-j+...+zI---n-log(z+n) tendtoaItmitasn--7+oo? 182.Foreveryfixed:::with0<ffi(z)<I, [1 1 1 hm1+2z+3z+...+z- n~oo n exists(cf.Ex.135). 183.Thefunction (1-z).sin(logl~.J mayheexpanded inapower series1:anznforI:::I<I,jfwctaketheprinCipal valueforthelogarithm. Show ~hatthissenesstillconverges absolutely forjzJ=1. 454 Chapter XII.Seriesotcomplex terms. 184.If6tendsto+1fromwithin theunitcircle,and"within the angle", wehave 1a)l-z+z4_Z0+Z16_+"'_-2-; b)(1-z)[1+z+Z4+zo+...)9-~-; 1 pp'p' 1c)-1--:c'-[z+z+z+z+...J-logp; ogl-z d)(l-z)P+l [z+2PZ9+3PZ3+...j_r(p+1); e)Ia"z~_Iim a~ 2b"z" b,,' provided therighthandlimitexists, b"ispositive forenchn,and2'b"is divergent. 185.InvE:stigate thebehaviour ofthefollowing powerseriesonthe circumference oftheunitcircle: c).2 z"; d)),.~. (n+a)a HP ""-'n101{.1' ee).2-1-"-z", wheree"hasthesamemeaning asinEx.47.nogn 186.IfIa"z"converges forIz:<1anditsslimisnumerically <1 forallsuchvaluesofz,then21a"19converges anditsSlllllIS<1. 187.Thepowerseries z"~2k-l a).2-, b).L)~-----,n 2k-I 2k-lz"c).5J(-I)k-l ~k-I' d))'(_1),,-1 - "-' n • z..1:(-1)"(-)-)z", e).2(-1)"(n+1)(n+2)'f) z"z2n g).2(n-1).(n+1),h)1:(2n-1).2~' allhavetheunitcircleascircleofconvergence. Onthecircumference, they alsoconverge ingeneral, i.e.withthepossible exee;-tion ofi~o1.lted points. Trytoexpress theirsumsbymeansofclosedexpressions mvolving elementary functions jseparate therealandimaginary partsbywritmgz=t'(cosX+isinx), andwritedownthetrigonometrical expansions soohlallled fort'<1andfor t'=1separately. Forwhichvaluesofxdotheyconverge? Whatarctheir sums?AretheytheFourzer seriesoftheirsUlns? 188.'Vhatarethesumsofthefollowlllg series: ~cosnxcosny. ~cosnxsinny a)4n' b)4 n ; c).2Si~~nSi~ I andofthethreefurtherseriesobta:ned bygivingthetermsoftheaboveseries thesign(_1)"7 ferclses onChaptE:'r XII. 455 IS9.Proceeding withthegeometric senes2,'z"asinEx.187,butleaving ,.<1,weobtaintheexpres~lOns '" 1 -rcosxa))'rncosnx=--- -------~ ;';;:'0 1 - 2,.cosx+,.9 00 YSlnXb)~r"Sillnx=-=--~---_:_ ___..'7::1 1-2rcosx+,.s· Deduce fromthemthefurther expansIOns c)1;_C~~_ =cos2x, "--1(2cosx)" d)y,sll~_nx __=sin2x ';;;:1(2co'>x)n dndindicate theexactintervals ofvalidity. 190.InExerCise 187athefollowlllg expansion willhavebeenobtained, among others. '",.n (rsinx)~'--Sillnx=tan-1-1------ • n~1n -rcosx Deduce fromittheexp,lDl>ions Cl) b)i'(-1)n--1rnSillnx.sinnx=tan-1(r+cotx)-(-~--x), ,,=t ,,,cosfl.T. ~lnnx 1t ..:..J----n---- =-2--x 11.==1 anddetcnlllne theexactintervals ofvahdity. 10).Detenmnc theexactregIOn!> ofconvergence ofthefollowing senes a))'(~~ ~z-l-n c)~';'~'" e)2,'[--~-+-~+~+...+~:~J,z--p"P,.P.. p,.1 b)LtT_z", d).2~, p" f)).(_~)IiOllnl ..:-.IPIt' where(P,,)isrealandg).2(:..)[lOll!ognl• increases monotonely to+00. 192.E"tnblish therelatIOns wherethesummatIon beginswith 11=1. 456 Chapter XII.Seriesofcomplex terms. 193.Corresponding toLandau's theorem (258)wehavethefollowing: The Dirichlet seriesE(-1)11-1a~andtheso-called binomial coefficient seriesEa"(Z-1)n" n Breconvergent anddivergent together, thepomtsz=I,2,3,•••bemgdisregarded. 194.ForwhichvaluesofZdoestheequation E(-1)"(Z)=0,,-0 n holdgood? 195.Determine theexactregions ofconvergence ofthefollowing in limteproducts: a)1I(1-:z)' b)Il(I-(-n~)K), c)1I(l+z9K+1), d)II(1+n9zK); e)lI(I-T~'»' f)1I[(1+ :)(1-1~r], g)1Il(1-~~)et+-HtJ+...+*(f,;)"J.IfIZn1-+ (X), h)1I(1-:), i)1I(1-(-n1)nz), k)n(1-~:~~z). 196.Determme, bymeansofthesineproduct, thevaluesoftheproducts a)"(I+~:), b)1I(1+~~), c)1I(1+~), forrealvaluesofx.Thesecond ofthesehasthevalue 1 - --22~-li[cosh(nxy 2)-cos(nxV2)J.:nx Doesthiscontinue toholdforcomplex valuesofx? 197.Thevalues oftheproducts 195,i)andk),canbedetermined inlhe formofaclosedexpressIOn bymeansofther-function. 19S.ForIzI<1 , 1 ~(I:;--_-z):-C("'I-_-z3c-) =(I---z---;-~-;-).-.-.=(I+z)(1+z2)(1+Zl)•••, 199.Bymeansofthesineproduct andtheexpansIOn ofthecotangent inpartial tractions, thefollowmg seriesandproduct maybeevalllated inthe +<Xl formofclosedexpressions; xBndyarereal,andthesymbol ~fen)indicates "=-00+<Xl thesumofthetwoseries ~fen) ..=0 product:+00 and~f(-k),andsimilarly forthe k=l c) d)ii(1---~~-\)."--COl (n+x)+00 1 a),,=~oo(n+x)9+y2' +<>-1 2J(x-n)I'n=-ODb) 71=-QC)1 I59.General remarks ondivergent sequences. 4:57 Chapter XIII. Divergent series. §59.General remarks ondivergent sequences andthe processes oflimitation. Theconception ofthenatureofinfinitesequences whichwehave setforthinalltheprecedmg pages,andespecially in§§8-11, isof comparatively recentdate;forastrictandirreproachable construction ofthetheory couldnotbeattempted untiltheconcept ofthereal number hadbeenmadeclear.Butevenifthisconcept andanyone general convergence testforsequences ofnumbers, sayoursecondmain criterion, wererecognized without proofaspractically axiomatic, it nevertheless remains truethatthetheoryofthcconvergence ofinfimte sequences, andofinfiniteseriesinparticular, isfarmorerecentthan theextensive useofthesesequences andseries,andthediscovery of themostelegant results ofthesubject, e.g.byEulerandhiScon­ temporaries, orevenearlier, byLeibniz, Newton andtheircontcm poraries. Tothesemathcmaticiallf, infiniteseriesappeared inavery natural wayastheresultofcalculation, andforcedthemselves into notIce,sotospeak: e.g.thegeometric series1+x+x'+...oc­ curredasthenon-termlOating resultofthedivision1/(1-x);Taylor's series,andwithitalmost alltheseriesofChapter VI,resulted from theprinciple ofequating coefficients orfromgeometrical considerations. Itwasinaslmilar manner thatinfimte products, continued fractions andallotherapproximation processes occurred. Inourexposition, thesymbolforinfinite sequences wascreatedandthenworked With; Itwasnotsoorigmally, thesesequences werethere,andthequestion was,whatcouldbedonewiththem. Onthisaccount, problems ofconvergence inthemodern sense wereatfirstremote fromthemindsofthcsemathematiciansl.Thus itISnottobewondered atthatEuler,forinstance, usesthegeometric series 1+x+Xli+...=1~x eventorx= -1orx= -2,sothatheunhesitatingly writes 2 11-1+1-1+-"'="2 1Cf.theremarks atthebeginnin~ of§41. \lThisrelation isusedbyJamesBernoulli (Posit.arithm., PartS,Basle1696) andisreferred tobyhimasa"paradoxon noninelegans". Fordetailsofthe VIOlent dispute whicharoseinthiSconnection, seetheworkofR.Re~1Imen­ tionedin69,R. 458 or similarly fromChapterXIII.Divergent series. 1 -2+22-2J+-...=!";} (}_)2= 1 + 2 x+ 3x2+...hededuces therelationI-x I1-2+3-4+-"'=4; andagreatdealmore.Itistruethatmostmathematicians ofthosetimes heldthemselves alooffromsuchresultsininstinctive mistrust, andrecog­ nizedonlythosewhicharetrueinthepresent-day sense 3.Buttheyhad noclearinsightintothereasonswhyonetypeofresultshouldbeadmitted, andnottheother. Herewehavenospacetoenterintotheveryinstructive discussions onthispointamongthemathematicians ofthe17thand18thcenturies 4. Wemustbecontentwithstating,e.g.asregardsinfiniteseries,thatEuler alwaysletthesestandwhentheyoccurred naturally byexpanding an analytical expression whichitselfpossessed adefinite value 5.Thisvalue wasthenineverycaseregarded asthesumoftheseries. Itisclearthatthisconvention hasnoprecisebasis.Eventhough, forinstance, theseries1 - 1+1 - 1+-..,resultsinaverysimple manner fromthedivision 1/(1-x)forx= -1(sceabove),andthere- foreshouldbeequated to~,thereisnoreasonwhythesameseriesshould notresultfromquitedifferent analytical expressions andwhy,inview oftheseothermethods ofdeducing it,itshouldnotbegivenadifferent value.Theaboveseriesmayactually beobtained, forx=0,fromthe functionf(x)represented foreveryx>0bytheDirichlet series 00{_l)n-1 I I If(x)=1:----x-=1 --2:<+;;X-4>;+...,n-ln ... l+x_1-x'_12+x3"+68+orfromI+x+-x.-1-_x3- - X -X"x-x-••• puttingx=1.Inviewofthislattermethodofdeduction, weshouldhave totake1 - 1+...=~,andinthecaseoftheformerthereisnoim- mediateevidence whatvaluef(0)mayhave;itneednotatanyratebe+~. 3Thusd'Alembert says(Opusc. Mathem., Vo!'5,1768,35; Memoire, p.183): "Pourmoi,j'avouequetouslesraisonnements etlescalculsfondcssurdesseries quinesontpasconvergentes ouqu'onpeutsupposer nepasl'etre,meparaitront toujours tressuspects". •Fordetails,sceR.Reijj,lococit. •InalettertoGoldbach (7.VIII.1745)hedefinitely says:"...sohabeich dieseneueDefinition derSumme einerjeglichen serieigegeben: Summa cujusque serieiestvalorexpressionis illiusfinitae,excujusevolutione tl1aseriesoritur". §59.General remarl{s ondIvergent sequences. 459 Euler's pnnciple istherefore Insecure inanyca.,e,anditwas only Euler'~ unusual lll;,tlllCt forwhatismathelllallcally correctwhich ingeneral s.lvedhimfromfalseconclusions inspiteofthecopious usewhIchhemadeofdivergent seriesofIhistype6•Cauchy and Abe1werethefirsttomakeIheconcept ofconvergence clear,andto renounce theu!'>eofanynail-convergent senes;Cauchy inhisAnalyse algebrique (1821),andAbetinhispaperonthebinomial series(1826), whichi'iC'xpressly basedonCauchy's treatise. Atfirstbothhesitated to takethisdeCISIve step7,butfinallyresolved todoso,asitseemed unavOIdable iftheIrreasoning weretobemadestrictandfreefromgaps. WearenowinaposItIon tosurveytheproblem fromabove,asit were;andthematteratoncebecomes clearwhenweremember that thesymbol foranmfinite<'equencc ofnumbers -inwhatever formit ISgivcn,scquence, senes,product orotherwIse --has,andcanhave, nomeaning wh;ttever initself,butthatameanlI1g wa5onlyassigned toitbyus,byanarbItrary conventIOIl. Thisconvention consisted firstlyinallowin:.c: onlyconvergent sequences, i.e.sequences whose termsapproached adefinite anduniquenumber inanabsolutely de­ fiIlltesense;secondly, itconsisted inassoclatmg thisnumber withthe infinitesequence, asitsvalue,orinregardmg thesequence asno morethananother symbol(cf41,1)forthenumber. However ob· viousandnatural thi.,c1efiIlltlon maybe,andhowever closelyitmay beconnected Withthewayinwhichsequences occur(e.g.assuc­ cessive approximatIOns toaresultwhichcannot beobtained directly), adefll1llIOn ofthiskindmustnelertheless IIIallcIrcumstances becon­ Sidered asanarbitrary one,anditmightevenbereplaced byquite clIffcrent c1efil1ltions. SuitabIlIty andsuccess aretheonlyfactors which candctermllle whether oncortheotherdefinition istobepreferred; inthenatureofthethingitself,thatistosay,inthesymbol (sn)of anmfimtesequence 8,thereisnothmg whichnecessitates anypreference. Wearetherefore quitejustified ina~kingwhether thecompli­ catlonwhichourtheoryexhibits (inpartsatleast)maynotbedue 6Cf.ontheotherhandp.133,footnote 6. 7SofarasCallchy ISconcerned, cf.thepreface tohisAnalyse algibrique, inwhich,amongotherthings,hesays:"JemeSUISvuforced'admetlre plusieurs propOSItions quiparaitront peut-Nre unpeudures,parexemple qu'uneseriediver­ genten'apasdesomme". Asregards Abel,cf.hiSlettertoHolmboe (16.I.1826), 111whichhesays:"Lesseriesdivergentes sont,engeneral, quelque chosedehien fatal,etc'estunehontequ'onoseyfonderaucune den~onstration".-As already mentIOned (p.458,footnote 3),:7.d'Alembert hadexpressed himselfinasimilar senseasearlyas1768. 8(sn)mayheassumed toheanygivensequence ofnumbers, inparticular, therefore, thepartialsumsofaninfimte series2:anorthepartialproducts ofan infinIteproduct. Weusetheletters,withitsremmder oftheword"sum", because mfi11ltesenesarebyfarthemostimportant meansofdefining sequences. 460 Chapter XIII.Divergent series. toourinterpretation ofthesymbol (sn)'asthelimitofthesequence, assumed convergent. beinganunfavourable one,--however obvIOUS andready-to-hand itmayappear. Other cOl1ventiol1~ mightbedrawn upinallsortsofways,amongwhichmo(esuitable onesmightper- 261.bapsbefound.Fromthispointofview,thegeneral problem which presents itselfisasfollows: Aparticular sequence (sn)isdefined in someway,eitherbydirectindication oftheterms.orbyaseriesor product, orotherwise, Isitpossible toassociate a"value" swithit, inareasonable way? ..Inareasonable way"mightperhaps betakentomeanthatthe number sisobtained byaprocess closelyconnected withthepreviott~ concept ofconvergence, thatistosay,withtheformation oflimsn=s. Thishasbeenfoundsoextraordinarily efficacious inallthepreceding thatwewillnotdepartfromittoanyconsiderable extentwithout goodreasons. ..Inareasonable way"mightalso,ontheotherhand,beIllter­ pretedasmeanmg thatthesequence (sn)istohavesuchavalues associated withitthatwherever thissequence mayoccurasthefinal resultofacalculation, thisfinalresultshallalways, oratleastusually, beputequaltos. Letusfirstillustrate thesegeneral statements byanexample. Theseries 262. 2'(-1t=1-1+1 - 1+-"', i.e.thegeometric series2'x"forx= -1,orthesequence (5n)==1,0,1,0,1,0,"', hassofarbeenrejected asdivergent, because itstermssndonot approach asingledefinite number. Onthecontrary, theyOSCIllate unceasingly between 1and0.Thisveryfact,however, suggests the ideaofforming thearithmetic means (n=O, 1,2,...). 1Sinces.="2[1+(-It],wefindthat 1 •,(n+l)+"2[l+(-l)] 11+(_1). s,.= 2(n+1) =2+4(n+1), 1sothats,.'(intheformersense)approaches thevalue"2: I, , 1 ImS,.=2' Bythisveryobvious process oftakingthearithmetic mean,we haveaccordingly managed, inaperfectly accurate way,togivea meaning toEuler's paradoxical equation 1 - 1+1 -+...={,to §59.General remarks ondlvergent sequences. 461 associate withtheseriesonthelefthandsidethenumber }asits "value", ortoobtainth1Snumber fromtheseries. Whether wecan always equatethefinalresultofacalculation to~whenever itap­ pearsinthe{Olm2:(-It,cannotofcoursebedetermined off-h:md. Jrlthecaseoftheexpan~ion -11 -=2'x"forx=-1,itiscertainly-x(-1)"-1so;inthecaseofZ------nx- forx=0,itisequally true,asmay beshownbyfairly;,imple me<tn~(cf.Exercise 200);-andagreat dealmoreevidence canbeadduced toshowthattheas-ociation of thesequence 1,0,1,0,1,...withthevalue{obtained inthemanner described aboveis"rca"onablc" 9. Wemightthereforc, asanexperiment, makethefollOWing de· finition. If,andonlyif,thenumbers (n=0,1,2,...) tendtoalimitsintheprevious sense,thesequence (sJ,orseries 2'an,willbesaidto"converge" tothe"limit", or"sum",s. ThesuitabIlity ofthisnewdefinition hasalready beendemon­ strated inconnection withtheseries2:(-1)",whichnowbecomes . h ".I h 1 h'h convergent "mt enewsense, WIt1t esum"2'-WICseems thoroughly reasonable. Twofurther remarks wIllillustrate thead· vantagesofthisnewd:'fimtion: 1.Everysequence (s,,),convergent intheformer senseandof limits,issoconstituted, JJ1virtueofCauchy's theorem 43,2,that itwouldalsohavetobecalledconvergent "inthenewsense", with thesamellmlts.ThenewdetlDllion wouldtherefore enableusto accomplish atleastallthatwecoulddowiththeformer, whilethe example oftheseries2,'(-1)"showsthatthenewdefinition ismore far-reaching thantheoldone. 2.Iftwoseries,convergent intheoldsense,2:an=Aand 2'bn=B,aremultiplied together byCauchy's rule,givingtheseries 2:cn==.2"(aobn+a1bn-1+...+anbo)'weknowthatthisseriesIS notnecessarily convergent (intheoldsense). Andthequestion when 2:cndoesconverge presents verycon.,iderable difficulties andhasnot beensatisfactorily clearedupsofar.Thesecondproofoftheorem189, •Fromtheseries(seeabove)for--!...±~ alsowecanaccordingly de­ l+x+x' 2ducethevalue"3forx=1.\Vehaveonlytoobserve thattheseries,written somewhat morecarefully, is1+0·x-x'+X3+O·XI_Xb++-...,nndis therefore 1+°-1+1+°-1++-...forx=1. 462 Chapter XIII.Divergent series. however, showsthatineverycase Eo+C 1+...+C..-AB n+l ife..denotes thenthpartialsumof.4e...Themeaning ofthisis that~c..alwaysconverges inthenewsense,withthesumAB.Here theadvantage ofthenewconvention isobvious: Asituation which, owingtotheinsuperable difficulties involved, itwasimpossible tu dearupaslongaswekepttotheoldconcept ofconvergence, may bedealtwithexhaustively inaverysimple way,byintroducing a slightly moregeneral concept ofconvergence. Weshallverysoonbecome acquainted withotherinvestigations ofthiskind(see§61inparticular); firstofall,however, weshall makesomedefinitions relating toseveral fundamental matters: Besides theformation ofthearithmetic mean,weshallbecome acquainted withquiteanumber ofotherprocesses, whichmaywith success besubstituted fortheformerconcept ofconvergence, forthe purpose ofassociating anumber swllhasequence ofnumbers (sn)' Theseprocesses havetobedistinguished fromoneanother bysuitable designations. Insodoingitisadvisable toproceed asfollows: The former('oncept ofconvergence wassonatural, andhasstoodthetest ';0well,thatitoughttohaveaspecialnamereserved forit.Accor· dingly, theexpression: "convergence ofaninfimte sequence (series, product,...)"shallcontinue tomeanexactly whatitdidbefore. 1£ bymeansofnewrules,as,forinstance, bytheformation ofthearithmetic meandescribed above,anumber sisassociated withasequence (sn)' weshallsaythatthesequence (sn)islimitable* bythatprocess, and thatthecorresponding series 2.~anissummable bytheprocess, and weshallcallsthevalueofeither(orinthecaseoftheseries,its sumalso). When, however, aswilloccurdirectly, wearemaking useof severalprocesses ofthiskind,wedistinguish thesebyattached initials A,B,...,V,...,andspeakforin!>tance ofa'V-process 10.Weshallsay thatthesequence (sn)islimitable V,andthattheseries.4anissumm ableV;andthenumber sWillbereferred toasthe'V-limit ofthe sequence orV-sumoftheseries;symbolically V-limsn=s,V-.4an=s. Whenthereisnofearofmisunderstanding, wemayalsoexpress the •German: lzmd~erbar. 10Inthecaseoftheconcept ofintegrab~lity thesituation issomewhat similar anditwasprob.•blyinthisconnection thattheabovetypeofnotation wasfirstintroduced. 1hll';wesayafunction isintegrable Ror~ntegrable L according aswearereferring tointegrability inRzemann's orinLebesgue's sense. §50.Generalremarksondivergent sequences. formerofthetwostatements bythesymbolism V(s,,)-~s,463 whichmoreprecisely impliesthatthenewsequence deduced from(s,,)by theV-process converges tos. When,aswillusuallybethecaseinwhatfollows,theprocessadmits ofak-folditeration, orcanbegradedintodifferent orders,weattacha suffixandspeakofaVk-limitation process,aVv-summation process,etc. Intheconstruction andchoiceofsuchprocesses weshallofcourse263. notproceed quitearbitrarily, butweshallratherletourselves beguided byquestions ofsuitability. Wemustgivethefirstplacetothefundamental stipulation tobemadeinthisconnection, namelythatthenewdefinition mustnotcontradict theoldone.Weaccordingly stipulate thatanyV­ processwhichmaybeintroduced mustsatisfythefollowing permanence condition: 1.Everysequence (s,,)convergent intheformersense,withthelimits, mustbelimitable Vwiththevalues.Orinotherwords,limSn=smust ineverycaseimply 11V-lims"=s. Inorderthattheintroduction ofaprocess ofthiskindmaynot besuperfluous, wefurtherstipulate thatthefollowing extension con­ ditionistohold: n.Atleastonesequence (s,,),whichdivergesintheformersense, mustbelimitable bythenewprocess. LetuscallthetotalItyofsequences whicharelimitable bya particular process therangeofactionofthisprocess. Thecondition II implies thatonlythoseprocesses willbeallowed whichpossess a widerrangeofactionthantheordll1ary process ofconvergence. It isprecisely thelimitation offormerly divergent sequences andthe summation offormerly divergent serieswhichwlllnaturally claimthe greater partofourattention now. Finally,ifseveralprocesses areemployed together, sayaVprocess andaWprocess simultaneously, weshouldbeindangerofhopeless confusion ifwedidnotalsostipulate thatthefollowing compatibility condition shouldbefulfilled: Ill.Ifoneandthesamesequence (s,,)islimitable bytwodifferent processes, simultaneously applied, thenitmusthavethesamevalue bybothprocesses. Inotht;rwords, wemustineverycasehave V-hmsf!=WlimSn'ifboththesevaluesexist. 11Wcmightalsobesatisfied ifsomeconvergcnt sequences atleastare limitable withunaltered valucbytheprocess consider~d. Thisisthecasee.g withtheEI'.process discussed furtheron,provided thesutfiJ:piscomplex 464 Chapter Xl1l.JJivergent series. Weshallonlyconsider processes whichsatisfythesethreecun ditioDs. BesIdes these,however, werequiresomeinrncation whether theassociation ofavalueswiththesequence (s,,)effectedbyapartI· cularV-process isareasonable oneinthesenseexplained above (p.460).Herewidely-varying conditions maybelaiddown,andthe processes whIChareincurrentuseareofveryvarieddegrees ofef­ ficiencyinthisrespect. Inthefirstinstance weshouldnodoubtreqUIre thattheelementary rulesofthealgebraofconvergent sequences (v.§8) shouldasfaraspossible bemaintained, i.e.therulesforterm-by-term addItion andsubtraction oftwosequences, term-by-term addition ofa constant, andterm-by-term multiplication byaconstant, andtheeffectof afinitenumber ofalterations (27,4),etc.Nextwemightperhaps requirethatif,say.adivergent series:Ea"hasassociated withitthe values,andifthisseriesisdeduced, e.g.fromapowerseries f(x)=:Ecnx"bysubstituting aspecialvalueXlforx.thenthenumbers shouldbearanappropriate relation tof(xl)or10limf(x) forX-Xl; andsimilarly forothertypesofseries(Dirichlet series,Fourierseriesetc.). Inshort,weshouldrequirethatwherever thisseriesappears asthefinal remltofacalculation, theresultshouldbes.Thegreater the number ofconditions similartotheabovewhicharesatisfied bya ~64-.particular process-letuscallthemtheconditions F,withouttakmg painstoformulate themwithabsolute precision -andatthesame time,thegreatertherangeofactionoftheprocess, thegreaterwill beitsusefulness andvaluefromourpointofview. Weproceed toindicate afewoftheseprocesses oflimitation whichhaveprovedtheirworthinsomewayoranother. ~65.v1.TheCl-,Hl-,orM-process It.Asdescribed above,262,we formthearithmetic meansofthetermsofasequence (s,,): So+SI+...+SrI---n+T-- (n=0,1,2,...) whiehwewilldenotebye,.',h,.',ormn•Ifthesetendtoalimitsin theoldersense,whenn-00,wesaythat(s,,)islimitable Clor limitable HIorlimitable Mwiththevaluesandwewrite M-lim Sn=sorM(s,,)_s, orusethelettersClorHIinsteadofM.Theseries1:a"withthepartial sumss"willbecalled8ummable Clor8ummable HIor 8ummable M,andswillbecalleditsCl-,Hl-,orM-sum. Thesequence ofunitsI,1,1,...maybeconsidered tobethe simplest convergent sequence wecanconceive. Theprocessdescribed aboveconsistsincomparing, ontheaverage,theterms srIofthesequence 11ThechoiceofthelettersCandHisexplained inthetwonextsub-sections. §59General remarks ondIvergent sequences. underconsideration withthoseofthesequence ofunits:465 This"averaged" compari50n of(5,,)withtheunitsequence willbemet withagaininthecaseofthefollowing processes. Theusefulness ofthisprocess hasalreadybeenillustrated above byseveral examples. Wehavealso ~eenthatitsatisfies thetwocon­ ditions263,Iand11,andIIIdoesnotcomeunderconsideration at themoment. In§§60and61itwillfurther beseenthatthecon­ ditionsF(264)arealsoinwidemeasure fulfilled. 2.lltiltler's process, ortheHp-process 13.Ifwithagiven sequence (5,,),weproceed fromthearithmetic meanshn'Justformed totheirmean h"=ho'+ht'+...+hn' n n+l(1£=0,1,2, ...) andifthesequence (h,,")hasalimitintheordinary sense,limhn"=s, wesaythat14thesequence SnlSti1nitable H2withthevalues. By43,2,everysequence whichislimitable:' HJ(andtherefore alsoeveryconvergent sequence), isalsolimitable HII'Withthesame value.Thenewprocess therefore satlsfies theconditions 263,I,IIand Ill;moreover, itsrangeiswiderthanthatoftheH1 -process, for theseries 1;(-It(1£+1)=1-2+3-4+-.... •=0 Iforin'itance, issummable H'Jwiththesum4-'butnotsummable Hlnorconvergent. Infact,wehavehere (5n)=:1,-1,2,-2,3,-3, and 2 3(h,,')=1,0,-3-'0,5'0,.•.• Thesesequences arenotconvergent. Ontheotherhand,thenumbers h,,1 ."1I I d Th". "-4"'asISeastycacuate.ISISprecisely thevaluewhich oncwouldexpectfrom (1)'J ao--=-=..2(n+1):v"1xn=O forx= -1. laHolder,0.:Grenzwerte vonHeihen anderKonvergenzgrenze. Math. Ann.,Vol.20,pp.535-549. 1882.Hercarithmetic meansofthekinddescribed areforthefirsttimeintroduced foraspecial purpose. 14Therestofthenotation isformed inthesameway,H.-Iims"=5, H.-~'an=S, H.(sn)_S, etc.buthereafter weshallnotmention itspeciall)·. 466 Chapter XIII.Divergent series. Ifthenumbers h,,"donottendtoauniquelimit,weproceed to taketheirmean h,,,=ho"+ht"+...h,," " n+1(n=0,1,2,...) or.10general, for15p~2,themean ( )h(P-1)+h(p-1)++h(p-1) hP=_0__ 1 .., "(n=0,1,2,...) " n+1 between thenumbers h,~P-l)obtained attheprevious stage;ifthesenew numbers h~P)_s,forsomedefinite p,wesaythatthesequence (s,,)is Hmitable R1Jwiththevalues. Itiseasytoformsequences whicharelimitable H1Jforanyparticular givenp,butfornosmallervalueofpthanthis16.This,together with 43,2,showsthattheH1J-processes notonlysatisfytheconditions 263, I-Ill,butthattheirrangeofactioniswider for eachfixedp~2than forallsmaller valuesofp.Asregardstheconditions F,wemustagain referto§§(iOand6l. 3.Cesaro's process, ortheCk-process 17.Wefirstwrite '":="s~°l,andalso,foreachII>1, Solk-I)-I- Si(k-I)-I-.•.-I-S(k-I)=S(k), (°1 2 ) 11 11n=." ... andwenowexamine thesequence Qfnumbers IS (k) ')S~k) C= -----~,"(n1k) foreachfixedk.If,forsomevalueofk,C,~k)_S,wesaythatthesequence (s,,)isli1nitable Ckwiththevalues. InthecaseoftheH-process, wecannotobtainsimpleformulae giving h(p)directlyintermsofs'"forlargervaluesofp.InthecaseoftheC-process, "thisiseasilydone,forwehave (k)_(n+k-1)(n+k-2) (k-1)S"-k_1So-I-k_1SI-I-...-I-k_1Sn, '"Orindeedforp~1,provided weagreetoputh~O)""snandtaketheHo­ processtobeordinary convergence, asweshalldohereandinallanalogous cases infuture. 16Wnte,forinstance, (h~-1») ==1,0,1,0,1,...andworkbackwards tothe valuesofs".Otherexamples willbefoundinthefollowing sections. 17Cesdro,E.:Surlamultiplication desseries. Bull.dessciences math.(2), Vo!.14,pp.114-120. 1890. 18Thedenominators oftherighthandsideareexactlythevaluesofS.:kl obtained bystarting withthesequence (s,,)0=1,1,1,.••,i.e.theyindicate how manyofthepartialsumssvarecomprised inS~k).ThustheCk-process againin­ volvesan"averaged" comparison between agivensequence (s,,)andtheunit sequence. §59.General remarks ondivergent sequences. 467 orifwewishtogobacktothesNies ~:all'withthepartialsumssn' s~)=(ntk)ao+(n+~-1)a1+...+(~)a,. Thismaybeproved quiteeasilybyinduction, orbynoticing that, by102,iJS~k-l)x"=(1-x).fS~k)xn, "=0 "=0 sothatforeveryintegral k~0 co(!)n 1 co 1 co..25"x=(1_X)h..2snx"=---X,+1 ..2anx","=0 "~O (1-x)"=0 whence, by108,thetruthofthestatement follows 19• Inthefollowing sections wef-hJllenterindetailintothisprocess also,whichbecomes identical withthepreceding one(h,,'=e..')for p=1. v4.A'Jel'sprocess, ortheA-process. GivenaseriesIaftwith thepartialsumss",weconsider thepowerseries f(x)=2:anxn=(1-x)Isnx". Ifitsladiusis~1,andif(forrealvaluesofx)thelimit lim2:anx"=lim(1-x)Isnx"=S ",-..1-0 ",-+1-0 exists,wesaythattheseries2-'anis20summable A,andthatills sequence (s,,)islimitable A,withthevalues;insymbols: A-2:an=s,Aohmsn=s. Inconsequence ofAbel'stheorem 100,thisprocess alsofulfils thepermanence condItion I,andsimpleexamples showthatitfulfils the"extension condition" II;forinstance, inthecaseoftheseries I(-I)" already used,thelimitforX-1-0 lim(I(-l)" x")=lim__1_=~l+x 2 exists.ThusEuler's paladoxical equation (p.457)isagainjustified 18Inviewoftheselastformulae, itisfairlynaturaltoallownon-integral values > -1forthesuffixkalso.SuchlimItation processes ofnon-mtegral orderwere firstconsistently introduced andInvestigated bytheauthor(Grenzwerle vonReihen beiderAnnaherung andieKonvergenzgrenze, Inaug.-Diss., Berlin1907). We shallhowever notenterintoth,squestion, eitherhereinthecaseoftheC-process, orlaterinthatoftheotherprocesses consIdered. 20Iftheproduct (1-x)Es"x"iswritteninthefonn Xs"x" ~Xx"·_' weseethatitisagainan"averaged" comparison ofthegivensequence withthe unitsequence whichisinvolved, thoughinasomewhat different manner. 468 Chapter XIII.Diverg-ent series. bythisprocess. Ifwcnowusethemoreprecise form A-.2'(-I)"=-; orC1·.2'(-lr=-;, wethusindicate twoperfectly definite processes bywhichthevalue }maybeobtained fromtheseries.2'(-1)". o.E"le,.'s process, ortheE-process. Wesawin144thatif thefirstofthetwoseries Cl> 00Ak.2(-1)"anand.2" ~'!- 11=0 k=U2k+1 converges, thensodoesthesecond, andtothesamesum.Simple examples show,however, thatthesecond seriesmayquitewellcon· vergewithout thefirstonedoingso: 1.Ifan:=1,thenao=1andJkao=0fork~1.Accordingly, thetwoseriesare 11 - 1+1 - 1+-...and-If+0+0+0+... 1thesecondofwhichconverges tothesum-2-' 2.If,forn=0,1,2, a=n then•••J 1,2,3,4,...• andfork~2 Acc:ordmgly, theAa"=-I, -1,-1, -1, •••J Aka=0,0,0, 0,...."t\\0senesare 1 11 - 2+3 - 4+-...and"2-4-+0+0+.... 1thesecondofwhIchconverges tothesum4-' 3.Similarly fora"=(n+1)3wefindJao=-7,A2ao==12, ,;j:lao=-6, and,fork>3,LJ"ao=O. Thetwosenesarethus 171261 - 8+27-64+-...and1f-"4+-8-16+0+0+.." thesecond ofwhichconverges tothesum--~-. 4.Fora"=2",LJ"ao=(_1)". Thusthetwoserieo>are: 1 1 1 11 - 2+4 - 8+-...and1f--,f+8"-11;+-.... thesecondofwhichconverges tothesum~,i.e.thesumwhichweshould 1expectforx= -2fromr=--x=Ex". 5.Fora,,=(-I)"z", LJkao=(1+z)" ..Thetwoseriesaretherefore Cl>" <Xl(1+Z)k.2z and.2--k-+t' "=0 k=O2 thesecondofwhichconverges tothesum1~11'providedIz+11<2. §59.General remarks ondivergent sequences. 469 Ifwestartwithanyseries2:all'without alternately+and signs,thesenes an'=2n1 +1[(~)ao+(7)a1+...+(:)anjLL:an',with n=O willbeanEuler's transformation ofthegiven dlsuobtainasfollows: TheseriesL:anresults 2'a"X,,+lforx=1,hencefrom ':",(y)"+1 n~oanl-yseries,whichwema} fromthepowerseries weobtainEuler's transformation.1fory=2 1 Y=-2'Expanding thelatterinpowers ofy,beforesubstituting Infact kgakxk+1=k~ak(1~Jk+l=kI:oakA~(ktl)yl.+J.+l =n?otgG)a~}ynH=n~a,.'(2y)n+l. Inordertoadaptthisprocess forusewithanysequence (s,,)wewrite, devlatHlg somewhat fromtheusualnotatIOn, ao+a1+...+a"-1=snforn~1,and So=0, andalso an'+a/+...+a~-l=s,,' forn~l, and 50'=0. Itisnoweasytovenfythat21furcv,ry11_::0 Weaccordlllgly makethefollowing definition: Asequence (5,,)issaid tobelimitable Elwiththevalue 5,ifthesequence (s,,')justde­ finedtends 22tos.If,Without testlllgtheconvergence of(5,,'),wewrite <Xl OIl.2SkXk=(l-y).2 sn'(2y)". L-O n=O1 I-x21Fro1112'a"x"+1=2:a,,'(2y)n+l l-y 1_2y'thatitfollows, bymultiplication by Hence =n~2\[(~)so+(~)S1+...+(:)s"J(2y)", whence therelation mayatoncebeinferred. 88Herealsothedenominator 2nisobtained fromthenumerator byreplacing eachofthe5,,'Sby1.Thusweareagainconcerned withan "averaged" comparison, ofadefinite kind,between thesequence (s,,)andthe unitsequence. 4:70 Chapter XIII.Divergent series. 1Inthespecialcaseof}J-=- -1'nn-1-andingeneral, forr:::::1, s;:l=2\[(~)scir-I)+G)st-1)+ ...+(:)s;:-1)J, (n=O,1,2, ...), weshallsimilarly saythatthesequence (S,.)islimitable Erand rpgardsasitsEr'limit,if,foraparticular r,s;:)-s. Ourformer theorem 144(seealso44,8)thenshowsinany casethatthisE-process satisfies thepermanence condition J,andthe examp.es giventhereshowthatthecondition IIisalsosatisfied. This process willbeexamined furtherin§63. 6.Riesz's process, ortheRH'process lIJ.Formaking the principle ofaveraged comparison ofthesequence (S,.)withtheunit sequence morepowerful, -aprinciple which,aswesaw,hesatthe basisofalltheformer limitation pro-:esscs, - afairlyobvious pro cedureconsists inattributing arbitrary weights tothevarious termssn' If11-0'11-1>11-2'••:denoteanysequence ofpositive numbers, then I1-'0So+fLlSI+...f-fLnSnS..= ~~ 1-'0+1-'1+...+I-'n isageneralized meanofthiskind. wespeakofalogarithmic mean. Aswiththell-,C-,orE-processes, thisgeneralized method offor.n­ ingthemeanmayofcourseberepeated, writing, forinstance, asinthe C-process, U(O)=Snnand,\(0)=1, n andthen,fork2':1, (h) (h-I)+ (h-I)+ + (h-1)Un=p·oall 11-1UI• • • I1-nUll and ,\(h)_.:\(h-1) +.:\(hl)+ +,,(h-I)"--11-00 11-1I • • • I1-n" ' andthenproceeding toinvestigate, forfixedk~I,theratio (k) (h)_an ell-ilk) n forn---'>-+00.Ifthesetendtoalimits,wemightsaythat(s,,)was limitable 24R,'kwiththevalues.Thisdefinition, however, isnot inuse.Theprocess inquestion hasreached itsgreatimportance (>DIy bybeingtransformed intoaformmorereadilyamenable toanalysis, as 23Riesz,M.:SurlesseriesdeDirichlet etlesseriesentieres. Comptes rendus Vo!.149,pp.909-912. 1909. 2.HereweaddasuffixI-'toRhthenotation oftheprocess, asareference to thesequence (fLn)usedintheformation ofthemean. ForfLn~1,thiSprocess reduces exactlytotheCk-process. §59.General remarks ondivergent sequences. 471 >.(~) iLoSo+iLl$1+...+iLnSn=f$(t)dt ufollows:A(complex) function S(t)oftherealvariablet?:0isdefinedby s(t)=Svin,,~~~<tS"P)(v=0,1,2,•••;"~l=0) with $(0)=0;then anditisnaturaltosubstitute repeated integration fortherepeated sum­ mationusedintheformation ofthenumbers cr~k)and ,,~kJ.Ak-plem­ tegration 25gives fI1 lk-lh tufdtk-lf...fs(t)dt=(k-~Ififs(t)·(m-t)k-ldt u uu u instead ofa(k).Similarly, instead ofthenumbers l(kJ,wehaveton n takethevalueswhichweobtainbyputtings"=1 mtheintegrals justwritten down,i.e. Cd_I_J(_)k-ld_wk (h-I)!. mtt-hI' u Weshouldthenhavetodealwiththelimit(forfixedle) Cd HmhkJs(t)Cm-t)k-tdt. to-"+ CXl0)o Ifthislimitexistsand=s,thesequence (sn)willbecalledlimitable BJ.kwiththevalues. Herewecannotenterintoamoredetailed exammation ofthe question whether thetwodefinitions givenfortheRA/<·process are reallyexactly equivalent, orintotheelegant andfar-reaching appli­ cations oftheprocess inthetheoryofDirichlet's series. (Forrefer· encestotheliterature, see266.) 7.Bm'el's process, ortheR·process. Wehavejustseenhow Ricsz'process tendstoincrease theefticiency oftheH-orC-pro­ cesses, bysubstituting forthemethod ofaveraged comparison be· tweenthesequence (s,,)andtheunitsequence amoregeneral form ofthisprocedure. TherangeofAbet'sprocess maybeenlarged m asimilar waybymakmg useofotherseriesinsteadofthegeometric seriesthereusedforpurposes ofcomparison. Taking theexponential seriesasaparticular case,andaccordingly considering thequotient ofthetwoseries and 95Theequality ofthetwosides ISeasilyprovedbyinduction, usinginte· gration byparts. 4:72 Chapter XIII.Uivergent senes, that15tosay,theproduct ClO ~. F(x)=e-z..2s"nl,,=0 forx_+00,weobtaintheprocess introduced byE.Borel26,In accordance withitwemakethefollowing definition: Asequence (s,,) xn suchthatthepowerseries.2s"nlconverges everywhere andthe function F(x)justdefined tendstoaunique limitsasx-+00, willbecalledlimitable Bwiththevalues, Inordertoillustrate theprocess tosomeextent,letusfirsttake 2,'a"==I(-Itoncemore;thens"=1or0,according asniseven orodd.Accordingly xn x'x' e"+e-"2)s"nl=1+21+4"1+...=--2- andwehavetodealwiththelimit ._'"e"'+e-"hme.-----2 'Z-.+co 1whichisevidently 2"'Thus2,'(-1)"ISsummable Bwiththe Isum2"'Moregenerally, taking:Ea"==2,'zn,wehave,provided only thatz-++1, and xn1 ItF(x)=e-z,2)s-= - --e(z-l)z"n!l-zl-z which__1_whenx_+00,provided in(z)<1.ThusthegeometricI-It seriesIz"issummable Bwiththesum-11throughout thehalf­-.p!ane 27~(z)<1. Thisprocess alsosatisfies thepermanence condition; forwehave (e-z.2)s":~)-s=e-Z •,2(s"-s):~. Ifs"- sintheordinary sense,wecanforanygivenechoo~emso .8Surlasommation dess~riesdivergentes, Comptes rendus, Vol121,p.]]25. 1895,-andinmanyNotesinconnection withit.Aconnected account isgiven inhisLe~onssurlesseriesdivergentes, 2nded.,Paris1928. 27BytheC-processes, asshewnin268,8,thegeometric seriesissummable, beyondIzI<I,onlyfortheboundary pointsoftheunitcircle,+1excepted; byEuler'sprocess ItISsummable throughout theCIrcleIz+II<2,whichen­ clo&estheunitcircle,WIthawidemargin; byBoret'sprocess itissummable inthe wholehalf-plane ffi(z)<I,-thevalueinthiSandtheprecedmg casesbemgevery- 1 where1-:tt' §59.General remarks ondivergent sequences. 473 largethatIsn-sI<le::foreveryn>m.Theexpression ontheright handsideistheninabsolute value Cl) xn m X"e ;;5e-Z•EISn-si·I~r'".EISn-si·,+ 2~'n'O n. n~O n. forpositive x's.Nowtheproduct ofe-'"andapolynomial ofthemth degreetendsto0whenx--++00;wecantherefore choose ~solarge thatthisproduct is<le::foreveryx>~.Forthesex'sthewholeex­ pression isthen<e::inabsolute value,andourstatement isestablished. 8.TheBr-process. Therangeoftheprocessjustdescribed is,in acertainsense,extended bysubstituting otherseriesforE:~,inthefirst instanceE(~~n)"say,whererissomefixedinteger>1.Weaccordinglyrn. saythatasequence (sn)islimitable Brwiththevaluesifthequotient ofthetwofunctions rLJxrn a:>x""" 00 rnESn~(~)'andE(-)I'i.e.theproduct rrZEs-,);--n-Orn. n_.orn. n_on(r1l)!' tendstothelimitswhenx--++00.(Wemust,ofcourse,assumeagain herethatthefirst-named seriesiseverywhere convergent.) Thusthe B-process, forinstance, isquiteuselessforthesequence Sn=(-l)nn!, xnsincehereESn-,.=E(-l)nxndoesnotconverge foreveryx;whereasn. xrn theseriesESn(~)'alreadyconverges everywhere 28whenwetaker=2.r11• 9.LeRoy'sprocess. Wchaveusuallyinterpreted thelimitation processes bysayingthatbymeansofthemwecarryoutan"averaged" comparison between thegivensequence (sn)andtheunitsequence 1,I, I,...Wemaylookatthematterinaslightlydifferent way.Ifthenumbers SnarethepartialsumsoftheseriesEan>wehavetoexamine, forinstance intheCcprocess, thelimitof So+SI+...+Sn n+l =ao+(1-n~1)a1+(1-n11)a2+...+(1-n+-l)an. Herethetermsoftheseriesappearmultiplied byvariable factorswhich reducethegivenseriestoafinitesum,oratanyratetoaseriesconvergent intheoldsenst'".Bymeansofthesefactors,theinfluence ofdistantterms isdestroyed ordiminished; yetasnincreases allthefactorstendto1 andthusultimately involveallthetermstotheirfullextent.Thesituation issimilarinthecaseofAbel'sprocess, wherewewereconcerned with thelimitofEanxnforx--+1 -0;heretheeffectdescribed aboveis 18ThiSdoesnotmeanthattheBr-process (r>1)ismorefavourable than theB-process foreveryseqUl.'nce (sn)'Onthecontrary, therearesequences th::t arehmitable Bbutnotlimitable BI• 474 ChapterXIII.Divergent series. brought aboutbythefactorsx",which,however, increase to1asx--+1 -O. Thisprinciple appearsmostclearlyasthebasisofthefollowing process 29: Theseries ~r(nx+1) .£Jnla",,=0 isassumed convergent for°<x<1.Ifthefunction whichitdefines inthatinterval tendstoalimitsasx-+1-0,theseries:Ea"may becalledsummable Rtothevalues. Thismethod isnotsoeasilydealtwithanalytically, andforthis reasonitisofsmaller importance. 10.Themostgeneral formofthelimitation processes. Itwill havebeennoticed thatalltheprocesses sofardescribed belong es­ sentially totwotypes: 1.Inthecaseofthefirsttype,fromasequence (s,,),withthe helpofamatrix(cf.Toeplitz' theorem221) T=(ak,,) anewsequence ofnumbers sk'=akOsO+aklsl+···+akns,,+.'" (k=0,1,2, ...) ISformed bycombination ofthesequence so'SI'...,sn'...withthe succeSSive rowsakO'akl,...,aknl..., -theassumption being,of course, thattheseriesontherighthandsiderepresents adefinite value,i.e.isconvergent (intheoldsense) 30.Thesequence so',SI"•••• Sk"•••willbecalledforshorttheT-transformation 31ofthesequence (sn)anditsnthterm,whenthereisnofearofambiguity, willbedenoted byT(sn)'Iftheaccented sequence (s,:)isconvergent withthe limits,thegivensequenceissaidtobelimitable Twiththevalues.In symbols: T-lims"=sorT(s,,)-s. ULeRay:Surlesseriesdlvergentes, Annales delaFac.dessciences deToulouse (2),Vol.2,p.317.1noo. 10IfeachrowofthematrixTcontains onlyafinitenumber ofterms, thiscondition isautomatically fulfilled. Thisisthecasewiththeprocesses 1,2,:3andfi. 31Theseries2ak',ofwhichthesk"saretheparllalsums,maysimilarly becalledtheT·transformation oftheseries ~:anwiththesn'sasitspartial sums.Thuse.g.theseries a, a,+2a.+...+nan aO+r:2+···+ n(n+1) +... istheCl,transformation oftheseries2"an'Inthissense,allT-processes givemoreorlessremarkable transformations ofseries,whichmayveryoften heofuseinnumerical calculations. (Thisisparticularly thecasewiththe E,process). Thetransformation oftheseriesmayequally, ofcourse, bere. gardedastheprimary process andtbetransformation ofthesequence ofpArtial sumsmaybededuced fromit.Indeeditwasinthiswaythatwewereled totheE-process. §59.General remarks ondivergent sequences. 475 Itisatonceclearthattheprocesses 1,2,3,5,andthefirstone described in6belongtothistype.Theydifferonlyinthechoiceofthe matrixT.Theorem 221,2alsoimmediately tellsuswithwhatmatrices wearecertaintoobtainlimitation processes satisfying thepermanence condition 32. 2.Inthecaseofthesecondtype,wededueefromasequence (sn), bycombining itwithasequenceoffunctions (CfJn)==CfJo(x), CfJl(x),. . . • CfJn(x),..•• thefunction F(x)=CfJo(x)So+CfJl(x)SI+...+CfJn(x)Sn+..., whereweassume, say,thateachofthefunctions CfJn(x)isdefinedforevery x>Xoandthattheseries1:CfJn(x)Snconverges foreaehofthesevalues ofx.InthatcaseF(x)isalsodefinedforeveryx>xo,andwemayin­ vestigate theexistence ofthelimitlimF(x).Ifthelimitexistsand=s, x-':>-+"" thesequence (sn)willbecalled 33limitable cpwiththevalues. Byanalogy with221,2,weshallatoncebeabletoassignconditions underwhichaprocessofthistypewillsatisfythepermanence condition. Thiswillcertainly bethecaseifa)foreveryfixedn. limCfJn(x)~0, x---+·j ao ifb)aconstant Kexistssuchthat ICfJo(x)I+ICfJl(x)I-I-.•.-I-ICfJn(x)I<K foreveryx>Xoandalln's,andifc)forx---+-I-00 lim{1:CfJn(x)}=1. Itwillbenoticedthattheseconditions correspond exactlytotheassump­ tions 34a),b)andc)oftheorem 221,2.Theproof,whichisquiteanalogous tothatofthistheorem, maytherefore belefttothereader. Boret'sprocess evidently belongs tothistype,with CfJn(x)=e-Z•:~. ThesamemaybesaidofAbet'sprocess, iftheinterval 0...-I-00 32Theimportance oftheorem 221,2lieschieflyinthefactthatthecon­ ditionsa),b)andc)ofthetheorem arenotmerelysufficient, butactually necessary foritsgeneral validity. Wecannotenterintothequestion (v.p.74,footnote 19), butwemayobserve thatinconsequence ofthisfact,theT-processes whosematrix satisfies theconditions mentioned aretheonlyoneswhichfulfilthepermanence condition. 23Inallessentials thisisthescheme bymeansofwhichO.Perron(Beltrage zurTheorie derdlvergenten Relhen, Math.Zschr.Vol.6,pp.286-310. 1920) classifies allthesummation processes. a.LikethesetheyarenotonlysuffiCIent, butalsonecefSGry forthegeneral validityofthetheorem. Further details InH.Raff,Lineare Transform.ttionen bes­ chriinkter integrierbarer Funktionen. Math.Zeitschr. Vol.41,pp.605-629. 1936. 476 Chapter XIII.Dlvergenl series. isprojected intotheinterval 0...1whichisusedInthelatter,that is,iftheseries(1-x)Is"x"isreplaced bytheseries F(x)=1:x"J:s"(1~xr andthelatterisexamined forX-++00.-Inanequally simple manner, itmaybeseenthatLeRay'sprocess belongs tothistype. Thesecond typeoflimitation process contains thefirstasapar­ ticularcase,obtained whenxassumes integral values ~0only (cp"(k)=ak").Wemerely useacontinuous parameter IDtheonecase, andadiscontinuous oneintheother.Conversely, inviewof§19, def.4a,thecontinuous pa<;sage tothelimitmaybereplaced bya discontinuous one,andhencethecp-proces5es maybeexlllbiled as asub-class oftheT-processes. Theseremarks, however, areofhttle use:infurther method., ofinve~tigatioll thetwotypesofprocess nevertheless remain essentially dilTerent. Iti.,notourintention toInvestigate alltheprocesses whichcome underthesetwoheadings fromthegeneral pointsofviewmdicated above. Letusmakeonlythefollowmg remark.,. Wehavealready pointed outwhatcondltlons thematllxTor~equence offunctions (cp,,) mustfulfil,inorderthatthelimitation process basedonitmaysatisfy thepermanence condition 263,I.Whether theconditions 263,Il an,lIIIareal'ofulfiled,willdepend onfUTtherhypotheses regarding thematnxTorsequence (T,,);thisquestion ISaccordll1gly betleft to<iseparate investIgation ineachcase.The que~tlOn astotheex­ tenttowhichtheconditions F(264)arefulfilled, cannotbeattdcked inageneral wayeither,butmustbe~pecially exammed foreach process. Oneimportant property alone IScor"mon toalltheT­ andcp-proce'>Ses, namely theirlmearcharacter: Iftwosequences (5,,) and(t,,)arelimitable inaccordance withoneandthe~ameprocess, thefirstwiththevalues,andthesecond Withthevaluet,thenthe sequence (a5"+btJ,whatever theC0:1St;111ts aandbmaybe,isalso limitable bythesameprocess, withthevalueas+bt.Theproof follows Immediately fromthewayInwhichtheproce.,s i.,constructed. Owingtothistheorem, allthesimplest rulesofthealgebra ofcon­ vergent sequences (term-by·term addition ofacon5tant, term-by-term multiplicatIon byaconstant, term-bytermaddition orsubtractIOn of twosequences) remain formally unaltered. Ontheotlll-'rhanJ,wemust expressly emphasize thefactthatthetheorem ontheinfluence ofa finitenumber ofalteratIOns (42,7)doe-notneces5arily remainvalId ~15, 35Forthis,thefollowing simpleexample relating totheB-process wasfirst givenbyG.H.Hardy:Let$nbedefined bytheexpansion • 00xn sm(eX)=L:$"" n-O11. ~inl;ee-".sin(eX)~0asx~+00,thesequences So.$11s,.'••isIimltabl~lJ §59.General remarks ondivergent sequences. 477 Ifwewi~hedtogiveageneral andfairlycomplete surveyofthe prescnt stateofthetheoryofdivergent series,weshould nowbe obliged toentermtoamoredetllled investigation oftheproce~ses which wehave de~cnbed. Tobeginwith,weshouldhavetodealWiththe questions whcther, andtowh:ltcxtent,theindividual proces~es do actually satisfythestipulatIOns 203,If,IIIand204;weshouldhave tooot.linnecessary andsufficient conchtlollS foraseriestobesummable byaparticular proce-s; weshouldhavetofindtherelations between theways 111whichthevarious processes act,andgofurthermtothe questIOns indicated inNo.10,etc.Owingtolackofspaceitisof courseoutofthequpstion toinvestigate allthisindetail.Wemustbe content Withexamllling afewoftheprocesse., moreparticulary; ­ wechoosetheH·,C"A·,andE-proce..,ses. Atthesametimewe willsoarrange thechoicC'ofsubjects tb,lt2.5faraspossible all questions andallmethods ofproofwhichplayapartinthecorn· pletetheorymayatleastbeindicated. Fortherestwemustrefertotheorigmal papers, ofwhichwemaymen·'l66. tionthefollOWing, inadditIOn tothosementIOned inthefootnotes ofthis sectionandofthefollOWing sectIOns: 1.ThefollOWing giveageneral surveyofthegroupofproblems: J3orel,E.:Leo;ono; surlessenesdivergentes, 2'Jcd.,PansH)28. Bromwlch. T.J.l'A.·AnintroductIOn tothetheory ofinfinite series. London 1908:2nded.lU2fi, Hardy,G.H.,andS.Chapmall Ageneral viewofthetheoryofsummab~e series.Quarterly Journal Vol.42,p.181.1911. Chapman, 5.:Onthegeneral theoryof~umll1ab\lity, Withapplications to Four:er's andotherseries. Ibid.,Vol.43,p.11911. Carmlchael, R.D.:General aspects ofthetheory ofsummable series. Bull.oftheAmerican Math.Soc.Vol.25,pp.97-131. 1919. Rnopp,K.:NeuereUntersllchungen inderTheorie derdlvergenten Reihen. )ahresber. d.Deutschen Math.·Ver. Vol.32.pp.43-67. 1923. 2.Amoredetailed account oftheRI.,•.process, whichisnotspecially considered inthefollowing sections, isgivenby Hardy, G.H.,andM.Rlesz.Thegeneral theoryofDirichlet's series. Cambridge 1915. TheB.process isdealtwithInthebooksbyBorelandBromwlclt mentioned under1.,andalsoinmoredetllllby Hardy, G.H.:TheapplicatIOn toDlrlchlet's seriesofBorel's exponential method ofsummation. Proceedings oftheLond.!\lath.Soc.(2)Vol.8,pp.301 to320.190U. w.ththevalueO.BydifferpntiatlOn oftherelation above,we"b~ain <Xlx"cos(e3)=e-.....2SIt+1nl; n=O ~ thisshows,sincecos(e3)tendstonolimitwhen:.r:.....+00,thatthesequence S,Is••sa'•••isnotlimitable BataUI 478 Chapter XIII.Divergent senes. Hardy,G.11.,andJ.E.I.ltllewood: Therelations between Borel's and Cesaro's methods ofsummation. Ibid.,(2)Vol.11,pp.1-16.1913. Hardy.G.H.,andJ.E.Llltlcwood: Contributions tothearithmetic theory ofseries.Ibid.,(2)Vol.11,pp.411-478. 1913. Hardy,G.H.,andJ.E.Llttlcwood: Theorems concerning thesllmmability ofseriesbyBorel's exponential method. Rend.delCircolo Mat.diPalermo Vol.41,pp.36-53. 1916. Doetsch. G.:EilleneueVernllgemeinerung derBorelschen Summabllitilts­ theorie. Inaug.•Diss.,r.ottingen 1920. 3.Apartfromthebooksmentioned under1.,afullaccount ofthetheory ofdivergent seriesistobefoundin Bieberbach. L.:Nel1ere Untersuchungen libcrFunktionen vonkomplexen Variablen. Enzyklop. d.math.Wissensch. Vol.If,PartC,No.4.1921. 4.Finally, thegeneral question oftheclassification oflimitation processes isdealtwithinthefollowlIlg papers: Perron,0.:Beitrag Zl1rTheorie dt'rdivergenten Reihen. ;\Iath.Zeitschr. Vol.6,pp.286-310. 1920. Hattsdor//, F.:Summationsmcthoden nndMomentenfolg'en Iundn.Math. Zeit_chr. Vol.9,p.74seqq·andp.280seqq.1920. Knopp,K.:ZurTheone derLlmltlerungsverfahrcn. Math.Zeltschr. Vol.31; 1stcommunication pp.97-127, 2ndcommUllleatlOn pp.271l-305. Hl29. §60TheC-andH-processes. Ofallthesummation processes brieflysketched inthepreceding section,theC-andll-proeesses -andespecially theprocessoflimitation byarithmetic meansofthefirstorder,whichisthesameinboth-are distinguished bytheirgreatsimplicity; theyhave,moreover, provedof greatimportance inthemostdiverseapplications. Weshallaccordingly firstexamine theseprocesses insomewhat greaterdetail. 267. InthecaseoftheH-process, Cauchy's theorem 43,2showsthat, forp~I,h;;-J)--+simplies 36h~)--+s,sothattherangeoftheIIp-process contains thatoftheHp_I-process. Thecorresponding factholdsinthe caseoftheC-process: Theorem 1.Ifasequenceislimitable Ck-Iwiththevalues,(h~I), itisalsolimitable Ckwiththesamevalue.Insymbols: From C~-I--+s, itfollowsthatc~k)-~s.(Permanence theoremfortheC-process.) 3SCf.p.466,footnote 15.BytheOthdegreeofatransformation, higher degreesofwhichareintroduced, wemeantheonglllal sequence. §60.TheC-andH-processes. 479 Bydefinition (v.263,3) 5(.1:) 5(k-1)+...+5(.1:-1) fa 0 faProof. c1:) ==C1k)=(~==~)+...+(n;~~l) (k-l)clk-1)+•..+(n+k-l) c(k--J)k-l 0 k-l fa -(~==D+...+Cf~-~?)--. whence by44-,2thestatement immediately follows. Accordmgly, toeverysequence whichislimItable Cp'forsome suitable suffixp,therecorresponds adefinite integerksuchthatthe sequence islimitable Cl<butisnotlimitable CI<_l'(Ifthesequence isconvergent fromthefirst,weofcoursetakek=0.)Wethensay thatthesequence isexactlylimitable Cl<' 268.ExampIesoftheCh-limitationProcess87, Ivalue1f'Proofabove,262. 1C.I:+1,tothevalues=2k+1'1.i(-I)"issummable Clwiththe n=O -;;,- n+k)2.:E(-1)"( isexactly sllmmable n=O k ..(n+k) 3 Infuct,foran==(-1)k Iwehaveby263, ~5(.1:)x"=(_I_)k+l..2(_Ir(n+k)x"=(_I__)k+l(_I_)k+1 ~faI-x kI-x I+x =(_I__)k+l=~(V+k)2"1-x" ~kx• Accordingly 5~k)=(V1k)or=0,according asn=2"or=2,,+1. Hencebothforn=2vandforn=2,,+I, S~k+1)=(~)+Ctk )+...+C1k )=(V1~iI), whence thestatement follows Immediately. 3.Theseries.E(-I)"(n+l)k=I-2k+3k_4k+_...,summable Clto thevalue-}fork=0IbyExample 1.,isforeachk~Iexactlysllmmable Ch+1 2k+1_1 tothesums=k+IBk+1,ifB"denotes the"thofBernoulll's numbers. Thefactofthesummability indeedfollowsdirectly fromExample 2.Forthe moment denoting theseriestheresummed by2.'h,wcatoncesee,fromthe linearcharacter ofourprocess (v.p.476),thattheseries,obtained from.Eh 37Asaresultoftheequivalence theorem established immediately below theseexamples hc'dunaltered fortheHk·limitation processes. Onaccount oftheexplicit fOllllulae for5~k)andc~k),givenin265,3, towhichthereis noanalogue inthcH-process, theC-process isusuallYoreferred. 480 ChapterXIII.Divergent series. bytenn-by-term addition, ofthefonn co.Eo+cl.El+...+ck.Ek isexactlysummable Ck+1ifCo,Ch•••,ckdenoteanyconstants, withCk*'O.Now thec"mayobvIOusly bechosensothatweobtainprecisely theseriesE(-1)"(n+l)k. ThevaluesismosteaSilyobtained byA-summation; see288,1. 4.Theseries;+cosx+cos2x+...+cosn x+...ISsummable Clto thesum0,provided x'*'2k11". Proof. By20I, 1 Sin(n+~):Ie s"=2+cosx+cos2x+...+cosnx=-------;--' 2sln2 foreachn=0,I,2,•••;hence 1(Cl:x.(2 )x)'0+'1+...+,"=--'xSin~+SIn3 2+...+Sinn+1 2 2~ln2 andconsequentlysin'(71+1); "2sm22 I~+~Sl+...+'nI<::__1.. . on+1 I-n+1 2sm2~ Forafixedx*,2k11",theexpreSSiOn ontherighttendsto0asnIncreases, which proveswhatwasstated.-ThiSisourfirstexample ofasummablc serieswith vanable terms.Thefunction represented byits"sum" ""0 meverymterval not containing anyofthepoints2h11".Attheexcluded pomts,thcsenes ISdefinitely divergent to+tX:J! 5.Theseriessmx+sm2x+sm3x+...isobviously convergent WIth thesum0,forx=k?T.Forx*'k?Titisnolongerconvergent, butItISsummable Cl'anditthen'8hasthe"sum" -~cot-;. Proof. Fromtherelation 1 :le .1""=sinx+...+sin11x=2cot2:lecos(2n+1)2 ----._~-----,x2sm2 thestatement follows asin4. 6.cosx+cos3x+cos5x+...issummable Cltothesum0,for,'I:*'k11". 7.smx+sin3:1e+SIn5x+ ."isalsosummable Cltothesum2 -~.., forx=l=k11". SInX 8.1+z+ZI+...issummable Clanthecircumference IzI=I,ex­ 1ceptmg onlyforz=+1,andthesumis1 _z.(Examples 4and/)resultfrom thISbyseparating realandimaginary part•.)Here,infact, 1 Z"+1sothat .1"0+~1~'.±.!n=1'n=1-=-';;;-1~z' n+1 1=·z whence thestatement canbeinferred ataglance.1z(I-zn/-l) n+1-(1-=Z)2... 3.Thegraphofthisfunction thusexhibits "mfimtely gre.J'lumps" atthe points2k11". §60.The6-and!i-processes. 481 1 ~(n+k-l)9.Thesenes(1-_';)k-£., Z'''remainq summable Cktothe 11-0k--l sum(1~z)konthecircumference IZ'I=1,provided onlyz'1=+1.Forthe correspondmg quantities S~klare,by265,3,thecoefficients ofx"mtheexpansion of 1 1 a (C-x)k+I'(1-xz)k=(1-xjk+l+..., (thefighthandsidebemgtheexpansIOn mpartialfractions ofthelefthandside). AllthepartialfractIOns aftertheonewntten downcontain inthedenominator thek'hpowerof(1-x)or(1-:1;z)atmost.Hence,multlplymg by(1-x)k+l andlettmgx.....1,weatonceobtaina=(1~z)k'Accordmgly ~S(kl,,_~[__1__(n+k)J" £., 11x-£., (1_z)kk+...x, n0 ,,-a whereitissufficient toknowthatthesupplementary termswithinthesquarebracket Involvebmmmal coefficIents oftheordernk-1withrespecttonatmost.Therefore, asn.....+00, S(k) 11 1(;;tk).....(1-z)k' q.e.d. SincetheI/-process outwardly seemstobearacertainrelationship totheC-process, itisnaturaltoaskwhether theireffectsaredistinguish­ ableornot.Weshallseethatthetworangesofactioncoincide completely. Indeedwehavethefollowing theorem, duetotheauthor 39andtoW. Schnee 40: Theorem 2.Ifasequence(sn)is,forsomeparticular k,limitable 41Hk tothevalues,itisalsosummable Cktothesamevaluesandconversely. In269. symbols: h~k)_salwaysinvolves C(k)_S "' andconversely. (Equivalence theoremfortheC-and!I-processes.) Manyproofshavebeengivenforthistheorem 42,amongwhichthat ofSchur 43isprobably theclearestandbestadapted tothenatureofthe 39Cf.thepapercitedonp.467,footnote 19. 40Schnee, W.:DieIdentltilt desCesaroschen undH61derschen Grenzwertes. Math.Ann.Vo!.67,pp.110-125. 1909. USmcefork=1thetheorem istflvial,wemayassumek~2inthesequel. <2Adetailed bibliography, forthistheorem anditsnumerous proofs,may befoundmtheauthor's papers: 1.ZurTheorie derC-undH-Summierbarkeit. Math.Zeitschr. Vo!.19,pp.97-113. 192:3;H.Obereineklassekonvergenz­ erhaltender Integraltran,formationen unddenAqUlvalenzsatz derC-undH-Ver­ fahren,ibid.Vo!.47,pp.229-264.1941; IH.OberemeErweiterung desAquiva­ lenzsatzes derC-undH-Verfahren undeineKlasseregularwachsender Funktionen, Ibid.Vo!.49,pp.219-255. 1943. ..Schur,I.:OberdieAquivalenz derCesaroschen undHolderschen Mittel­ werte.Math.Ann.Vo!.74,pp.447-458. 1913.Also:EinigeBemerkungen zur Theofle derunendllchen Relhen, Sitzber. d.Ber!.Math.Ges.,Vo!.29,pp.3-13. 1929. 482 ChapterXIII.Divergent series. problem. Comhined withaskilfulartificeofA.F.Andersen 44,theproof becomes particularly simple. Wenextshowthattheequivalence theorem iscontained inthefol­ lowingtheorem, simplerinappearance: 270. Theorem 2a.If(zn),fork~1,islimitable C"withthevalue,.the 270ifh.h' ,Zo+Zl+...+Z".I',sequence 0t eantmetlcmeansZn=-~_..---------- ISImlt-n+lableCk-1withthevalue ~,andconversely. Bythistheorem, eachofthekrelations c,~k)==Ck(sn) --*S Ck-1(h,,')--*s C2(h~k-2))--*S Cl(h~k-l) ==h,~k)--*s isinfactaconsequence ofanyoftheothers;inparticular, thefirstisa consequence ofthelast.Butthatiswhattheequivalence theorem states. Itsuffices,therefore, toproveTheorem 2a.Butthisfollowsimmedi­ atelyfromthetworelations connecting theCl.-andCk_l-transformations ofthesequence (z,,)withthoseofthesequence (zn'),viz. (1)Ck(zn)=k Ck_l(zn')-(k-I)Ck(z,,'), (ll)Ck(z')-1C('")+(1__1)f'kJ.:,.O) +_~k(z,L+"_,_·_+f',.(z,,)-1n-kk~n k n+1 . Forif,inthefirstplace,wehaveCk-1(zn')~"then,byTheorem I,wc havealsoCk(zn')--*,.Henceby(I), Cdzn)--*k~-(k-IH=~. If,inthesecondplace,Ck(zn)--*"then,by43,2,sodothearithmetic means Ck(zo)+Ck(z,)+...+Ck(zn),. n+1 --*'"' and,withequalease,(ll)provides that45 Ck-l(Zn')--*il:+(1--})~=1:. Accordingly allreducestoverifying thetworelations (I)and(ll), andthismaybedoneforinstance asfollows: UAndersen, A.F.:Bemerkung zumBeweisdesHerrnKnoppfUrdieAqui. valenzderCesaro-undHo'lder-summabilitiit. Math.Zeitschr. Vol.28,pp.356-350. 1928. ..IfMdenotestheoperation oftakingthearithmetic meanofasequence, theaboverelations (I)and(11)maybewrittenintheshortandcomprehensive form (I) Ck=kCk_1M-(k-1)CkM, (11) Ct=k Ck_1M-(k-1)MCl<' EachofthesefollowsfromtheotherifitisknownthattheCk-transformation and theprocessoftakingthearithmetic meanaretwocommutable operations. §60.TheC-andH-processes. 483 In265,3,theiterated sums S~h)wereformed, todefinetheCt­ transformation ofasequence (sn).Letusdenotethesesumsmoreprecisely byS~h)(s),andusethecorresponding symbols whenstarting withother sequences. Theidentity 1 00 J 00 (l-~)k};Znyn=(1:.:.:--)~-l};(zo+ZI+...+zn)ynYn-O Yn-0 1'"(+1),n =-(-1_---)k=l £.Jnzny Yn-O thenimplies S(I.)(~)=(11+k-2)1~'+ _L("+k-2-I') (+I)~,+n~ k_2 . ~o...1- k_2 v ~v••. +(~=~)(n+1)z,,'. Herewrite v+1=(n+I~)--(n+k-1-v), andobserve that ("+:~:-v)(11-I-k_1_v)=(k_1)(11+~=~-v). Itthenfollowsfurtherthat (*) S~h)(z)=(11+k)S~h-I)(z')-(k-1)S:~)(z'). Dividing by(";k),wededuceatoncetherelation (I). Ontheotherhand,bythedefinition ofthequantities(h)S,.,wehave S(h-I)=S(h)_S(h) n n n-l(h)(n=0,1,•••;S_l=0). Substituting in(oil),weget (**) S~h)(z)=(11+I)S;,h)(z')-(11+k)S~~I(z'), andhence,dividing by(11t'~), Ck(zn)=(11+1)Cdzn') -nC~(Z'n-I). Substituting inturn0,I,..•,nforninthisrelation, andadding, we obtainfinally Putintowords,thisrelation signifies thatthearithmetic meanoftheCt­ transformations ofasequence isequaltotheCk-transformation ofits arithmetic means,or,aswesayforshort,theCk-transformation andthe processofforming thearithmetic meanaretwocommutable operations 46• ••Cf.preceding footnote 45. 484 ChapterXIII.Divergent series. 271.Nowifwesubstitute forCk(zn')in(I)ti.eexpression justfound,we obtain(II)atonce.Thiscompletes theproofoftheEquivalence Theorem. Afterthusestablishing theequivalence oftheC-process andthe H-process, weneedonlyconsider oneofthem.AstheC-process iseasier toworkwithanalytically, onaccountoftheexplicitformulae 265,3for theS~k)'s,itisusualtogivethepreference toit. Wenextinquirehowfaritsrangeofactionextends, i.e.whatare thenecessary conditions tobesatisfied byasequence inorderthatitmay belimitable Ck•Usingthenotation, whichwasintroduced byLandau andhasbeengenerally adopted, Xnc=0(n<», 0(real,toindicate thatthe sequence (:~)isbounded, andXn=0(n<»toindicate that(~~)isanull sequence 47,wehavethefollowing theorem, whichmaybeinterpreted by sayingthatsequences whosetermsincrease toorapidlyareexcluded from Ck-limitation altogether: Theorem 3.IfL:an,withpartialsumss'"issummable Ck,thenI._ an=0(nk)andSn=0(nk). Proof. Fork=0,thestatement isaconsequence ofTheorem 82, I,whichwearcgeneralizing. Fork~],withthenotation of265,3, thesequence ofnumbers 8~k-l)+...+8,:4--1) isconvergent. Since(n+:-1)'"(ntk),thesequence S~k-I)+...+8~~~1) (n-;;k) •••tisconvergent, withthesamelimit.Thedifference ofthetwoquotients, viz.S;-1/(n-;;k),therefore formsanullsequence. As(n;I,)'"n4 thiSimpliesthatS~k-1)=0(nk).Itfollowsthat S~k-2)=S~k-l) _S~~;l)=0(nk)+0(nk)=0(nk), andsimilarly 48 S~k-3)=0(nk) , ••Thefirststatement thusimphesthatthequantities [xnIareofatmOff thesameorderasconst..n",thesecondthattheyareofsmallerorderthann",In thewayInwhIchtheyIncrease to+00• ••ThereaderwIllbeabletoworkoutquiteeasilyforhimselftheverysimple rulesforcalculations WIththeordersymbols 0and0whIchareusedhereandin thesequel. §60.TheC-andH-processes. 485 272.Theintermediary result S,~k-l)=0(n/c)justobtained intheproofmay beinterpreted asanevenmoresignificant generalization ofthetheorem inquestion. Infact,itmeansthat (nt~~1)aa+(n;~~2)al+...+(~=~)an-------+0.(n;Il) Weaccordingly havethefollowing elegantanalogue of82,1: Theorem 4.InaseriesEay"summable C\,'Wenecessarily have C/c-lima"=o. Moreover, evenKronecker's theorem 82,3hasitsexaetanalogue. thoughweshallconfineourselves tothecasePn=n: Theorem 5.InaseriesEamsummable Cl.,1fenecessarily have 273. C-lim(~J_±~a'L ._.~__-.t-"!l") =0,. n-I-1 . Infact,itfollowsfromthecorollary to270thatC,.(s,,)-~sinvolves Ck-l('0IS\zl+:-i.:-.:L!n) --+s,andtherefore bytheperm,lI1enee theorem C/Ceo1-S\z~~~!n) --+s.Subtracting thisfromC"(s,.)-»s,weatonce obtainthestatement CI ( So1-s,-I-•..-I-Sn)~,C_-lim(/1.±2lI,1_-•.•--1-_11_all) ,-~0.,.-lmSn---;-+1- - - I. n_1_1 . Bymeansofthesesimpletheorems, therangeofactionoftheCk­ processisstakedoffontheoutside,aswemightsay,forthetheorems inform ushowfaratmosttherangemayextendintothedomainofdivergent series.Wherethisrangeproperly beginsisamuchmoredelicatequestion. BythiswemeanthefolIo\\'ing:Everyseriesconvergent intheusualsense tothevaluesisalsosummable Cl.(foreveryk2:0)tothesamev.llues. Whereisthehoundary line,intheaggregate ofallserieswhicharesummable Cbbetween convergent anddivergent series?Onthispointwcha\-cthe following simpletheorem, relatingsolelytotheC\-process: Theorem 6.IftheseriesEa"isCl-Sllmmablc toSlIms,andif274. '"a,+211.--t--•••-In11"0I P••fi .I0"=-----n-1--1-- --+•tlen'"anISZ1lactconvergent 'lI.'ltISlIm s.For(v.supra) s_s,,-j:__~_l-I-•__..:.±s"=~!..±_2".+..:-...:...-l:..!!._'lll :-Il",z+l 11+1 n. whencetheproofofthestatement isimmediate 49.Thelastexpression ••Withreference to262,1(or43,Theorem 2).andto82,Theorem :1,\\cmay express thetheorem asfollo\lls: Asenes2,'anconH'rgcs If,andonlyIf,ItISC1­ summuble with8n--+o. 486 ChapterXIII.Divergent series. tends,Inparticular, to0ifan=0(~).Amuchdeeperresultisthefact thatan=0 (~)suffices, i.e. Theorem 6a.Ifaseries1:anissummable Ckandifitstermsansatisfy thecondition then1:anisconvergent. (O-Ck--+K-theorem) 50. Aproofofthistheorem maybedispensed withhere,sinceitwill followasasimplecorollary ofLittlewood's theorem287.TheJirectproof wouldnotbeessentially easierthantheproofofth~\ttheorem. A p PI1cation.Theseries1;an==E111~>i' ex~0,is.lOtconverg('nt, ,,=1 n=l asItiseasyto,enfl',byanargument modelled ontheproofonp.442,footnote '4,thatforn=1,2,•••, ;)'ll={~[;}-(n:l)'"J-~~ with (~11)bounded. Further, fortlusseries(nan)i.bounded, h,mcetheseriescannot besummable Cl'toanyorder. Closelyconnected withthepreceding, wehavethefollowing theorem, whereforsimplicity weshallconfineourselves tosummation ofthefirst order. 275. Theorem 7.Anecessary andsufficientconditionforaseries1:an>with partialsumsSn>tobesummable Cltothesums,isthattheseYles (A) 60Hardy,G.II.:Theorems relating totheconvergence andsummability ofslowlyosclllatmg series.Proc.Lond.Mat!J.Soc.(2)VO!.S,pp.3UI-32U. ItlUtl. Cf.alsotheauthor's work1.quotedonp.481,footnote 42.-Thetheorem deduces convergence (K)fromC-summablhty. Weaccordmgly callita C_Ktheorem for short,andmoreprecisely ano-e-Ktheorcm, smcean0(thatis,thebounded­ nessofacertamsequence) isemployed inthedetermming hypotheSIS. Atheorem ofthiskindwasfirstprovedbyA.Tauber,-inhiscase,fortheA-process (v. 286);forthisreason,Hardygivesthenameof"Tauberian theorems" toalltheorems illwhIchordmary convergence isdeduced from some typeofsummability. 'Vc shallcallthemconverse theorems or,moreprecIsely, hmltlzing converse oraveragmg j;onverse theorems. §GOTheC-andH-processes. shouldbeconvergent andthatforitsremainder __(/n+l-l_!!J!:t!+en-11+2In+3..487 (n=0,1,2,...) therelation (B) sn+(n+1)en--+-s holds51. Handenotes thepartialsumsoftheseries(A),andaitssum,then (B)assertsthat (B') 1-Sn-(n+1)(a-an)~0, i.e.thattheerror(s-sn)isntimesaslargeastheerror(a-un),except foradifference thatdecreases to0withn. Proof. 1.IfEa"issummable Cl'wehaveby183,sineeav-=sv-Sv-lt "~ay=== ~~_+EfJSv .__+_s~_t-P_ v-~1v+1n+2 .'~"1-1(v+l)(v+2)n+p+2' and,sinceSv=S/-S/-l'onagainapplying Abel'spartialsummation thisbecomes Sn Sn' 11+2-(ti-=-t-2f(n-+-3) +2v_~~(v+1*~~-2f(~+:l)+1/-:1-"/;")+2+(1/+P+;j't:-+p+:l)" Asn~+00,allfivetermsoftherighthandsidetendto0,whatever thevalueofp,forbytheassumed Cl-summability andtheorem :~,Sn=0(n) andSn' =ccc0(n).Hence(A)holds.Atthesametime,keeping nfixed andlettingp-,+00,wcobtain Sn' ~ Sv' Sn+(n+2)en= - 71+3+2(n+2)"';:1 (v+1)(vI2)(~+3)" S'Thistendstos,by221,because n+1~s.Hence(B)alsoholds, sinceen~O.Thus(A)and(B)arenecessary. n.Suppose conversely theconditions (A)and(B)holdgood.Then, ifwewrite Tnfortheexpressions ontheleftin(B),wehave Tnl-l-T"=an+!+(n+2)en+!-(n+1)en =en+all+!+(n+2)(en+!-en) =em andhence Tn='n+(n+1)(Tn+!-Tn). "'KJlopp, K.:CberdieOszillationen einfach unbestimmter Reihen, Rit­ zungsber. Ber!.Math.Ges.,Vol.XVI,pp.45-50. 1917. IJardy, G.H.;Atheorem concerning summable series. Proc.Cambridge Phil.Soc.Vo!.20,pp.304-:!07. 1921. Another proofis~ivenintheauthor's workI.quoted infootnote 42,and another againmG.Lyra.VbereinenSatzzurTheorie derC-summierbaren Reihen. Math.Zeltschr. Vo!.45,pp.559-572. 19:!!J.Thislatterworkhasfurnished the aboveproofofthesufficiency of(A)and(B)fortheCl-summabihty of1:an' 488 Consequently andthereforeChapter XIII.Divergent series. Butowingto't"n-'?s,itfollowsfromthisthatthesequence (sn)islimitable Cltothe\ulues,asrequired 52. WeshJIlcontentourselves withthesegeneraltheorems onC-sum­ mability 53andweshallnowproceed toafewapplications. Amongtheintroductory remarks (pp.461-t(2), itwaspointedout thattheproblem ofmultiplication ofinfiniteseries,whichremained very difticultandobscureaslongastheoldconceptofconvergence wasscrupu­ louslyadheredto,maybecompletely solvedinanextremely simplemanner whentheconcept ofsummability isadmitted. Forthesecondproofof Abel'stheorem (p.322)provides the 276. Theorem 8.Cauchy's productECn=:=E(aubn-I-a1b"_l+...-I-anbu) oftwoconurgent seriesEan=AandEbn~Bisalwayssummable C\ tothevalueC=A.B. Overandabovethis,wenowhavethefollowing moregeneral 277. Theorem 9.IfEanissummable Ca.tothevalueAandEbnisSU11l­ 1I.'ableCfltothevalueR,thentheirCauchyproduct ECn=E(aubn-I-albn_1+...+anbo) iscertainly summable CltothevalueC=A.B,wherey=ex-I-f3-I-1. Proof. I,etusdenote by A(a),B(fll,cry)thequantities whichin 11 1l 1l thecaseofourthreeseriescorrespond totheS;,khsofthegeneralC-process asdescribed in265,3.ForIxI<1,since Eanxn.Ebnxn=Ecnxn, wehave54 1 ~n1 ~ln_1 ~.n (l~-x)~-ti ~{InX•(1_x)9+1~)nX-(1_x)YH~CnX • Hence,by265,3, CCy)=A(>:)B'fl)-I-At")BCfl)+...-I-A(")B(fl). n 0 11 1n-1 n0 .2Thetheorem maybeestahlished similarly forsummablhty e,.;cf.the paperquoted Infootnote 42,p.481. 63Averycomplete accountofthetheoryisgivenbyAndl!Tsen, A.F.:Studier overCesaro's Summablhtetsmetode, Kopenhagen 1921,andE.Kogbetliantz, Som­ matlOn desseriesetintfgrales dIVergentes pardesmoyennes arithmc!tiques et typiques, Memorial desSciences math.,Fasc.51,Paris1931. 6.Sincea"=0(n"),bn=0(n(J),thepowerseriesemployed areabsolutely convergent forIxI<1. §60.TheC-andH-processes. 489 Butfromthisthestatement required followsimmediately, byTheorem 43,6.Weneedonlywrite A(a<j 11X=------- n(n;1-a<)' inthattheorem, sothat nythehypotheses made,wehaveXII;.A,Yn-~n,andthea""clearly satisfythefourrequirements oftheth::orem. HencethelastexpressIOn tendstoA Basn~+00. Examples andRe'lJarks. I.Ifthe,enesE(-1)11ismultlphed byItself(k-1)timcsinsuccessmn, weobtamthesenes (k-I)!];(- 1)1I(Il+k-l) 1111 k-1(k-I,2,. ,,). Theonginal series(k-~1)bemgsummable Clby262,its'lJuareis(certaml}) summablc C3,Itscubesummable C"etc,However, by268,2,weknowtbatthe kthofthcsesene'IS(exactly) summable Cl;' 2.Theseexamples showthattheorderofsumm,lblhty oftheproduct-senes givenbytheorem !JISnotnecessanly thee\'llctorder,andthatInspeCialcasesIt mayactually betoohigh.Thisisnotsurpnsmg, masmucll aswealrcduy know thattheproduct oftwoconvergent ,encs(k=0)maystillbeconvergent. The determinatIOn oftheexactorderofsummablhty oftheproduct seriesreqUires a specialinvestlg,ltlO n meachcase. Inconclusion, wewillinvcstigate onemoretheorem whichmaybe materially cxtended byintroducing summahihty inplaeeofconvergenc'~, -namely Abel'slimittheorem 100anditsgeneralization 233; Theorem 10.Ifthepowerseries.f(x)=.EaTtz"isofradius1andis27S. summable Cktothevaluesatthepoint+-1ofthecircumference oftheunit circle,then foreverymodeofapproach ofzto+-I,inwhichzremainswithinanangle ofvertex+-1,bounded bytwofixedchordsoftheunitcircle(v.Fig.10, p.406). Proof. Asintheproofof2.33,\Vcchooseanyparticular sequence ofpoints(zoo,z!,...,ZA,•••)withintheuniteircleandtheangle,and tendingto+1aslimit.Wehavetoshowthatf(.;A)~s.ApplyTuepfitz' 490 ChapterXIII.Divergent series. theorem 221tothesequence 55an=S~k)/(n~k),whichbyhypothesis converges tos,usingforthematrix(aAn) _(n+k)(1 ~)k+l~naAn- k -~A •'"A' Wededuceatoncethatthetransformed sequence alsotendstos: 00 00 a;'=EUAnan=(l-ZA)kj.1 •ES~k)•Z~-+S.,,-0 ,,-0 SinceES~,k)zn=(1_I Z)k+1fez),thisisexactlywhatourstatement Im­ plied.Forthisprooftobecorrect,wehave,however, stilltoverifythat thechosenmatrix(aAn)satisfies theconditions (a),(b)and(c)ofthe theorems 221.Since ZA->1,thisisobvious for(a);and,since A-·Ea _(I_~)k+IE(Il+k)..,.n_.(l_~)k+I'_!.-k+i-lA-,,~o An-- ~A ,,~Ok ~A- -A (1--ZA)+1--, (c)isalsofulfilled. Thecondition (b)requires theexistence ofaconstant K'suchthat (I1-ZAI)k+l, ~IaAnI=1_IZAI<K foreveryA.Bytheconsiderations onp.40G,thisisobviously thecase withK'=Kk+t,ifKhasthemeaning therelaiddown. Fork=0,thisisexactlytheproofofAbel'stheorem ascarriedout onpp.406--407, inthegeneralized formofStolz.Fork=1,weobtain anextension ofthistheorem, firstindicated byG.Frobenius 56,andfor k=2,3,...weobtainfurtherdegrees ofgeneralization, dueinsub­ stancetoO.HOlder 57-takingHk-instead ofC,,-summability and approaching alongtheradiusinsteadofwithin the angleonly-andfirst expressed intheformprovedabove(though withentirely different proofs) byE.Lasker 58andA.Pringsheim 50. Bythistheorem 10,wehave,inparticular, lim(Eanx")=s,for realx'sincreasing to+I,andaccordingly wecanexpresstheessential content ofthetheorem inthefollowing shortform,whichismorein keeping withthecontext: 279. Theorem 11.TheCk-summabilityof aseriesEantoavaluesalways involvesitsA-summability tothesamevalue. &&kisnowthefixedorderoftheassumed summability . ••Joum.f.d.remeu. angew. Math.Vo!.89,p.262.1880. '7Cf.thepapercItedonp.4H5,footnote 1:1• ••PhI!.Trans.Ray.Soc.,Senes(A),Vo!.l!}6,p.431,London 1901. 69Actamathematica Vol.28,p.1.1904, §60.TheC-andH-processes. 491 Withtheexception oftheC\-summation ofFuurier senc."\\11Ich willbeconsidered morefullyinthefollowing section, furtherapplications oftheC,,-process ofsummation mostlypenetrate toodeeplymtothe theoryoffunctions topermitustodiscusstheminanydetail.Weshould, however, liketogivesomeaccount, without detailed proofs,ofanappli­ cationwhichhasledtospecially elegant results. ThisistheapplicatIOn ofC,,-summation tothetheoryofDirichlet series. TheDirichlet series ~_00(_I)n-l f(~)--L:.-n'111 is611convergent foreveryzforwhich ~ll(z)>0,divergent foreveryother, Atthepoint0,ho\ve\er, whereitreduces tothesenes2:(--1)"-1, 111 itissumm.lble C\tothesum~;atthepoint-1,whereitreduces to N IE(-1)11-1n,itis(cLp.{(j!j)summahlc C2tothesum-1;.1I1dthein- /I1 dications givenin268,:3showthatforz~-(h-1)theseriesissum- "k1rnab1cCIetothesum-;;BI.'foreveryintegral valueofI~-~2. Thisproperty ofbeingsummable Cl"~forasuitable l~,outside its regionofconvergence 9i(z)>0,isnotrestricted tothepomtsmentioned; itcanbeshownbyrelatively simplemeansthatourseries ISsummdble Ckforeveryzwith·m(z)> -k.Moreover theorderofsummabI1ity isexactlykthroughout thestrip -I~<!H(z)< - (/~-I). Thusinaddition tothehoundary ofc01wergence, wehaveboundaries ofsummability ofsuccessive orders, thedomam inwinchtheseriesis certainly summablc toorderkbeing,infact,thehalf-plane m,(z)> -k (h=0,1,2,...). Whereas formerly itwasonlywitheachpointoftherighthandhalf-"" .p(_I)Il-1planeffi(z)>°thatwecouldassociate asumofthesenes"'"..-•--,n wenowassociate suchasumwithe'verypointoftheentireplane,thus defining afunction ofzillthezoholeplane.Inawayquiteanalogous to thatusedforpointswithinthedomainofc01lvergence ofDirichlet's series, further investigations nowshowthatthesefunctional valuesalsoreprc- 60f(z)-(1-22.)•'(z),where'(z)-=1;I.isRielllllnn's '-function. (Cf. n111- 256,4,g.IOand11.) 492 ChapterXIII.Divergent series. 8(;ntananalytic function inthcdomain ofsummability -i.e.111the wholeplane.Ourseriestherefore definesanintegralfunction 61. Quiteanalogous propertlcs ofsummability belongingeneral to everyDiriclzlet series 62 Bcsides theboundary ofconvergencem(z)='\,-or.\0,asweshall nowprefertowrite,sinceconvergence coincides withCo-summability, -wehavetheboundaries gt(z)"'="kforCk-summability, I~~=1,2,.... Theyaredefinedbythecondition thattheseriesiscertainly summable tothehthorderforffi(z)>"bbutnolongersoform(z)<"~.Weof coursehave"0~"12;"2~...,andthenumbers Aktherefore tend eitherto-00ortoadefinite finitelimit.Denoting thisineithercase byA,thegi\enDirichlet seriesissummable Ckforeveryz\vithm(z)>A, \\herehissuitably ehosen, anditssumdefinesananalytic function which ISregular inthisdomain.IfAisfinite,thestraight linem(z)=A iscalledtheboundary ofsummability oftheseries. Fortheinvestigation ofthemoregeneralDirichlet series,(v.p.111, footnote 52) ithasbeenfoundmoreconvenient touseRiesz'R,'k-summation. Cr. thetractbyHardyandRieszmentioned in266,2. §61.Application ofCl-summation tothetheory ofFourier series. Theprocesses described abovepossess theobvious advantage of 211summation processes, namely, thatmanyinfinite serieswhichpre­ viouslyhadtoberejected asmeaningless arehenceforth givenauseful meaning, withtheresultthatthefieldofapplication ofthetheoryof infiniteseriesisconsiderably enlarged. Apartfromthis,theextremely satisfactory natureoftheseprocesses fromatheoretical pointofview liesinthefactthatmanyobscure andconfusing situations suddenly be­ comevcrysimplcwhentheseprocesses areintroduced. Thefirstexample ofthiswasafforded bytheproblem ofthemultiplication ofinfiniteseries (seep.461-2,alsop.488).Buttheapplication ofCl-summation which 81Fromthisitfollows fairlysimplythatforRlemann's '-function thediffer­ ence'(z)-_J_IISanintegral function, -animportant result.z- 82Bohr,H.:OberdieSummabilitiit Dirichletscher Reihen, Gott.Nachr. 1909,p.247,and:HidragtildeDmchletske Rakkers Theon, Dissert., Kopenhagen 1910. §61.Application ofCl-summation tothetheoryofFourierseries.493 is,perhaps, themostelegantinthisrespect,aswellasthemostimportant inpractice, istheapplication tothetheoryofFourierseries,duetoL. Fejh 63.Aswehaveseen(pp.36!J-:370), thequestion ofthenecessary andsufficient conditions underwhichtheFourierseriesofanintegrable function converges andrepresents thegivenfunction isoncwhichpre­ sentsverygreatdifficulties. Inparticular, itisnotknowne.g.whattype ofnecessary andsufficient conditions afunction continuous atapoint XomustsatisfyatthatpointinorderthatitsFourierseriesmayconverge thereandrepresent thefunctional valueinquestion. In§49,C,wcbecame acquainted withvariouscriteriaforthis;butalloftheseweresufficient conditions only.Itwasforalongtimesupposed thateveryfunction f(x) whichiscIJntinuous atXopossesses aFourierserieswhichconverges at thatpointandhasthesumf(xo)there.Anexample givenbyduBois­ Reymond (sec216,1)wasthefirsttodiscredit thissupposition. The Fourierseriesofafunction whichiscontinuous atXomayactually diverge atthatpoint. Thequestion becomes stillmoredifficult, ifwerequireonly-as theminimum ofhypotheses regardingf(x)-thatthe(integrable) func­ tionf(x)shouldbesuchthatthelimit liml(f(xo+2t)+f(xo-2t)]=s(xo) t-'>-I 0 exists.Whatarethenecessary andsufficient conditions 1chiclzmustbeful­ filledbyf(x)inorderthatitsFourierseriesmayconverge atXoandhave thesums(xo)? Aswaspointedout,thisquestion isnotyetsolvedbyanymeans. Nevertheless, thisobscure andconfusing situation isclearedupvery satisfactorily whentheconsideration ofthesummability ofFourierseries -Cl-summability isquitesufficient -issubstituted forthatoftheir convergence. Infactwehavethefollowing elegant Theorem ofFejer. Ifafunction f(x),whichisintegrable in2S0. 0:;::;x<2TTandperiodicwithperiod2TT,issuchthatthelimit lim~[f(xo+2t)+f(.vo-2t)]=s(xo) 1-'>-0 eXists,thetzitsJi'ourier serzesisalwayssunzmable Clatthispoint,tothe values(xo). Proof. Let .aFejir,L.:Untersuchungen uberdieFourierschen Reihcn. Math.Ann. Vol.58,p.51.1904. 494 ChapterXIII.Divergent series. betheFourier seriesofI(x)atthepointxo;weknow,frompp.356-­ 859,thatthenthpartialsummaybeexpressed by 2 2f1 ()] sin(2n+1)tSn=Sn(Xo)= - 2-[f(xo+2t)+IXo- 2t~--t--d tw ~n o (n=0,1,...). Consequently, forn=1,2,.••• So+SI+...+Sn-l 2 =~f~U(xo+2t)+I(xo-2t)]sint+sin3t+~i~'t+sin(2n-1)tdt. o Nowby201,5,wehave,fort'*k7T,.•tsint+sin3t+...+sin(2n-1)t=sl~_lI_,SInt andthiscontinues toholdfort=k7T,ifwetaketherighthandsidetohe inthiscasethelimitoftheratiofort-+k7T,whichisevidently O.Hence 64 So+SI+...+sn-lan-1= n n 2" =n2nf~[f(xo+2t)+f(xo-2t)](9~~t)2 dt. u Ascontrasted withDirichlet's integral, thecritical factorsinn~occurssInt tothesecond powerintheaboveintegral-whichiscalledFejb's integral forshort-andtherefore thelattercanneverchangesign;to thisandtothefactthatthewholeismultiplied by~thesuccessofthen subsequent partoftheproofisdue.Ifthenthelimit lim~[f(xo+2t)+I(xo-2t)]=s(xo)=$ t~+O exists,Fejb'stheorem simplystatesthatUn-+s. Weobserve that since 85theintegrand is 1:sin(2.v-1)t v-I srnt ItHerethearithmetic meanofthenumbers Inisdenoted byUn=Un(x) insteadofbyIn'=In'(x),toavoidconfusion withthenotation fordifferentiation. 15ThevalueoftheintegralmayalsobeinferreddirectlyfromFejir'sintegral itself,forf(x)=1,forwhich Qv=2andtheremaining Fourierconstants =O. §61.Application ofCl-summation tothetheoryofFourierseries.495 andeachtermofthis,whenintegrated from0toi'contributes thevalue TT •2,-smce sin(2v-I)t 2 ( 1._.-.-- =1+2cos2t+cos4t+...+2cos2v- )t.smt Hencewemaywrite "2 s=.~.Js.(si~_n~)2dtnTT smto andtherefore ":i o-s==2J[1~~o..±J..!LtJ(X,,-.=-_!.t) -sJ.(si~nt)2dt. "-1 nTT 2 smt o Byhypothesis, theexpression insquarebrackets tendsto0whent-++O. Inordertoprovethat17"_1ora"-+s,itistherefore sufficient toshowthat Ifcp(t)isintegrable in0..•;and 281. limcp(t)=0, 1-++0 then "ii2 (Sinnt)2 ~TTJcp(t)··sini·dt-+0 o asnincreases. Nowthisfollowsfromaverysimpletrainofinequalities. Ascp(t)-+0, wecandetermine cS<iIforagivene:>0,sothatIcp(t)I<~forevery tsuchthat0<t<cS.Then sincethelastintegral hasapositive integrand, andtherefore remains lessthantheintegralofthesamefunction overthewholerange0toi. Ontheotherhand,aconstant MexistssuchthatIcp(t)Iremains<M throughout°<t<t7t.Consequently "2 I~.f'~(:).(~~nt)\2dtI~2"'[. TT•.-!-.nTT' smt nTT2sm'8 d Ontherighthandside,everything butnisfixed,andwecantherefore 49G ChapterXIII.Divergent series. If(x±2t)-f(x)I<~choosenosolargethatthisexpression becomes<lEforevery 11>no. Wethenhave IUn-l-si<E forthesen's;henceUn-+s.ThusFejer'stheorem iscompletely estab­ lished 66. 282. Corollary 1.If((x)iscontinuous intheinterval 0::::::x::::::2'TT, andif further f(O)=f(2'TT),thentheFourierseriesoff(x)issummable Clto thesumf(x),Jor everyx.Forthehypotheses ofFryer'stheoremarenowcer­ tainlyfulfilledforeveryx,ands(x)=f(x)everywhere. Weassume,asusual, thatthefunction f(x)isdefinedintheintervals 2k'TT:::::: x::::::2(k+])'TT,for k=±1,±2,...,bymeansoftheperiodicity condition,f(x) =f(x-2/w). Wenowfurtherstate: Corollary 2.Withtheconditions ofthepreceding corollary, theCl­ summability, whichhasbeenestablished forallx's,is,moreover, ulII/orm forallx's,i.e.thesequenceoffunctions Un(x)tendsuniformly tof(x)for allx's.Inotherwords:Given E>0,wecandetermine onenumberN suchthatforeveryn>N,irrespective oftheposition ofx,wehave67 IUn(X)-f(X)I<E. Proof. Wehaveonlytoshowthattheinequalities intheproof ofthetheorem canbe arranged soastoholdforeveryx.Now 1 1cp(t)=cp(t,x)=2[f(x+2t)-f(x)]+2[f(x-2t)-f(x)]; sincef(x)isperiodic andiscontinuous everywhere, itisuniformly con­ tinuousforallx's(cf.§19,theorem 5),and,given E,wecanchooseone S>0suchthat foreveryItI<S,andeveryx.Thisimpliesthatforalltheset's Icp(t)I=Icp(t,x)I<~ irrespective ofx;hence,asbefore. 8 1_2J(t)(si~~!)2d tI<E:•n1Tcp Sint 2 o Further, since.f(x)isperiodicandiscontinuous everywhere, itisbounded, sayIf(x)I<Kforeveryx.Itfollowsatoncethatforallt'sandallx's. Icp(t)I=Icp(t,x)I<2K 88Noteinpassingthatthecurvesofapproximation y=Un(x)donotexhibit Gibbs'phenomenon (v.216,4).(Fe,jt'r,L.:Math.Annalen, Vol.64,p.273.1907.) 17Thecorresponding statement holds,moreover, inthecaseofthegeneral theoremofFe,jirforeveryclosedintervalentirelycontained, togetherwithitsend· nnint ingnnnpnintprv!:ll inUThirh'1",'ie.rnntlnllnUG §61.Application ofCl-summation tothetheoryofFourierseries.497 andhence,asbefore,,. 2 l~fm(t).(~i_~_n..~)2dtI51. 2~. n7r" Sint nSin'11 o Nowwecanactually determine onenumberNsuchthatthelastex­ pression remains<!e:foreveryn~N.Forthesen'swctherefore have Ian-l-si<e:,sothat,asasserted, wecanassociate witheverygivene: onenumberNsuchthat Ian(x)-f(x)I<E foreveryn>N,irrespective ofthepositionofx. Asaneasyapplication, thefollowing important theorem resultsfrom theabovetheorems: Weierstrass's Approximation. IfF(x)isafunction continuous2S2a. intheclosedintervala;:;:x~h,andife:>0isarbitrarily assigned, then thereisalwaysapolynomial P(x)withtheproperty that,ina<x<h, IF(x)-P(x)I<e:. Proof. PutF(a+b-.::ax)=f(x).Thenl(x)isdefinedand continuous in°;:;x<7T.In7T;:;X:::::27T,writeasin§50,2ndmethod. f(x)=f(27T-x).Defincf(x)forallotherxb¥theperiodicity con· ditionl(xj-217)=f(x). Thenf(.r:)iseverywhere continuous. Now, forthisf(x),letan(x)havethemeaning laiddowninthestatem:::nt of thepreceding theorem. Anindexmmaythenbefoundsuchthat If(x)-am(x)I<~ forallx.Thisam(x)isthesumofafinitenumberofexpressions ofthe formacospx+hsinqx;henceitcanbeexpanded inapowersenes convergent everywhere, bymeansofthepowerseriesof§24.Let Co+ClX+...+Cnxn+... denotethisexpansion. Sinceitconverges uniformly in0<x::::::17,we candetermine afiniteksothatthepolynomial Co+ClX+...+Ckxk=P(x) satisfiestheinequality Puttingfinallythroughout 0<x<17.Henceitsatisfies If(x)-p(x)1 <e:. p(~=:7T)=P(x), weseethatP(x)isapolynomial oftherequired kind,since,throughout a;:5x<h, IF(x)-P(x)I<E. 498 ChapterXIII.Divergent series. §62,TheA-process. Thelasttheorem of§60hasalreadyshownthattherangeofaction oftheA-process embraces thatofalltheCk-processes. Inthisrespect itissuperior totheC-andH-processes. Also,itisnotdifficulttogive:: examples ofserieswhicharesummable Abutnotsummable Cktoany orderh,however large.Weneedonlyconsider1:a1lxn,theexpansion inpowerseriesof 1 f(x)=e1-O: atthepointx= -1.Sinceobviously limf(x) existsforx-*-1+0 and=ye,theseries1:(-l)nanissummablc Atothevalueve.-If, however, itweresummable Ck,forsomespecifich,by271wcshould requiretohavean=0(nk).Nowaparticular coefficient anisobtained byaddingtogether thecoefficients ofxnintheexpansions oftheindi­ vidualtermsoftheseries,whichisuniformly convergent forIxIse<1: -...!:..- 1 1 1 1 1 e1-0:=1+l=-~+21(1-X)2+...+;y(1-x)v+.,. (v.249).Asallthecoefficients intheseexpansions arepositive, anis certainly greaterthanthecoefficient ofxnintheexpansion ofasingle term.Pickingoutthe(h-+2)tbterm,weseethat 1(n+k-+-1)nk+1 an>(k+2f! k+1>(k+2)!(k+1)!' Forafixedh,an/nktherefore cannottendto0;onthecontrary, ittends to+00. Although theA-process isthusmorepowerful thanalltheCk-processes takentogether, itis,nevertheless, restricted bytheverysimplestipulation thatinorderthatitmaybeapplicable toaseries1:amtheseries1:anxn and1:Snxnmustconverge forIxI<1: 283. Theorem 1.Iftheseries1:an,withpartialsumsSmissummable A, ecessarily have Hmvra:-I ;51andlimvlSJ;;:;1 or,whatcomestoexactlythesamething, an=0«1+e)n)andSn=0«1+e)n)I foreverye:>0,howeversmall. Inthiswehaveacompanion totheorem 3of§60;buttheorems 4:and5ofthatsectionalsohaveliteralanalogues inthisconnection: 28'J. Theorem 2.Inaseries1:amwhichissummable A,wenecessarily have A-Hman=0a"dindeedA-Hm(Q1+2Q~--~';' +~~)=o. §62.TheA-process. 499 Proof. Thefirstofthesetworelations indicates that(1-x)Eanx" musttendto0asx-+1-O.Thisi<;almostobvious, sincebyhypothesis Eanxn-+s.Thetruthofthesecondstatement follows, onthesame linesastheproofof273,fromthetworelations (&)(1)... and(1)...So+Sl+...+Sn~ -X'':''Snxn-+S -x.~ n+1x"-+$ bysubtraction; thefirstoftheseisnothing morethananexplicitfonn ofthehypothesis thatEanissummable A,whilethesecondisquite <:<lsilydeduced fromit.Infact,from(1-x)ESnxn-+S,wefirstinfer that (1-X)2E(so+SI+...+sn)xn-+S, hy102.Thatthesecondoftherelations(*)followsfromthis,isaspecial caseofthefollowing simpletheorem: Auxiliary theorem. If,forx-+1-0,afunctionf(x),whichis2Sl'i. integrable in0<x<I,satisfiesthelimitingrelation (1-X)2f(x)-+s, then,forx-+1 -0,wealsohave x (1-x)ff(t)d t=(1-x)F(x)-+s. () Theprooffollowsimmediately fromtheruleknownasI'Hospital's, bywhich . F (C\:) .F'(C\:) hmG(x)=hmd~(x)'X-+I-0 x--'>-l-O provided therighthandsideexistsandG(x)ispositive andtendsto +00asx-+1-O.Thedirectproofisasfollows 68: Put(1-X)2f(x)=S+e(x).Thefunctione(x)tendsto0as x-++1 -0,andsoforanygiven E>0,wecanassignanXlin0<Xl<I, suchthatIe(x)Iremains<~forXl<X<1.Wethenhave,forthese valuesofx, X, 1(1-x)F(x)-sI>(1-x)'1S1+(1-x)./f(I~~x~)2dxI+~. o Fromthisthestatement followsintheusualway. Bythesetheorems 1and2wehavetosomeextentfixedouterlimits totherangeofactionoftheA-process. Asbefore(cf.thedevelopments 68Theproofisonquitesimilarlinestothatin43,1and2.-Themeaning oftheassertion underconsideration, thatthefirstoftherelations (.)impliesthe second,mayalsobestatedthus: A-limsn=Simplies AC1-limIn=I. Forinthecaseofthesecondrelationweareconcerned withthesuccessive appli. cationfirstoftheC,-orocess. andthenoftheA-orocess. forthelimitation of(s_). 500 Chapter XIII.Divergent series. 011p.4HfJ.fj),th-:question a~toth~point,beyondtheregionofserieswhich actually converge, atwhichitsactionbeginsisamuchmoredelicate one. Inthisconnection wehavethefollowing theorem duetoA.Tauber 89: 286. Theorem 3.Aseries.Ean,whichissummable A,andforwhichnan~0, i.e.forwhich an=0C} isconvergent intheusualsense.(o-A->-K-theorem.) Proof.Ifweare everyn>110 e: a)Inan1<3'given e;>0,wecanchoose 110>0sothatfor (Hcrea)andc)canbesatisfied byhypothesis, andb)byreferring to 43,2.)Forthesen'sandforeveryp03itive ,x<I,wethenhave n ~ sn-s=f(x)-s+ .Eav(1-xv)-.Eavxv• v-1 vllH Ifwenowobservethatinthefirstofthesums (1-xv)=(1-x)(1+-x+-...+-xV-1)~IJ(I-x), cl·h clI I Ivavle:· CIIhanIIIt esecon av=v<an'1t10owstat 11 e:ISn-sI~If(x)-si+(1-x),_~IIJavI+-:l~;-U-.::--;Y foreverypositive x<1.Choosing, inparticular, x=1 -1,weobtain, 11bya),b)andc), forevery 11>no.HenceSn->S,q.e.d. Inthisproof,ifweinterpret e;asbeing,notanarbitrary prescribed positive number, butasuitably chosen(sufficiently large)onc,thenwe mayinferthefollowing corollary: Corollary. Aseries.Ean>summable A,with(nan)bounded, i.e.one forfvhich hasboundedpartialsums. Onaccountofthegreatsimilarity between thistheorem andtheorem 6of§60,itappearslikelythatanO-A~K-theorem alsoholds,i.e.one 89Tal/ber,A.:EinSatzausderTheorie derunendliehen Relhen, Monats· heftef.Math.u.Phys.,Vo\.8,pp.273-277. 1897.Cf.p.486,footnote GO. §62.TheA-process. 501 whichdeduces theconvergence ofEanfromitsA-summahility, hyas­ suming, asregardsthean's,mcrelythefactthattheyarc0(~).This theorem isactually true.Itgoesverymuchdeeper,however, andwas provedforthefirsttimein]910,byJ.E.Littlewood 70: ITheorem 4.AseriesEamwhichissummable A,andwhoseterms2S7. satisJytherelation -i.e.Jorwhich(nan)isbounded, -isconvergent intheordinary sense. (O-A--*K-theorem.) Beforegoingontotheproof,wemaymention thatthistheorem contains, asacorollary, Theorem 6of§60,asalreadystatedthere.For ifaseriesissummable Ck,thenbyTheorem 11,§60,itisalsosummable A.Everyseries,therefore, thatsatisfiestheassumptions ofTheorem 6, §60,alsosatisfiesthoseofLittlewood's theorem juststated,andistherefore convergent. Previously knownproofsofLittlewood's theorem wereverycom­ plicated, inspiteofthenumberofresearches devoted toit71,tillinH)30 J.Karamata 72foundasurprisingly simpleproof.Weshallprefacehis argument withthefollowing obvious lemma: Lemma. Leteandebearbitrary realnumbers, andletJ(t)denotethe following function 73,definedandintegrable (intheRiemann sense)overthe interval°<t:::::::1(v.Fig.13): fOin0:::::::t<e-(l+el J(t)= .~ine-~+J)::::::: t<e~\ loine-1<t<1. Thenthereexisttwopolynomials p(t)andP(t)Jorwhich (a) (b)P(t):::::::J(t)<P(t) inO<t<I, 1J(P(t)-P(t))d t<e. o 10Theconverse ofAbel'stheorem onpowerseries:Proc.Lond.Math. Soc.(2)Vol.9,pp.434-448. 1911. 11Besidesthepaperjustmentioned, cf.E.Lalldau, Darstellung u.Begrun­ dungeinigerneuererErgebnisse d.Funktionentheorie. 1"ed.pp.45--46. 1916; 2nded.pp.57-62. 1929. 12Karamata, J.,DberdieHardy-Littlewoodsche Umkehrung desAbelschen Stetigkeitssatzes. Math.Zeitschr. Vol.32,pp.319-320. 1930. 13Thetheorem holdsunaltered foreveryfunction integrable inthesenseof Riemann. 502 Chapter XIII.Divergent series. Proof. LetOAA'B'BE (cf.theroughdiagram, Fig.1:3)bethegraph ofthefunctionf(t), sothatAandA'havetheabscissa e-(1lg),whilethat ofBandB'ise-l•Nowchooseapositive 8lessthantheabscissa ofA, lessthanhalfthedifference between theabscissae ofAandB,andfurther­ more e:<4:e-(lle). Onthegraph,markthepointsAI'A2withtheabscissae e-(1+e)±8,and thepointsBI,B2withtheabscissae e-1±8.ThenthelinesOAA2BIBE o\ \\ \\\\ Fig.13.E andOAIAIB'B2E(withA2B1andA'B'takenalongthecurve ~,theother portions beingstraight) arethegraphsoftwocontinuous functions g(t) andG(t)respectively, forwhich,obviously, (a') (b/)g(t)~f(t)< G(t)inO~t~1, I J(G(t)-g(t»dt<~. o (282a),thereexistsapolynomial p(t) thecontinuous function g(t)-~inthe 4ByWeierstrass's approximation thatdiffersbylessthan~from interval 0<t:'S1: Ig(t)-~-P(t)I<~III0~t<1. Similarly thereexistsapolynomial P(t)thatdiffersbylessthan:from G(t)+~there: IG(t)+~-p(t)1<~in0::;;;t<1. Thesepolynomials clearlysatisfytheconditions (a)and(b)ofthelemma. §62.TheA-process. ProofofLittlewood's theorem.503 1.Bythecorollary totheorem 3,thesequence (sn)'underpresent hypotheses, iscertainly bounded. Inproceeding withtheproof,itwillbenorestriction toassume t:le termsoftheseriesEantobereal.For,oncethetheorem isprovedforreal series,itcanbeinferred immediately forseriesofcomplex termsbysplit­ tingtheseupintotheirrealandimaginary parts. n.Letebegiven>0,andput[(1+e)n]=k(n)=k.LetSndenote asusualthepartialsumsofEamand,forn>!,write74e MaxISv-SnI=fLn(e), n<v~_k and limfL"(e)=fL(e)· ,,-)-I7J ThenfL(e)--+0ase--+O. Indeed, forn<JJ::;k,wehave ISv-s"I-=Ia"H+a"+2+...+avI <(k-11)Max(Ia"+lI,Ia1112I,.., IakI)· IfIarIbethismaximum, itfollowsfurtherthat ISv-SnI:s;k_-n.rIarI~e.'"IarI. T Now,(11a,,)beingassumed bounded, thereexistsaconstant Ksuchthat nIani<Kforall11,andso fLn(e)=MaxISv-SnI::::::eK. n-:"v'5.k Thus whencethestatement follows.fL(e)::::::eK, Ill.Suppose thesequence (sn)is (i)bounded ononeside,saySn2-M,(M>0); en (ii)limitable A,say(1-x)ESvXV--+s,forX-~1 -O. v-o aThesymbolMax(tl't2••••,tf)'orMaxt,-(1~J~p).denotes thelarge'lt ofthenumbers tlot2•••••tf)(assumed real). 504 ChapterXIII.Divergent series. Then 75Iff(t)denotesthefunction definedinthelemma, ~ 1 ('it') (1-x).Es"f(x")•x"~sIf(t)dt,i.c.=es. ,,-0 0 For,by2,wehave,foreveryintegerk>0, ""(1-Xk+I).Es".(xk+l)"~S, ,,-0 asx~1 -0,so NowifQ(x)=ba+hIx+...+haxqisanypolynomial, itfollowsat oncethat (1-x)Es"Q(X")XV-)-S(~o+~'+...+_ba_)=s.jQ(t)di. ,,-0 q+1 0 NowletEdenoteanypositive number. Thenapairofpolynomials p(t),P(t)canbeassigned, bythelemma,sothat (a) (b)p(t)<f(t)<P(t) inO<t<l, 1f(P(t)-P(t»d t<E. o FirstassumeM=0,sothats">°forallv;then (Xl 00 (LJ (1-x).ESlIP(X").x"<(1-x)Es"f(x")· x":::;;(1-x)Es"P(xv).x". ,,~O ,,-0 ,,~O Forx~1 -0,itfollowsby(U),that 1 'l) 1 s·Jp(t)d t<lim(1-x)Es"f(x").x"<sJP(t)dt. o -- v-u u By(a)and(b),theintegrals ontheleftandontherightdifferfromeach 1 otherandfromJf(t)dtbylessthane.Hence o Ili~(l-x)£s,,/(x")· x"-sJf(t)dil<s·E.,,-u U Since E>°wasarbitrary, itfollowsthatthestatement ('it')IStruefor non-negative Sn' 7'ThISis:7.Karamata's MainTheorem. Boththeorem andproofapplyun­ alteredtoanyfunction integrable IntheRiemann senseover0~t~I,-except 1 forthespecialvaluepsoftheintegralJf(t)d tinourcase. o §62.TheA-process. 505 1/however M>0,applythetheorem assofarprovedtothetwo sequences tn0:=sn+MandUn:=MinsteadoftoSn'Subtracting the results,weget(*)initsfullgenerality. 1IV.InIII('I),putx=e-nonthelefthandside,andreferbacktothe definition of/(t)inthelemma; writingasbefore[(1+e)1l]=k(n)=k, weinferthatasn-+<Xl, 1k (1-e-ii)1:Sv4-eS. v=n+1 Since wehavethen Writing, therefore, wehavean_0,andI k--1:Sv4-S. k--tlv-'n11 k k~;/1:Sv-s=an. v-n11 Hence Making n_+<Xl,wededuce limIs-SnI:s:;p.(e); andasthisholdsforeverye>0,itfollowsbyIlthat,fore_+0, limIS-SnI=0, I.e. Thiscompletes theproof. 28S.506 ChapterXIII.Divergent series. Examples andApplications. 1.Everyserieswhichissummable Cisalsosummable A,tothesamevalue. Thiqoftenenables ustodetermine thevaluesofserieswhicharesummable C. Thusin268,3wesawthattheseries1:(-1)"(n+1)karesummable Ck+1;by meansoftheA-summation processwecannowobtainthevaluesoftheseseries, whIchoccurconveniently asthekthderivatives ofthegeometnc serIes1:(-1)"x", whentheexponential function ISinserted bysubstitutmg x=e-t•Inthiswaywe obtaintheseries convergent fort>O.Thesumofthisseriesis e-t 1 =1~t:e=c =;F+-iet+1 - 2 1 2-e2t-=-I-- ~"et--1 -e2t__l Forasufficiently smallt>0,theselastfractions maybeexpanded inpowerseries by105,5;thefirsttermsofthetwoexpansIons canceleachotherandweobtam -t_~2t _ _n-(n+l)t __ 002n+1-I ne e+...+(1)e+...-E(-1)'RnIIt. n~O 11+- Differentiating ktimesinsuccession wIthrespecttot,wefurtherobtain 002,,+t- 1=(-1)k+tE(--+1)-'-Bnel•n(11-1)...(n-k+1).t"-k. n.kn . Now,letting tdiminish and-++0,weatonceobtainontherighthandSIde'B 2k+t-1(-1)k+tk+1BU1. Puttinge-t=xonthelefthandside,weseethatwcaredealingwithapowerseries ofradIUS1;whenIdecreases to0,xincreases to+1.Thevaluejustobtamed is therefore bydefimtion theA-sumoftheseries 1 -2k+3k-+...+(-1)"(n+1)k+... forintegralk~O.Andasthisserieswasseentobesummable Ck+1in268,3, wehavethusobtained itsCk+csum also,by279. 2.Ifthefunction represented byapowerseries1:c"z"ofradiusrisre1!ular atapointZIofthecircumference oftheunitcircle,limf(xZI)forpositive in­ creasing x-++1certainly existsand=f(ZI)'Ateverysuchpointtheseries 1:an~1:CnZl"istherefore summable AanditsA-sumisthefunctional value f(z,), ,.Fork>0,thesign(-1)k+tmaysimplybeomitted, byfootnote 4,p.237 §63.TheE-process. 507 3.Combining thepreceding remark withtheorem 4,wegetthestatement: Iff(x)=.Ee"z"converges forIz[<1and(nen) i~bounded, thentheseries contInues toconverge (intheordInary sense)ateveryPOIntZ1'onthecircumference oftheunitcircle,atwhichf(z)ISregular. 4.Cauchy's product 2:c"=:E(aob"+...+a"bo)oftwoseries2:a"and 2:b",whicharesummable Atotbevalues AandB,isalsosummable A. tothevalueC=AB,asanimmediate consequence ofthedefinition ofA-sum­ mability. 5.Withregardtotheseriesi__1_.,et::z.0,wehavealready seenn1+u, ,,~l 10274thatthesedonotconverge, andthattheyarenotbummable Cktoany orderk.ByLzttlewood's theorem 4,wemaynowaddthattheycannotbe bummable Aeither. §63.TheE-process. 77 TheEl-process wasintroduced onthestrength ofEuler's trans­ formation ofseries(144). Starting fromanyseries2'a..(nothaving alternately+and-signs),weshouldhavetowrite 2}+1[(~)ao+G)al+...+(:)a,,]=a,,' andweshould havetoconsider .:f:a,,'astheEl-transformation of 1:an''Yehadagreedtodepartfromtheusualnatation sofarastowrite78 So=0andSIt=ao+a1+...+a"_lfor1l>0,-andsimIlarly for theaccented series.Then(v.265,5) 5,,'=~n[(~)So+G)51+...-I-(:)5,,] IStheEl,transfo,mation ofthesequence (5,,).Applying thisagain, weobtainfortheE:a.transformation, afteraneasycalculation, ~a,,", witha""=4}+i[(~)3"ao+G)3n-lal+ ..·+(:)a,,] thepartialsumsofwhicharenow "+"++""aoal• • •a"-1=5" =4\[(~)310So+(7)3,,-151+...+(:)510],(n>0). 77Adetailed investigation ofthisprocess istobefoundintwopapers bytheauthor,Obe"dasEulersche Summlerungsvl'r!ahren (I:Mathemat. Zeitschr. Vol.15,pp.226-253. 1922;11:ibid.,Vol.18,pp.125-156. 1923). Complete proofsofallthetheorems mentioned inthissection aregiventhere. 18Itmaybeverified without muchdifficulty thatinthecaseoftheEl­ process "afinitenumber ofalterations" isallowed, asinthecllseofcon­ vergent series. (Aproof, intowhichwesllllllnotenterhere,isgiven in thefirstofthetwopapers mentioned inthepreceding footnote.) Conse­ quently theshifting ofindices hasnoeffectontheresultofthelimitation process. 508 ChapterXIII.Divergent series. Forthegp-transformation weobtaininthesamewaytheseries Ea(P) n withterms a~P)=(2P~n+l[(~)(2P-l)nao+(~)(2P-l)n-lal+...+C)an] andpartialsums 79(n>0) arip)+alp1+...+a;;~1=s<,:') =(2~)"[G)(21'-l)nSo+G)(21'-1t-lSI+...+(:)Sn]. Theexamples givenin265,5havealready illustrated theaction oftheEl'process; thelastofthemshows thattherangeofthe El'process isconsiderably widerthanthoseoftheC-andH-processes. Byanalogy withthatexample, wemayformtheEp-transformation of thegeometric series ~zn,andweshallobtain ao[1 n(11)l'n-"]1'"(21'- 1+z)n~(2p)n+-l,.~ v(2-1)z,.=21'n~ 21'. Thisseriesconverges 80,-tothesum1~'-if,andonlyif,-zIz+(21'-1)I<21',i.e.ifzlieswithinthecircleofradius21'round thepoint-(21'-1).Evidently everypointinthehalf-plane lR(z)<1 canbemadetolieinsidesuchacircle,bytakingtheexponentp sufficiently large.Wemayaccordingly say:Thegeometric series ~:zn issummable Eptoasuitable orderpforeachpointzinterior tothe half,plane lR(z)<1.Thesumisineverycase1~z'i.e.itisthe analytical extension ofthefunction defined bytheseriesintheunit circle. Thecaseofanypowerseriesisquitesimilar, butinordertocarryout theproofswerequire assistance fromthemoredifficult partsoffunction theory. Weshalltherefore content ourselves withindicating themosttangible results 81: ThepowerseriesIenz·isassumed tohaveafinitepositive radius of convergence, andthefunction whichitrepresents inItscircleofconvergence isdenoted byf(z).Thisfunction wesuppose analytically extended alongevery rayamz=rp=const.untilwereachthefirstsingular pointoff(z)onthis ray,whichweshalldenotebyC'P'(Ifthereisnosingular pointatallonthe ray,itmaybeleftentirely outofaccount.) Foraparticular integerp>0 wenowdescribe thecircle IZl'Ip1,;+2-1<2, whichcorresponds totheoncoccurring inthecaseofthegeometric series, andwhichweshalldenotebyK'P'Thepointscommon toallthecirclesKtp 7.Theformula ofthistransformation sugg"ests that,fortheorderp,the restriction tointegers ~0mightberemoved. Here,however, weshallnot entermtothequestionOfthesenon-integral orders. Cf.p.467,footnote 18. 80Thisseriesisthen,moreover, absolutely convergent. 81Asregards theproof,seep.507,footnote 77. §63.TheE·process. 509 will,inthesimplest cases(Le.whenthereareonlyaslllallnumber ofsingular points), makeupacurvilinear polygon whospboundary con~i~t~ ofarcsof thedifferent cIrcles, andineverycasetheywillC01111adefinite setofpoints whichwedenoteby(»p.Wethenhavethe Theorem. Foreachfixedp.2.'Cnzn~ssummuble Epateverymtenor pointof289. (MpandtheEp-transformahon of~Cnzn~sIndeedabsolutely convergent atthat POint.Thenumerical l'aluesthusassociated WitheveryInterior pmntof(Mpform theanalytical extension oftheelement 2."Cnznmtotheentenor of(Mp"Outside (Mp, theEp'transformahon of~CnznISdIVergent. Afternotingtheseexamples, wenowreturntothegeneralquestion, withinwhatbounds therangeofactionoftheE-process lies;inthis asinfurtherinvestigations, weshallrestrictourselves tothefirstorder, i.e.totheEl'process. ThecasesofE-summation ofhigherorders are,however, quiteanalogous. SinceE1·summability ofaseriesIa"means,bydefinition, the convergence ofitsEl,transformation Z2.1+1[(~)ao+...+(:)an]' thegeneral termofthelattermustnecessarily tendto0: and272 here.For s,itsEl"to82,1 analogue thevaluewhichwemaynowwrite,forshort, E1,lima,,=O. Inthisform,weagainhaveanexactanalogue or284.Kronecker's theorem82,3alsohasits If(s,,)isasequence whichislimitableElwith transformations El(sJ==s,,'--tos.Thearithmetic means so'+s/+...+SN' --n+l ofthelattertherefore alsotendtos;wemaydenotethemforshortby ClEl(s,,),sincetheyareobtained byapplying insuccession firstthe El,transformation andthentheCl·transformation. Nowitiseasyto showbydirectcalculation -weprefertoleavethistothereader-, thatweobtain exactly thesameresultifweapplytheCl'transfor­ mationfirst,andthentheEl-transformation, i.e.ifweformthe EC()E(So+s,+...+SN) )sequence 1 1S"=1n+l :WehaveClE1(sn=E1Cl(Sn); thetwotransformations arecompletely identical 82.Thuswealso 8.Bycalculation, theidentity tobeproved isatoncereduced tothe relation (k+1)+(k+1)+...+(k+1)=(k)+2(k-1)+...+2k-n(n)n+ln+2 k+lnn nI for0:;;;;n:;;;;k,whichiseasilyseentobetrue.-Onaccount oftheproperty Inqllestion, theEr-andC,·transformations arcsaidtobecommutable. The corresponding property holdsgoodforEp'andCfl'transformations ofany order;inel:erycase,E'I)Co(s,,)==CqE'J}(sn).Cf.p.482,footnote 45,andp.433. 17- (<:51) 510 haveChnpter XITIDivergent series. E(30+"'i~._.+Sn)--.S 1 n+1 ' andsubtracting thisfrom £1(Sn)-+S, weobtain,exactlyasonp.~85,therelation stated,namely E.lim ~l+2a,+-'-'.±n an=0 1 11+1 ' asanecessary condition fortheEl'summabllity oftheseries2a,,' Todetermine further whatthecondition El-Jiman=0implies as regards theorderofmagnitude ofthetermsa",wededuce from a'=-!--[(n\)a-t-...+(n)a]n2,,+1 0 0 11n theexpression forthean'sintermsofthean"s: an=(-1)".2[(~)ao'-(7)2a/+ -...+(-1)n(:)2"au'], whence, asa,,'-+0,itatoncefollows, by43,5,that I!.'!-+O3"or Ifwecarryoutthecorresponding calculation larsnands,,',we~imi· ladyfindthats..=0(3").Summing up,wetherdore have 290. Theorem 1.Thelourconditions Eli 0E-llnllll +_2..,'l~.i:.." .+11U"-_0, 1 -ma"=, 1 --11-I-I an=0(3") ands"=0(3") arenecessary inorderthattheseries2;an,withpartialsumss". maybesummable El' Acomparison ofthistheorem withthetheorems 271and283 andtheexamples for265.5showsthattherangeoftheEeprocess is considerably moreextensive thanthoseoftheC-andA-processes; the El-process isagooddealmorepowerful thanthese.ThequestIOn, how­ ever.whichinthecaseoftheC-andA-proc('sses ledtothetheorems 274and287,hererevealswhatmaybedescribed asalossofsensitiveness intheEl-summation process, ascompared withtheC-andA-processes. Weinfacthave 291. Theorem 2.11theseries:Eall'withpartialsumss",issum- mableEltothevalues.sothatthenumbers (A) s,,'=2\[(~)So+G)SI+...+(:)s,,]-+s, andil,besidesthis,wehave fB) an=0(v~), thentheseries2anisconvergent withthesums·(a-Et-+K-theoremJ. §63.TheE-process. Proof. 'Veformthedifference511 andwesplituptheexpression ontherighthandsideintothreeparts: 1'1+1'2+Ta·1\istodenotethepartfromv=0tov=n,Tathe partfrom ~,=:311tov=111,and1'2thererr.aining partinthemiddle. Invirtueof(B)therecertainly exists-roughly estimated - aconstant KlsuchthatISItI:5K1v'n.-Hencetherealsoexistsaconstant Ksuch that ISv-S2n[:SKv'n forevery11,provided 0s:;:v~411.HenceI1'1iand:Ta[areboth I - ",(4n)Kv'n(1n)<2'".K·V11vL,ov< -24n(n-I-1)n • Nowforeveryintegerh>-1,wchave83 eet<h!<e1~(:/. andaccordingly 84 I(In) I4114m(IG)" 24r1 11<vi,'e:J'"<211:n. Therefore 1'1andTJbothtendto0asnincreases. In1'2'i.e.for11<V<311,\\ehave,by(B), ISv-S2"I<-:~nI2n-vI,vII ifEndenotes thelargestofthevaluesIa..+11'\in-=,-=-1,Ian+21vn+2,.••; Enmusttendto0as1Iincreases. Therefore Il'I<:~,,_~:111;)I')_ I(411)<2~n_~~(2_ )(4n)2 0-'H" £.., ~11V 'Hn.,.£.., 11V •\11-)Jlltl v -v11vO v Thislastsumishowever ea~ilyseentohavethevalue 11(:n);itsuffices ~71 toseparate 211-vinto2nand-v.Thus ITI:<:::::2C:n~/n(4n). 2-2'" 271 88Thissomewhat roughestimation for!l',",hich,however, isoftenuseful, ismostsimplyobtained bymultiplying together allthemequahtles (I+-~y<e<(I+DV+l (sce46a)forv-I,2,•..,!l-1. 8'Substitute(4n)=_~4(3n)')Ianduseinthenumerator theupperc:'stimaten 71.n. fork',inthedenominator thelowerestimate. 512 ChapterXIII.Divergent seriea. Thus,as8..-0,wehave,by219,3, T'.l-O. Summing up,wetherefore have Tl+T'J+Ta=s:n-s2ft-O. Nowbyhypothesis s:n-s,hence S2"-salso;andfurther, since a._0by(B),itfinallyfollows that q.e.d.85• Inconclusion, weshallalsoconsider thequestion oftheE-summa­ bilityoftheproduct oftwoserieswhicharesummable El'aswell asthatoftherelation oftherangeofactionoftheE-process tothat oftheC-process. Asregards themultiplication problem, wehavetwotheorems, whicharetheexactanalogues ofMertens' theoremISSandAbet's theoremIS9.Weconfine ourselves tothedevelopment oftheformer andwetherefore proceed toprove 292. Theorem 3.LetthetwoseriesXanandXbnbeassumed tobe summable El'i.e.lettheirEl-transformations, whichweshalldenoteby Ian'and2,'bn',beconvergent. 11oneatleast01thetwolatterseries converges absolutely, thenCauchy'sproduct XCn=Z(aobn+atbn-l+...+anbo) isalsosummable El'andbetweenA.B,andC,theEl-sums 01the threeseries,wehavetherelationA·B=C. Proof. By265,5,forx=-1~-andforallsufficiently small-y valuesofx(v.theorem 1),wehave fl(x)=2.'a"X,a+l=2.'all'(2y)n+t, f'J(x)=2,'b"xn+1=Xbll'(2y)n+t, fa(x)=ZCnxn1-1=Xcn'(2y)n+l. Ontheotherhand fl(x).f2(x)=x·fa(x). Thuswehavetheidentity (2y)·:£(ao'bn'+aI'b~_l+...+an'bu')(2y)n+l=1~y2.'cn'(2 y)nH, 81Thetheorems 274and287suggest thatanO-E-+K-theorcm may alsoholdhere,i.e.onewhichenables ustoinfertheconvergence ofl:an fromitsE1-summability Iprovided thatan=0(J;). Thisisactually the case,buttheproofissomuchmoredIfficult thantheabovethatwemust omitithere.(Cf.thesecondofthepapersreferred toonp.507,footnote 77.) §63.TheE-process. 513 whence, besidesco'=2aa'bo',weobtainthcgeneralformulaforn~1: c'=2(a'b'_La'b'-1-'"+a'b')--(a'b'1-j-...-I-a'1b')11 an1 In-l n0 0 n- n- 0 • Sincebyhypothesis oneatleastofthetwoseries2'an'and2'bn'is absolutely convergcnt, Cauchy's product, .2:(ao' bn'+..,+an'bo'),of thesetwoseriesisconvergcnt and=A·E,byISSFromthelast· obtained expression forcn',theconvergence of2'cn'followsatonce, andforitssumCweobtain C=2AB-AB=AB, q.e.d.86 Finally, weshallexamine thequestion oftherelation between theC·andE-processes. Itisveryeasytoshow,inthefirstin· stance,thattheprocesses fulfilthecompatibility condition 263,IlI; i.e.thatwehave Theorem 4.Itaseriesissummable Clandalsosummable El'the293. :twoprocesses giveitthesamevalue. Proof.If(cn')istheCl·transformation and(s,,')theEl,tram· 'formation of(sn)'boththesesequences areconvergent, byhypothesis: saycn'-+c',s,.'-+s'.Sincebothprocesscs satisfythepermanence -condition, therl'trans[ormation of(sn')alsoconvcrges tos': so'+st'+...+SIl' ,---n+1 -+s :amithcEl·transformation of(cn')converges toc': 21 "[(~)Co'+...+(:)cn']-+c'. Withtheabbreviated notation, thesetworelations are ClEl(s..)-+s'andElCl(s..)-+C'• But,aswaspointed outonp.509,thesetwosequences areidentical, sothats'mustbeequaltoc',q.e.d. Wehavealreadyseen(p.508),fromtheexample ofthegeometric series2'z",thattheEl'process isconsiderably morepowerful than theCl-process. Infact,wecannotsumthegeometric seriesbythe latteranywhere outside theunitcircle,whiletheEl'process enables ustosumitateverypointofthecircleIz+11<2.Butthismust notbeinterpreted tomeanthattherangeofactionoftheE1,process 'Complctely includes thatoftheCl'process, muchlessthoseofall Ck,processes. Onthecontrary. itiseasytogiveaninstance 01ase· 88Theformofthisproofsuggests Ihatincertain casesitwi1lbecon­ venienttointroduce theconcept ofabsolute summability: AseriesIa..willbe saidtobeabsolutely summable ElifIa..'.itsEl'transformation, converges absolutely. 514 Chnpter XIII.Divergent series. quence(Sn)wlzichislimitable Ct,butnotlimitable Et'Thesequenc(' (sn)==0,1,0,0,2,0,0,0, 0,3,0,... isofthistype,wheres.,=yandeveryothersn=°(i.e.forevery indexnwhichisnotaperfectsquare). 1Thissequence islimitable Clwiththevalue"2'Forthelargest valuesofthearithmetic means So+5,+'1'.+5"areobviously attamedn+ forn=y2andtheleastfor1t=y2--1.Thelatter=v(~~.!l,theformer (v+1)v .1. 1=20-2+ 1)'bothofwluch-"2; 1c.Cl(sn)-i' Ifthesamesequence werealsolimitable Elweshouldt!1f'fcfore require tohaveEl(sn)-i:but,forn=y2,the(2n)thtermufthe Et·transformation is 2~"[(20n)So+...+C:l)sn+...-I-(~~)S2"J~i,,,(2:)V;. TheexpressIOn ontherighthandsidetendsto~~by2lU3,so V·7 , thatforallsufficiently largen'sthetermsremain>%>-~andEl(sJ 1cannot-"2'Thesequence (s,.)istherefore notlImitable El'Wemay accordingly state: 294. Theorem 5.0/thetworanges.that0/theCl"processawlthat 0/theEt-process. neithercontains theotherentirely. 17tereareseries whichcanbesummed bytheCt"process. butnotbytheEl-process. andconversely 87• Thiscircumstance raisesthefurtherquestion: whichseries, Sll1ll1l1­ ableCl'canbesummed bytheEl'process? Littleisasyetknownon thissubject, andweshallcontent ourselves withmentlonmg thefol.ow­ ingtheorem: 295. Theorem 6.1/2,'anissummable Cl'with So+S.+...!-3..=st0(__~) n+1 .jn' thenXanisalsoslI11ll1lableE l.(o·CI-El,theorem88.) .7Thestatement remains thesamewhenhigherordersofbothprocesses areconSidered . ••Thecorrespondmg O-theorem doesnothold,ashasalreadybeenshown bytheexample ontheorem 6,wherethcarithmetic meanISactually 21+0(-!-).VII §63.TheE-process.515 Proof.Writmg s..- s=an'thesequence (a..)ISlimitable Clwith thevalueO.Put t10+t1,+...+t1n In+1=a..j bythehypotheses, wethenhavenotonlya..'-+0,butalsoY;an'-+O. Nowwehavethefollowing general inequality, duetoAbel:If b ' (1u+(1,+...+t1"( )00,01,•••,a"areanynumers,a"=v+1 ' v=0,1,...,n . thecorresponding arithmetic means; if,further, 7:isanumber greater thanallthe(n+1)quantitiesIa,,'I,forv=0,1,"..,n,and'r"a number greaterthanallthequantitiesIa,,'I,fork~v<n;then,given anysetof(n+1)positive numbers ao'al'...,a..,whlchincrease monotonely toth::termamanddecrease monotoncly fromthatterm on,wehavethefoIlowing inequality 89,if()/P.,.--m.-:1l: 1~(J(1n+a.,_"'+"'+Cl""" I<7:+tPO:p±itp+tlll)ma". ClO+Cl,+...+Cl"= m a(l+Cl,+...+Cln Applying tlus,forafixedn>8,tothenumbers a",a.:intro­ ducedaiJove,andtakinga=(n), V=0,1,2," ..,n,wecanchoose.." forpthegreatest integer<~-,i.e.p=[~-J,andwemaysimilarly taketn=~[i].Wethenobtain ~;;-[(~)00+(;)0,+...-I-C)a..]<7:".+2TnP(;)+22~~111(,:)" Thisinequality willholdafortiori ifwea~sume 'l:tobegreaterthan allthequantitiesIa..'Iand'r"greater thanallthequantitiesIa..' withv::2::k"Now,by219,3,thegreatest term(:)ofthevalues (/)"£1h1'"1""'n(1I) 1h'"'n(1I)' satlsest eImltmg reatlon"" -+-=,sotat'In IScer- If .., 111"ITT .... 111 tainly..:::1fromsomestageon.Sincefromsomestageonwchavcalso VII<3vi,itfollowsthat,forallsufficicntly largen, ~;:m(11)<TfJV',t<3Tp,/ji. ~m .9Infact,av=(v+1)a~-va~_landtherefore, taking 1X..+1=0, n" p-·lm-l 11 EIXvav~E(v-I-1)a;.(-x,.-1X"+1) ~.E-+E+E. v··o v~ll v-ll v~pv-m Notingthat(IXv-IXV+1)isne!':ative inthet1rstandsecondsumsandpositive in thethird,Itfo\lowsthat "IEIX"UvI;:;TpIX"-I-T"711IX",f-Tm(711"'m+IXm+1X",+1+...-t"'..). I'{} whence theaboverelation fo\lowsdirectly. 516 Chapter XIII.Divergent series. Since Tp"';pwastotendto0,since,further,pand1ntendto+00 asndoes,andatthesametimet"(;)-+0,itfollows thatwhen n-+oo i.e. El(Sn)-+S, q.e.d. Exercises onChapter XIII. 200.Withthehelpofexample 119(p.~70),provethefactmentioned onp.461,namely that • Cl>,(-l)n-1 1hm)-----= - "'....+0n71 n'" 2 Wh.ttsummation process related totheA-process mightbededuced from this?Define ItandIOdlcate someofitsproperties. always Involves (1-x)2'5"X" __S201.Isthecondition C I ( a,+2al+...+na")0 h-lm---n+1 = , givenin273.substantially equivalent a)toCk+1-lim(na,,)=O, b)toCk_ltm(~1+2a.: .•.+na,,)=0? 202.Withreference totherelations (*)intheproofof284,showthat ingeneral Alim5n=5always involves ACh-lim5"=5,ie. '"S(kl (I-x)n~o(nI-k)xn-- s. (forx--1-0). 203.Showsimilarly thatB-lim 5"=salwaysinvolves BCk-llln 5"=O.i.e. '"Slk) n-2:Xln X,£.i---·---I "=0(nkk)n! forx--+oo. 204.Aretheconclusions mentioned in202and203reversible, e.g. 5+5+"'+5doestherelatIOn (1-x).2 0 I 1 "x·-+simply, conversely, thatn+(1-x)::E5nX"--s? 205.Theseries1:(nk~~1)z"issummable CkforIz1=1,z=!,+1. n=O Putz=cosq;+isinq;,separate therealandimaginary partsandwritedownthe trigonometrical seriessummed inthisway,aswellastheirrespective values. E.g., 1 11+2cosx+ 3cos2x+4cos3x-l-•••=2"---­ 4.•:llS1l1-2 1-cosx+2cos2x+3cos3x+···=- ,etc. ".•x•5111Jf :n:log2' withtheslims,thefollow-200.If(a,,)isapositive monotone nullsequence, andifweput ao+at+a.+...+a"=b", theseri~s bo-bl+b,-b3+-... 1issllmmable Cltotheslims=2.4(-1)"a". 207.Ifwewrite1+_I.+~+...+~=hn,itfollows fromthepreceding2 3 n exercise thattheC,-sum 1hi-h~+h3-h,+-...=2log2, andsimilarly thattheCl-sum 1log2-log3+log4 -+...=2 208.If:sanisconvergent orsllmmable C, mgseriesisalways convergent withthesums: ~~,i·2al+...+na"a+ ---~.._----- =s. on-;;;;-~ n(n+l) 209.If~a"isknowntobesummable Cland};n,a"I'isconvergent, then ~:a"ISit,elfconvergent. 210.Provethefollowing extensions ofFrobenlus' theorem (p.490):Ifian isslInunable C,tothesum5,thenforz-++1(within theangle) "=1 00 00Ianznl-+sand 2,'anZ,,1-+S n=1 n=U andingeneral, foreveryfixedintegerp>1, 00 pIa"z"-+s. 71=0 But~a"z"1 doesnotnecessarily tendto5,asmaybeshownbytheexample i(-I)"X"I 71=0 forrealx-+l-0. (Hint:ThelDaximum ofI-tn,andthevalueoftfor whichitisattained, both-++1fromtheleftasnincreases.) 211.Foreveryreal s~ries1:an>forwhich1:anxnconverges in0;Sx<I, wehave (Cf.Theorem 161.) 212.Withreference toFe;er'stheorem, showthatthearithmetic means an(x)considered theredonotexhibit Gtbb5'phenomenon. (Cf.p.496,footnote 66.) 213.Theproduct oftwoserieswhicharesummable E,islnvariably summable E,Cl 214-.IfIallissummable E"then~anx"isalsosummable Elforo<x<1and HmEl-Ia"X"=EI·Iall' z-+1-0 21~.Giveageneml proofofthecommutability oftheEp'andCq.processes. 216.Deduce theo·Ep-+K-theorem (whatisitsstatement?) byinduction fromthe0 -E~-+l<.theor~m. 618Chapter XIV.Eulcr'ssummation formulaandasymptotic expansions. Chapter XIV. Euler's summation formula andasymptotic expansIOns. §64.Euler's summation formula. A.Thesummation formula. Therangeofactionofallthesummation processes withwhichwe became acquainted inthelastchapter waslimited. Itisonlywhenthe termsat,ofEan>thedivergent seriesunderconsideration, donotincrease toorapidlyasnincreases thatwecansumthescrics.Thusinthecase oftheB-process, itisnecessary thatE~~xnshouldbeconvergcnt every- wher<>,i.e.thatn/1anlor1Vra:Ishouldtendtozero.Hencethe'Vn! n B-process cannotbeuscdc.g.fortheseries <Xl E(-l)nn!=1-1!+2!-3!+4!-+...+(-l)nn!+... n-O Serieslikethisone,andevenmorerapidly divergent series,occurred, however, inearlyinvestigations ofthemostvariedkind.Inordertodeal conclusively withthembythemethods usedhitherto, weshouldhaveto introduce stillmorepowerful processes, suchastheBr-process. How­ ever,noessential resultshavebeenobtained inthisway. Atafairlyearlystageinthedevelopment ofthesubjectothermethods wereindicated, whichincertaincasesleadmoreconveniently toresults usefulbothintheoryandinpractice. Inthecaseofthenumerical evalua­ tionofthesumofanalternating seriesE(-l)nan>inwhichtheat.'s constitute apositive monotone nullsequence, weobserved (seepp.250 and251)thattheremainder Tnalwayshasthesamesignasthefirstterm neglected, and,moreover, thatitislessthanthisterminabsolute value. Thusinthecalculation ofthepartialsumsweneedonlycontinae until thetermshavedecreased downtotherequired degreeofaccuracy. A somewhat similarstateofaffairsell.istsinthecaseoftheseries -:z:___ ~~_ _nx~e-1x+2+...+(1)-I+...,x>0,11. §64.Euler'ssummation formula. -A.Thesummation formula. 519 sincetheterms .~,likewise decrease monotonely whenn>x.Wecan11. therefore write foreveryn>x,where3-standsforavaluebetween 0and1,depending onxandn,butisotherwise undetermined. Itisimpossible inpractice, however, actually tocalculate rXfromthisformula whenxislarge,for 103000 e.g.whenx=1000,thethousandth termisequalto-WOOl' As1000I isanumber with2568digits(forthecalculation seebelow,p.529),the termunderconsideration isgreaterthan1043\sothattheevaluation of thesumoftheseriescannotbecarriedoutinpractice. Fromthetheoretical pointofview,ontheotherhand,theseriesfulfilsallrequirements, since itsterms,which(forlargevaluesofx)atfirstincrease veryrapidly, never­ thelessendbydecreasing tozero,andthatforeveryvalueofx.Hence anydegreeofaccuracy whatever canbeobtained intheory. Thecircumstances areexaetlythereverse, ifweknowthatthevalue ofafunctionf(x)isrepresented bytheformula 0<&<1. foreveryn.ThesenesL:(-l)n~~,whosepartialsumsappearinthis formula, diverges fureveryx:butincontrast tonearlyallthedivergent seriesmetwithi1/thelastchapter, thetermsoftheseries(forlargevalues ofx)atfirstdecrease \-eryrapidly-theseriesatfirstbehaves likeacon­ vergentone-anditisonlylateronthattheyincrease rapidlyandwithout limit.Hencewccancalculate e.g.f(1000)toabouttendecimal places withgreatcase;wchaveonlytofindannforwhicht~!2i<~10-1°. Asthisistrueevenforn=3,thevaluesoughtisgivenby 126 1-103+10"-10' tothedesireddegreeofaccuracy. Thusithappens herethatanexpansion inpowers, whichtakestheformofaninfiniteserieswhichisdivergent everywhere andveryrapidly so,nevertheless yieldsusefulnumerical results,becauseitappearsalongwithitsremainder. Wearenotinaposition, however, -notevenintheory-toobtainanydegreeofaccuracy what- 520ChapterXIV.Euler'ssummation formulaandasymptotic expansions. everintheevaluation ofI(x),sinceI(x)isgivenbyitsexpansion only withanerroroftheorderofoneofthetermsoftheseries.Thedegree ofaccuracy therefore cannotbelowered belowthevalueoftheleastterm oftheseries. (Aleasttermcertainly exists,seeingthatthetermsfinally increase.) Astheexample shows,however, insuitable circumstances all practical requirements maybesatisfied. Seriesofthetypedescribed wereproduced forthefirsttimebyEuler's summation formula \whichweshallnowconsider moreclosely. Ifthetermsao,aI'•••,an>•••ofaseries 2arethevaluesofafunc­ tionI(x)forx=0,1,...,n,...,wehavealready provedbythein­ tegraltest(176)thatincertaincircumstances thereisarelation between thepartialsumssn=ao+a1+...+anandtheintegrals n In=ff(x)dx. o Euler'ssummation formula throwsfurtherlightonthisrelation.IfI(x) possesses acontinuous differential coefficient in0::;:xSn,then,for JI=0,1,...,n-1, I'll 1'+11'+1 I(x-JI-~)!,(x)dx= [(x-v-!)/(x)lv -II(x)dx. V v Now,foreachofthevaluesv,wecanputv=[x]intheintegrand onthe left,atleastforv::;;x<v+1.Since,however, by§19,theorem 17, theonevaluex=v+1doesnotmatter,weget 1'+1 1'+1 ~(Jv+1v+1)=II(x)dx+I(x-[x]-!)!'(x)dx. v v (Tosimplify thewriting, wedenotebyIvandI~k)respectively thevalues ofI(x)andofitsderivative/k)(x)forintegral valuesx=v.)Adding theserelations fortherelevant valuesofv,andadjoining thetermt(fo+In), wefinallyobtaintheformula 1Withregardtothesummation formula cf.footnote 3,p.521.-Thepheno­ menondescnbed abovewasfirstnoticed byEuler(Commentarii Acad.sc.Imp. Petropolitanae, Vo!.11(year1739),p.116,1750);A.M.Legendre gavethename ofsemi-convergent seriestoserieswhichexhibit thisphenomenon. Thisnamehas survived tothepresent time,especially inastronomical literature, butnowadays itisbeingsuperseded bytheterm"asymptotic series",whichwasintroduced by H.Poincare onaccount ofanother property ofsuchseries. IInthesubsequent remarks allthequantities aretobereal. §64.Euler'ssummation formula. -A.Thesummation formula. 521 n n fo+h+···+f ..=ff(X)dx+~(fo+fn)+ J(X-[X]-~)f'(x)dx. 296. o 0 ThisinfactisEuler'ssummation formulainitssimplest form3.Itgives aclosedexpression forthedifference between thesum10+11+...+In n andthecorresponding integralfI(x)dx. o Weshalldenotethefunction whichappears inthelastintegrand byPi(x): 1Pi(x)=x-[x]-2' Thisisessentially thesamefunction astheonewhichwemetwith inoneofthefirstexamples ofFourier expansions (seepp.351,375). Itisperiodic, withperiod1,andforeverynon-integral valueof zwehave Asimple example tobeginwithwillillustrate theimportance ofthis formula. Iff(x)="1_1_,weobtain,byreplacing nbyn-I,+x 11 1+~+...+~=logn+_1.+.!--f PI(x)dx.2 n 22n x9 1 ..-1 \Vemaysubstitute thelatterintegral forf(r~(;~dx,sincePI(x+1)=P,(x). o AsPI(x)isbounded inx2;;I,theintegral obviously converges whenn-..00, andwefindthul QC hm(1+-~+...+~-IOgn)=c=~-JPI(Xld:e. n"~.. 2 n 2x-- 1 •Theformula, initsgeneral form298,originated WithEuler,whomen­ tioneditIIIpa~sing intheCommentarit Acad.Petrop., Vol.6(years1732-3, publbhed 17:3il)andillustrated itbyafewexamples. InVo!.8(year1736, publbhed 1741)hegivesaproofoftheformula. C.Mac/aurin usestheformula IIIseveralplace, IIIATreatlse ofFluxion~ (Edinburgh 1742),andseemsto havedIscovered itmdependently. Theformula became well-known, especially through Euler'sInstitullol'es calcuhdifferentialis, inthefifthchapter ofwhich Itisprovedandillustrated byexamples. ForlongitwasknownasMaclaurm's formula, ortheEuler-Maclaurin formula; itisonlyrecently thatEuler's un· doubted priority hasbeenestablished. Theremainder -whichismostessential -wasfirstaddedbyS.D.Poisson (v.l\1cmoires Acad.sClenc.Inst.France, Vo!.6,year1823,published 1827).Thll particularly SImpleproofgIveninthetextisduetoW.W~rtinger (Actamathe. matica, Vol.26,p.255,1902). Anup-to-date, detailed, andexpanded treatment istobefoundinN.E.Nor­ lund'sDifferenzenrechnllng, Berlin1924,especially inchaptersII-V. 522Chapter XIV.Euler'ssummation tonnuht andasymptotic expan~jon5. Wealready knowthatthislimitexists,from128,2.Nowwehaveanew proofofthisfact,andInaddition wehaveanexpression intlwformufan integral forEuler'sconstant C,bymeansofwhichwecanevaluate thecon­ stantnumerically. Fromtheformula 296,i.e. n n (.)lu1-It+...+In=fI(x)dX+~(/0+Ill)-1-f1\(.r:)l'(x)d:c, u 0 integration bypartsleadstomoreadvantageous representations. In ordertobeinaposition tocarryitout,wemustfirstassumethatI(x) hascontinuous derivatives ofalltheorderswhichoccurinwhatfollows; thenwehavetoselectanindefinite integralof1\(x),andanintegralof thelatter,andsoon.Bysuitable choiceoftheconstants ofintegration thefurthercalculations aregreatlysimplified. WeshallfollowWirtillger 4 and'let ( ),al,2__co~211"7J: P~x==-I n~l(2n7t)~ Then P~'(x)=PI(x),foreverynon-integral value 1 001 1 M P(). .=-22-.L;-2=-1-2.oreover,~x1Scontmuous :ren=ln hastheperiod1.Wenowproceed toset ~(x)=+i~~~2n:rexa n=l(2nn)3 ,ofx,andP.J(0) throughout, and whence wehavePs'(x)=P~(x)foreveryvalueofx,Pa(0)=-cc0, andingeneral 297.(a)P()(1)1-1~2co~2n;ra: 21X= - ..:;..-----.-'1-' 11=1(21t:il)" 00'>•2 P()_(_1)1-1 '\'~ 1I1r;~ 21+1X- ....(2)21+1'..=1n:il Then,for1=1,2,...,allthesefunctions arethroughout continuous andcontinuously differentiable, amihavetheperiod1;andwehave (b) forh,,\=I,2,...(cf.136).Asisimmediately obviousfromtheproof, intheinterval0~x~1andforh>2,thefunctions Pk(x)arerational integralfunctions. BesidesthefactthatPt(x)=x-~in0<x<1, ,Cf.thelastfootnote. §6t.Euler'ssummation formula. -A.Thesummation formula. 523 wehave,in0~x~1, Henceingeneral, asmayimmediately beestablished byinduction, x.Bxk-1Bxk-!! B P,.(x)~kT+T~(h---=--1)1+2:(h-':-2)1+...+k~ =:1{(~)xl'+(~)B1Xk-l+(~)B;lXk-2+ .•• or Ifweemploy thesymbolic notation already usedin105.Theseare theo>o-called Bernoulli's polynomialsfJ,whichplayanimportant partin manyinvestigatIOns 6.Wcshallmeetwithsomeoftheirimportant properties directly. Firstofall,however, weshallimprove theformula (*)bymeans ofthesepolynotl1tals. Integration bypartsgives n nJP1(x)f'(x) dx=[P.J'J~-JP.d"dx o u n =:~({n'-to')-[PJf"];+fPa1""dx o n =~~(f,,'-to')+JP.lflitdx o 6TheyfirstoccurinJamesBerno1tlh, Arsconjcctandi, Basle1713._. Therethepolynomials appear astheresultofthespecialsummation problem whichwillbedealtwithlaterinH,l. aManywnterscallthepolynomial 'P"(x)=(x+E)k-EkthekthBernoultl polynomial; others,again,givethisnametothepolynomial (X+B)k+1_Bk+l 'P"(x)= k+1 • Thesedifferences areunimportant. 524ChapterXIV.Euler'ssummation formula andasymptotic expansions. and,generally, n n fP2'1f'~,\-1)dx=B2A_(I.(2).-1) -I.(2A-I»)+fPf(2A+l)d:le A- (2.\)1 n 0 2'\+1 o 0 for">1.Hence,foreveryk>0,provided onlythatthederivatives off(x)involved existandarecontinuous, wecanwrite: n 298.fo+ft+...+fr.=ff(x)dx+~(fn+fa) o +~t(fr.'-fo')+~;(fn'"-fo''')+... +B.k(f,(2k-l)_r(2"-1)) -'t-R(2k)! n Jo Ik. whereweput 11 Rk=JP2k+1(x)f(2k+l) (x)dx o forshort.ThisisEuler'ssummation formula. Remarks. 1.Sinceinthelastintegratiqn byparts,namely n n- IP2kf<'k)dx= -[P2k+1f<'k)J:+fP2k+lf',k+1l dx, o 0 theintegrated partvanishes, onaccount ofthefactthatP2kt1(n)=P'k+l(0)=0, wemayalsowrite n Rk= -IP2k(x)!,'k)(x)dx o fortheremainder terminthesummation formula. 2.IfweputF(a+xh)=I(x),theformulatakesthesomewhat moregeneral form,inwhich F(a)+F(a+h)+...+F(a+nh) formsthelefthandside.Theformula maytherefore heusedforthesummation, ofanyequidistant valuesofafunction. 3.Withsuitable provisos, itispermissible toletn-+00inthesummation formula. According asElnconverges ordiverges, wethenobtainanexpression forthesumoftheseriesorforthegrowthofitspartialsums.Thestatement is. different (ontherighthandside)foreveryvalueofk. §64.Euler'ssummation formula. -B.Applications. 525 4.Ifweletk--+00,RkmaytendtoO.Weshouldthenhaveaninfiniteseries ontherighthandside,intowhichthesumonthelefthandsideistransformed. ThIScaseactuallyoccursveryseldom,however, since,asweareaware(v.p.237, footnote), Bernoulli's numbers increase veryrapidly. Theseries 1:112k_(l2k-l) _/2k-l) k~d2k)! n 0 willturnoutdivergent foralmostallthefunctionsf(x)whichoccurinapplications, nomatterwhatnmaybe.Thustheformula suggests asummatIOn processJora certaintypeofdivergent series.Cf.however theexample B.3below. 5.Provided thatthedifferences (f~2l\-1) -f~2l\-1» havethesamesign,the seriesjustdiscussed isanalternating series,sincethesignsofthenumbers Bol\ arealternating. Weshallseethat,inspiteofthedivergence, theabove-mentIOned evaluation oftheremainder ofthealternating senesremains valid.(Cf.theintro­ ductoryremarks tothissection.) 6.Theformula wIllbeusefulonlyinthecaseswhere,forasuitablevalueof k,R"issmallenoughtogivethedesireddegreeofaccuracy. AtfirstSight,wehave onlythemequahty atourdisposal fortheestimation ofRk•forIl?2:but.asweseesubsequently. theinequalIty alsoholdsfork=I,andby136ItcanbeputInthemoreprecise formIPk(x)I;;::;;J~~lforevenvaluesofk. B.Applications. 1.Itisobvious thatthemostfavourable resultsareobtained when299. thehigherderivatives off(x)areverysmall,andespecially whenthey vanish. Wetherefore firstchoosef(x)=xP,wherepisaninteger>I, andwehave Heretheseriesontherighthandsideistobebrokenoffatthelastpositive powerofn,for(f~k)-f~k»vanishes notonlywhenj<k)(x)==0,butalso (by297b)when/k)(x)isidentically equaltoanon-vanishing constant. 1'husbytransferring nPtotherighthandsidewehave lP+2P+...+(n-l)p =P~i {nP+l+(PiI)BlnP+(P~1)B2nP-1+...}, 526Chapter XIV.Euler'ssummation formula andasymptotic expansions. or-sincethereisnoconstant termappearing insidethebrackets ontherighthandside- tP+2P+...+(n_l)p=_l_{(n+B)p+l_ BP+l}.p+l 2.Thesumsdealtwithabovecanbeobtained inquiteadIffer­ entway.Ifweimagine thateachtermofthesum 1+ee+e2e+...+e(n-lje tll 15expanded Inpowersoft,thecoefficient ofpT15obviously 1"+21'+...+(n-1)P. ent-1 e1n+B)t_eBe___eBe=-----t tOntheother sumISequalhand,ifweusesymbolic notatIOn (cf.105,5), thefirst to ent-1 et-1 Henceweimmediately obtaintheexpression _1_{( +B)P+l_BP+l1P+1n 1 tP forthecoefficient ofPi. 3.Ifweput((x)=eaz,n=1,weobtain a1 kB}_(ea+1)=~+.l}~a211-1(ea -1)2 CC11=1(2,,)1 1 +a2k+lJPU+1(x)'eaxdx, o or 1 IX a kB2 IXU+2f--=1--+.l} -~a2"+-- PU+l(x)eazdx.ea_1 2 ,,=1(2v}1 ea_1 o Sincewecanimmediately prove,by29S,6, thattheremainder tends tozerointhiscase,provided onlythatIaI<21t,wehave,forthese valuesofa, whichistheexpansion statedin105. Similarly, byputting((x)=cosax,n=1,weobtaintheeXIJan cc ccSlOn115for2cot"2' §64.Ruler'ssummation formula. -B.Applications. 527 4.Ifweput{(x)=l~X'wehave,byreplacing nby(n-1). 1 1 1 1 B9(1 )B4(1 )1+-+...+-=logn+- +- +--1 --+-1 ---2 n 2 2n2 n94 n4 n +...+Bgk(1-~)_(2k+l)lfP2k+1(X)d2k n9k X9k+9 X. 1 Sinceherewemayletn_00,justasonp.521above,weobtainthe following refinedexpression forEuler's constant: ~ 1B2B.. R2k •fP2k+1(x)0=2"+2+T+ ...+2k-(2k+l)! X2k+2dx. I Inthiscasetheremainder certainly docsnotdecrease to0askin­ creases; andtheseriesL;~2:diverges rapidly. -sorapidlythateven thecorresponding powerseries.LJ:":x2j;diverges everywhere jfor, by136. IBI-2(2k)1'YIwhere 1<'YI<2. '.Ik-(2n:)2k", " Nevertheless, wecanevaluate Cveryaccurately bymeansoftheabove expression (cf.Rem.6).Ifwetakee.g.k=3,wehave,inthefirst instance. (a) Ifwetakeonlythepartofthcintegral fromx=1tox=4,the absolute valueofthcerroris ClO ./ 4fdX- 4·71 -8 ~71(2n:)' XS-(2n:)7.7.4.<10 • 4 Hence , C-145~_71fp.(x)d+-!Lwhere-2520 x8x108 , 1 Therequired evaluation oftheintegral isalsogivenbythefirst formula written down,forn=4,namely 4 fp.(x) 1 1 1-71-dx =1+-+-+--log4x8 234 1 1459 1t 1 1 -2520-2·4+12.49-120.4 4+252.48• 528Chapter XIV.Euler'ssummation formulaandasymptotic expansions. Hence 0·5772146<C<0·5772168. InthiswaywecaneasilyobtainCwithmuchgreateraccuracy than before,andtheoretically toanydegreeofaccuracy whatever. Thereason forthisfavourable stateofaffairsliessolelyinthefactthatwemayregard thelogarithms asknown. 5.Wenowputf(x)=log(1+x)andproceed justaswedidin theprevious examples onpp.525-7.Ifweagainsubstitute (n-1)for Il,wefirstofallobtain,from298withk=0, n n log1+log2+...+logn=Jlogxdx+~logn+J~l!X)dx 1 1 or n (I)JPi(x)dlognI=n+2logn-(n-1)+ -;;-x. I Integrating byparts,wehave n nJ~lX(X)dx=[~'x~)I+J~~~~}dx, 1 1 whichshowsthattheintegral converges asn----+00.Hencewecanput lognI=(n+~)logn-n+Yn> andweknowthat limy"=Y n----+'" exists.Itsvalueisobtained asfollows: by(1Io)wehave 2log(2. 4.•••2n)=2nlog2+2lognI =2nlog2+(2n+1)logn- 2n+2YI1 =(2n+1)log2n-2n-log2 + 2Yn and log(2n+1)1=(2n+:)log(2n+1)-(2n+1)+Y2n+1' Bysubtraction log_2'~" 2n._1_=(2n+1)log(1-2-_1+1)--21log(2n+I)1·3·5·...2n-12n+l n +1-log2+ 2Y,,-Y2n+1' §64.Euler'ssummation formula. -B.Applications. 529 Ifwenowtransfer theterm-}log(2n+1)tothelefthandsideand letn-00,weknow,fromWallis'product(219,3), that logVi= -1+1 -log2+2i'-i', 50that r=logy'2n. Hence,finally,wehave.. (U)lognl=(n+ ~)logn-n+IOgy'2n;- fptx(X)dX. n Ifwemultiply byM,themodulus oftheBnggtan loganthms (pp.256-7), anddenotethelatterlogarithms byLog,wehave Lognl=(n+~-)Logn-nM+ Logy'in-MjP~(X)dx. n Thisgives,e.g.forn=1000, Log1000!=3001'5-434'29448 ...+0'39908...-MfooPt:X)d:c. 1000 SinceIMJP'x(X)dxI<M[p·x(X)fooo +IMjP~~X)dxI 1000 1000 4M 4M 1 ~(2--"')2'iooo+(2~)'.1000<-10000' itfollows that Log1000!=2567'6046 ... Withanerror<10-'inabsolute value,sothat1000Iisanumber with2568 digits,whIchbegins withthefigures402.... Justasintheprevious example, wecannowimprove ourresult(ltlt) considerably bymeansofintegration byparts.Since f~Fl(x)d=Pl+t(0)+~fooFI.+l (x)d l.X ), Al+1X,X n x n n after2kstepsweobtain (1) /- B~1B.1logn!= 1'1,+2"logn-n+logl2.1l'+t.;·n:+/J.4,·n: '+ B2k 1 .fCY>P2k+l (:1:)...+(2k_I):'.k'n2k-1 - (2I.)!:I:210+1dx. n Asheretheremainder (forfixedk)islessthanacertainconstant divided byn2k,wecanalsowritetheresultintheform B~1 B2k 1An-.-+...+.--+-(n)"-- 1·2.. (210-1)210 ,.210-1 ,.:110nl=-y'2.1l'ne ,e 630ChapterXIV.Euler'ssummation formulaandasymptotic expansions. inwhichtheAn'salways(i.e.foreveryfixedk)formabounded sequence. TheresultineitherformisusuallyknownasStirling's formula 7. 6.Ifwetakethesomewhat moregeneral formf(x)=log(y+x), wherey>0,Euler'sformula fork=0gives,tobeginwith, logy+log(y+1)+...+log(y+n)=(y+n)log(y+n)-n n -ylogy+ 1{log(y+n)+logy}+j~d~)dx.2 y+xo Hencewecanobtainacorresponding expression forthegamma-function (v.p.385andpp.439-10) asfollows: subtract thisequation fromthe equation (**)inthelastexample, addlognYtobothsides,andweobtain n1nY(1) ( 1) y+n log---- =y- -logy-y+n+.log--y(y+1)...(y+n) 2 2 n n 00+logv'27t -j~l(x)dx-jPdx)dx.y+x xo n Ifn_00,thisrelationbecomes 00 (1)I I./-jP1(X)logF(y)=Y--ogy-y+ogv27t-----dx.2 y+xo Byintegrating thisexpression byparts2ktimes(orbyatonceusmg Euler'sformula foranyvalueofk),wededucethefollowing generalized Stirling's formula B: logr(y)=(y-~)logy-Y+logv'27t +B21+B.1+ + B2k 1172y34Y;...(2k-1)(2k).y'k-l 00P:'.k11(x) -(2k)!j(y+X)2k+1dx. o 7.Wenowputf(x)=(1~x)"'wherex>0andsisarbitrary. Aswehavealready dealtwiththecasess=I,-I,-2,.••,andthe cases=0istrivial,weshallconsider sasbeingdifferent fromanyof thesevalues.Ifweagainreplacenby(n-1),Euler'sformula nowgives 7:1.Stirling, Methodus differentialis, London 1730,p.135.Butthefact thattheconstant yislog\/-2.".wasnotdiscovered tilllater. 8Stirling (loc.cit.)givestheformula forthesum logx+log(x+a)+log(x+2a)+...+log(x+na). ~64.Euler'ssummation formula. -C.Theevaluation ofremainders. 531 I I 1 1 ( I)1(1 )1-I-2'+3'+...+n'o-cs-11 -nS-1+2nS+1 +Bo(S)(1I) Bok(S+2k-2)(1)-21 - n"+l+...+2k2k_11 -n,=t-Ok-1 n _(2k+-l)I(S+2k)f~2k+d~) d. 2k+1X"+Ok+1 X. I Ifs>1wecanletn~00,andweobtainthefollowing remarkable ex­ pression forRiemann's {-function (cf.pp.345,444--6, and491-2): ,(s)=_}_+~+!lo(S)+...+B21£(S+2k-2)1-1 2 2 1 2k2k-l QC _(2k+1)1(S+2k)fP~Y1 (~d. 2k+1,x-'+Ok+' X. 1 Sincetherighthandsidehasameaning fors> -2k,s=l=I,andsince kcantakeanypositive integral valuewhatever, weimmediately infer fromtheab0ve-thedetailsoftheproofbelongtothetheoryofcom­ plexfunctions -that 1'(s)--1-1 isanintegral transcendental function (cf.p.492,footnote 61).Further thisexpression givesthevalues 1'(0)= -2' andfors= -p(papositive integer), ifwesuppose that2k>p: ,(-p)= -P-~l-BI+~2(~P)+~.(-P/2)+.... Heretheseriesterminates ofitself,andwecanwrite ,(-p)= -p~f{l+(Pi1)Bl+(P-;-1)B2+(Pt1)Ba+...} = -P-+-l(1+B)1J+I= -:-f-f., wherethelaststepfollowsfromthefact(v.106)that (1+B)P+1-BP+!=O. C.Theevaluation ofremainders. Theevaluation oftheremainder inEuler'sformula, whichforprac­ ticalpurposes isparticularly important, wehave avoirie~ hitherto. Now, however, thequestion becomes imperative whether wecannotformulate iomegeneralstatement astothemagnitude oftheremainder inEuler's 532Chapter XIV.Euler'ssummation tormulaandasymptotic expansions. summation formula.Itmaybeshownthat,verygenerally, theremainder isofthesamesignas,butsmallerinabsolutevaluethan,thefirsttermneglected, -i.e.thetctmwhichwouldappearinthesummation formula, ifwe replaced kbyk+1.Thiswill,moreover, alwaysbethecase2jf(x)has aconstant signforx>0andiff(x)andallitsderivatives tendmonotonely to0asxtendsto+00. Inordertoprovethis,wemustexamine thegraphofthefunction y=Pk(x),k::2:2,intheinterval 0<x:-:;:1,somewhat moreclosely. Weassertthatthegraphisofthetyperepresented inFig.14;1,2,3,4, according askleavestheremainder 1,2,3,or0,whendivided by4:. D1"""=7- a. Fig.14. Moreprecisely, weassertthatthefunctions withoddsuffixeshaveexactly threezerosofthefirstorderat0,t,1,butthosewithevensuffixesexactly twozerosofthefirstorderwithintheinterval, and,moreovcr, thatthe functions havethesignsshowninthegraphs. Moreshortly: P2>..(x) isofthetypeofthecurve(-1)>"-1cos21TXandP2H1(x)isofthetype ofthecurve(-1y-lsin21Tx. Thesestatements areproveddirectlyforthesuffixes2,3,4,byusing themethods whichfollow,ortheycanbededuced fromtheexplicitfor­ mulaeonp.522.Wemaytherefore assume thattheassertions are proveduptoP2)..(x),A>2,inclusive. Itisimmediately obvious, by 297,thatP2)..!1(x)vanishes forx=0,t,1,andalsothat P2>'+1(1-x)= -P2).tl(x), sothatP2>..+1(x)issymmetrical withrespecttothepointx=~,y=O. ThusifP2)..+1(x)hasanother zero,itmusthavetwomoreatleast,i.e. fiveinall,andP2)..(x)musthaveatleastfourzerosbyRolle'stheorem (§19,theorem 8),whichiscontrary tohypothesis. ThesignofP2>..+1(x) in0<x<~isthesameasthatofP2>..+1(0)=P2)..(0),thatis,thesame asthatofB2)..;i.e.thesignisgivenby(-1»..-1. §64.Euler'ssummation formula. -C.Theevaluation ofremainders. 533 Since P2H~(x)=P2>.+1(x),P:!.I\II'(x)hasonlyonestationary value in0<x<:1,namelyatx~~.ItsvalueP:!.I\I':!.G)musthavet~eopposite signtoP:!.>'+2(0),forotherwise P2>.,2(x)wouldhaveaconstant signin o<x<1,andconsequently weshouldhave 1IP':l.I\+2(x)dx=[P21\T3(x)]~=!=0, o whichiscertainly notthecase,becauseoftheperiodicity ofourfunctions. Fmally, sinceP21\+2(0)hasthesamesignasB'I.>'+2,i.e.thesign(-1)'\ allourassertions arenowestablished 9.Since (paninteger ~U),P2>.(1-x)=P2>.(x), P2>.(x)issymmetrical withrespecttotheline ,"X:=~. Now,ifh(x)isapositive andmonotone decreasing function forx:?0, p+lIP21\+1(x)h(x)d x p obviously hasthesignofP2H1(x)in0<x<~,i.e.thesign(-1)>.-1. For,onaccount ofthesymmetry ofthegraphofP2>',1(x)andthefact thath(x)decreases, wehave ptt p+lIIP~>'+1(x)h(x)dxI~IIP:!.>'+l(x)h(x)dxI. P p+. Hence "IP2>'+1(x)h(x)dx o alsohasthesignof(-1)>'-1,sothat,inparticular, thesignsarealter­ nating,if,\=0,],2,. . . .Theexactopposite signsoccur,ofcourse, whenh(x)isalwayslessthan0andincreasing. Nowifweassumethatf(x)isdefined forx>0,and,together with allitsderivatives, tendsmonotone1y to0asx-+00,eachofthesederiva­ tivesisofconstant sign10,andf(2k+1)(x)hasthesamesignasf(:!.k+3)(x). Theremainder inEuler'ssummation formula isgivenby "Rk=IP2k+l(x)f(2k+l (x)dx. o (G51)9Thefactthatonlyzerosofthefirstordercomeunderconsideration follows immediately fromtherelation P~l(x)=Pk(x). 10Thepossibility thatoneofthesederivatives isalways =0fromsomepoint onwards istobeincluded here. 18 53·1ChapterXIV.Euler'ssummation formulaandasymptotic expansions. HenceR"andRh+1haveopposite signs,andtherefore R"and(R"-R"H) havethesamesign,andwehave,moreover, IR"I~IR"-R"HI• Now,byEuler'ssummation formula, R-(I+1+ +I)fnf()d B2k(!.(2k-l) j,(2k-l»,,--la 1 • • • In-0x x-•••-(2kj!."-0 ' whenceitfollowsthat R-R= -B2"tL(f.(2k+l) _1(2k+1») le"+1(2k+2)!"la' Butthisisthe"firsttermneglected", sothatitssignalsoisthesameas thesignofR",whereas itsabsolute valueexceedstheabsolute valueof Rk>q.e.d. Thuswehavethe 300. Theorem: Iff(x)isdefinedforx?':0,and,togetherwithallitsderiva­ tives,tendsmonotonely to°asx-+00,Euler'ssummation formula maybe statedinthesimplified form n fa+f1+..,+fn=ff(x)dx+~Un+fa)+~~Un'-fa')+... o +B21e(1.(2k1)_f(2k1»+(,_~~k.~ (f.(2k+ 1)_1(2kI1l) • • •(2k)!n 0 v(2k+2)I"la' where0<a<1. Thusinthisformtheseries(divergent ingeneral), ofwhichthe firstfewtermsappearontherighthandside,effectively possesses the characteristic property ofalternating series(mentioned onp.518)which isparticularly convenicnt fornumcrical calculations. Remarks andExamples. 1.AsCauchy remarks, thecharacteristic property ofalternating seriesjust mentioned isexhibited bythegeometrical serieslit t2c-+t=c-c2+ca-+...• c>0,t>0, notonlywhenitconverges, butforarbitrary (positive) candt.For,ifwewrite ItWiththeremamder, i.e.intheformlIt t" tn+I 1__= - _-+...+(_l)n-._-.+(_1)'1+1 •__ C+tCc2C"+I C"+2 t '1+ _ C itistruewithout exception. Forany(positive) candt,thevalueofthelefthand sideisrepresented bythenlhpartialsum,exceptforanerrorwhichhasthesign ofthefirsttermneglected, butislessthanthistermInabsolutevalue. Ifwecarryoutthisprocesswiththefractions -:-;;--:2~ (I t2t4 )4v2..2+t2=2(2v..)2-(2~..)4+(2v..)8-+... §65.Asymptotic series. 535 o<~1<1.andadd,weseethatthevalueof 002(I1I)1 ,,:;4,,'11"2+t'=ei---::""1-t+2t foreveryt>0isalsoequaltothesum B.B.. B./r.k-. B.Ic+. k2!+-41t+...+(2k)!t+~(2k+2)!t'lJ whereallthatisknownabout ~isthatitliesintheinterval 0...1. Ifwenowmultiply bye-;x;tandintegrate from0to+00,itfollows, since that 00 J(-}----I+I)e-;x;tdt= _B2•!+B•.1+ +_--!!.~~~_~__._l_ et-1t2t1·2x3·4x3...(2k-1)(2k) ~'·,-1 o 2.Byn.IithefunctIOn logr(x)-{(x-Dlogx-x+logV2;) alsocanbcequated totheexpression foundinI,fortheremainder termusedin n.limaybereplaced bytheonejustwnttendown,by300.Butw~maynotconclude fromthis,without further examination, that (cf.301,4).Wehaveindeedprovedthatbothsidesagreeverycloselyforlarge valuesofx;butwemaynotconclude fromtheprevIOUS conSIderatIOns thatthey areactually equalforanyvalueofx.(Infact,however, theequatIOn written above istrue.) 3.Justasbefore,wecanalsobrieflyevaluate theremamder intheexamples 4,5,6,7ofsectionB.,fromthefactthattheremainder hasthesameSIgnas,but issmaller mabsolute valuethan,the"firsttermneglected". Foritisimmediately obvious thatthefunctionsf(x)usedintheseexamples satisfythehypotheses ofthe theorem 300. §65.Asymptotic series. Wenowreturntotheintroductory remarks of§64,A.Thesenes whichweobtainfromEuler'ssummation formula inexamples 4-7, bycontinuing theexpansion toinfinity insteadofwriting downthe remainder, aredivergent. Inthecaseswhentheyarepowerseriesin ~or~,wecansay,moreprecisely, thattheyarepowerserieswhich diverge everywhere. Inspiteofthis,theycanbeemployed 10 practice, sinceexamination oftheremainder showsthattheerror corresponding toaparticular partialsumissmallerinabsolute value 536Chapter XIV.Euler'ssummation formula andasymptotic expansions. than,antlofthesamesignas,thefirsttermneglected. Nowatfirst thesetermsdecrease, andbecome evenverysmallforlargevaluesof thevariable; itisonlylateronthattheyincrease toahighvalue.Hence theseriescanbeusedfornumerical calculations inspiteofitsdi· vergence; withlimited accuracy, tobesure,butwithanaccuracy whichisoftencloseenoughtobesufficient forthemostrefinedpractical purposes (inAstronomy inparticular) 11.Moreover, thelargerthevariable is,themorereadilydoestheseriesyieldtheresultsjustmentioned. Moreprecisely: if(asinB,5and6)theexpansion obtained from Euler'ssummation formula isoftheform f(x)=g(x)+ao+~+;~+.". notonlydowehave f(x)-(g(x)+ao+~+...+::)-.0 asx-00,foreveryfixedk,buteven xk[f(X)-(g(x)+ao+~+..,+:Z)J-.0. Ageneral investigation ofthisproperty oftheexpansions was ladealmost simultaneously byTh.J.Stieltjes 12andH.Poincare 13. 'ollowing theolderusage,Stielties callsourseriessemi·convergent, termwhichemphasizes thefactthatsofarasnumerical purposes reconcerned theybehave almostlikeconvergent series.Poincare, ntheotherhand,speaks ofasymptotic series,thusputting thelast­ lentioned property, whichcanbeaccurately defined, intheforeground. 'heoldertermhasnothelditsground, although itisoftenused, specially inastronomical literature. Thereasonisthatitclashes with leterminology whichiscustomary, particularly inFrance, whereby urconditionally convergent seriesarecalledsemi-convergent. We halltherefore adoptPoincare's term,andweproceed tosetupthe )llowing exact 11Euler,whomakesnomention ofremainders whatever, frankly regards lelefthandsideof298asthesumofthedivergent seriesontherighthand side.ThushewritesC={+~9+~4+...without hesitation, onaccount of 299,4. Thisinterpretation isnotvalid,however, evenfromthegeneral view­ pointof§59,fortheinvestigations of§64haveprovided noprocess bywhich thesuminquestion maybeobtained fromthepartialsumsoftheseriesbya convergent process, aswasalwaysthecaseinChap.XIII. 13Stielties. TA.J.Recherches surquelques seriessemi-convergentes, An­ nalesdel'Ec.Norm.Sup.(3),Vol.8,pp.201-258. 1886. 13Poincar~. H.:Surlesintegrales irregulieres desequations lineaires, Actamathematica, vo!.8,pp.295-344. 1886. §65.Asymptotic series. 537 Definition. A.~eriesoftheformao+~+!:+'"(whichneed301.xx notconverge foranyvalueofx)iscalledanasymptotic representation (orexpansion) ofafunction F(x)whichisdefinedforeverysuffi­ cientlylargepositivevalueofx,if,forevery(fixed)n=0,1,2,..., [F(x)-(ao+~+;:+...+;=)]xn-0 asx--+00:andweshallwritesymbolically F(x)'"ao+~+;:+..'. Remarks andExamples. 1.Herethecoefficients anarenotboundtosatisfyanyconditions, since theseries 2)~-neednotconverge. Theymaybecomplex, infact,ifF(x)x· isacomplex funetion oftherealvariable x.Thevariable mayalsobecom­ plex,inwhichcasexmustapproach infmityalongafixedradlllsamxc'con­ stant;fortheasymptottc expansion maybedifferent foreachradius. Inwhat follows we!>hallsetthesegeneralizations asideandhenceforward suppose all thequantities tobereal Ontheotherhand,Itfrequently happens thatthefunction F(x)isdefined forIntegral valuesofthevariable only;e.g. IP+2P+...+xl', Insuchcasesweshall u~uallydenotethevariable byk,v,n,...ThenF(x)simply represents asequence, thetermsofwhichareasymptotically expressed asfunctions oftheintegralvariables. 2.Iftheseries2)~ndoesconverge forx>R,andrepresents thefunc·x· tionF(x),theseriesisobviously anasymptotic representation ofF(x)inthis casealso.Thus.examples ofasymptotic representation canbeobtained from anyconvergent powerseries. 3.Thequestion whether afunction F(x)possesses anasymptotic re· presentation, andwhatthevaluesofthecoefficients are,isimmediately settled intheorybythefactthatthesuccessive limiting values(forz--+(0) F(x)--a oI (F(z)-ao)z--a1, (F(X)-ao-~) Z9__ag, mustexist.Infact,however, thedecision canseldom bemadeinthisway. butthesesimpleconsiderations showthatanyfunction canhaveonlyone asymptotic expansion. 4.'Ontheotherhand,forfez)=s-"',x>0,allthean'sarezero,since Zks-"'__0 538Chapter XIV. EuI~r'ssummation formula andasymptotic expansions. toreveryintegral k>0,whenx__00.Thus '"0 0 0,,-""+-+-+""xx9 aresultwhichshowsthatdifferent functions mayhavethesameasymptotic expansion. Thus,IfF(x)hasanasymptotic representation, e.g. F(x)+e-"', F(x)+ae-bx(b>O),.••, havethesameasymptotic representation. Itwasforthisreason thatwecouldnotInterthatthetwofunctions mentioned in300,2 wereIdentical. 5.Geometrically speaking, thecurves and havecontact ofatleastthentborderatmfiDlty; andthecontact become~ closerasnincreases. 6.Forapplications itisadvantageous tousethenotation F(x)--f(x)+g(x)(ao+~+~~+...) • wheref(x)andg(x)areanytwofunctions whicharedefined forsufficiently large valuesofx,andsuchthat,further, g(x)nevervanishes. ThiSnotation ISmtended essentially toexpressthat F(x)-f(x)+~+ill+g(x)--aoxx'•••• Someoftheexamples workedoutin§64,Bmayberegarded asgivingtheasymptotic expanSIOns, inthissense,ofthefunctions involved, forwemaynowwrite 1 1 I C 1 BI1B.1a)1+2+...+n'"-'ogn++2n-2.no-4.;;4-•••; c)log(r(x»--(x-~)logx_x+log'/2:;;:+B2.1+_B•.1t-2 V 1·2x3·4x"•.•; +1[~_1!1.($)~_B•.($+2)~-J.n"2 2 1 n4 3 n"•••, Inthelastformula wemusthave $=l=1;for$=1itbecomes theexpansion ina). §65.Asymptotic series. 539 Calculations withasymptotic series. Inmanyrespects wecanmakecalculations withasymptotic series justaswedowithconvergent series. Itisimmediately obvious thatfrom and G(x)",bo-+~+~-+"', thereresultstheexpansion aF(x)+{lG(x)'"ltao+{lbo+aa,~fJb1+aa';fJbg-+..., where IXand(lareanyconstants. Itisalmostaseasytoseethattheproduct ofthefunctions also possesses anasymptotic expansion, andthat F(x)G(x)'"Co+~+:gg+... if,asinthecaseofconvergent series, aobn+a,bn-l+...+anbo issetequaltoen'For,byhypothesis, wemaywrite(forfixedn), FCx)=a+~+~2+..,_1_an-t+~±oxx9 xn-1xn' G(x)=b-+~-+~9-+...-+bn-1+bn+1Joxx' x·-1x" ifbye=e(x)and'YJ='YJ(x)wedenotefunctions whichtendto0as x__-I-00.Butinthiscasewehave [F(x).G(x)-(co+~-+;~+...+;:)]xn=ao'YJ+bo8 -+a,(bn+1f)+a. b,,_,+...+(a,,+s)b,+...+(an+s)(b!!+'1) x :en, andthisobviously tendsto0asx--+00. Repeated application ofthissimpleresultgives Theorem 1.11each01thefunctions F,(x),F~(x),•••,Fp(x)302. possesses anasymptotic representation, andifg(Z"Z"•..,zp)isapoly­ nomial,or-ilweanticipate whatimmediately lollows-anyratIonal function whatever, 01thevariables %"%9'•••,%p'thenthelunctwn F(x)=g(F,(x),Fg(x),••.,F"(x)) (h=0,1,2,...)540Chapter XIV.Euler'ssummation formulaandasymptotic expansions. alsopossesses anasymptotic representation; andthisiscalculated exactly asifalltheexpansions wereconvergent series,provided onlythatthedenom­ inatoroftherational function doesnotvanishwhentheconstant termsof theasymptotic expansions aresubstituted forZI'Z2'•••,z1J' Further, thefollowing theorem alsoholds: Theorem 2.Ifg(z)=CXo+CX1Z+...+cx..z..+...isapowerseries withpositiveradiusr,ifF(x)possesses theasymptotic representation F(x),...."ao+~+~=+...,x x andifIaoI<r,thefunction ofafunction l/J(x)=g(F(x» -whichisobviously definedforeverysufficiently largex,sinceF(x)--+00 asx--++00,andsinceIaoI<r-alsopossesses allasymptotic repre- sentation, andthisisagaincalculated exactlyasifEa~wereconvergent.x Proof. Inordertocalculate thecoefficients oftheexpansion of tP(x),when.Ea:converges forx>R,say,wehavetosetF(x)=ao+f, ,). and-assuming onlythatIaoI<r-weobtain, inthefirstinstance, g(F)=--g(ao+f)=f30+f3d+...+f3dk+...,("") whereweput 1higCk)(ao)=f3k forshort.Thisexpansion ("'')converges wheneverIf(x)I<r-IaoI. whichiscertainly thecaseforeverysufficiently largevalueofx,whether .Ea:converges ornot,sinceinfactf(x)-?0asx-?00.Inaccordancex withthepartoftheorem 1whichhasalreadybeenproved,from wededucetheasymptotic expansion (j(X»k,...." ~k)+~:~!+... foreveryk=1,2,3,••••Herethequantities a~k)havequitedefinite values,obtainable bytheproduct ruleforasymptotic expansions (i.e. asforconvergent series). Wemustnowsubstitute theseexpansions ("""") in("")andarrangetheresultformally (i.c.againjustasiftheseries(••) §65.Asymptotic series. 541 1wereconvergent) inpowersof.Thusweobtainanexpansion ofthexform A-j-A1+A.+...,a x x' wherethecoefficients aregivenby Aa=(lo,Al=f3laI'Aa=f32aa+f32aa(2),•••, An=f3lan+f32an(2)+...+f3nan(n),.... h h"An...Itremains tosowtat4JnISanasymptotic expansIOnx IS,thattheexpression [ep(x)-(Ao-I-~1+~:+...+~::)] .xnofcP(x),that tendsto0forfixednasx--+00. NowifEl=El(x),Ea=Ea(x),...denotecertain functions which tendto0asx-+00,itfollowsfrom(oil)and(U)that cP(x)=f30+fJI(:1+...+:~+:~)+...+f3n(:~:l+:~) +r\-1•[f3n+l+f3n+2!+...]. Hence,since{nHmaybeputequaltoe:n~l,x cP(x)-(Aa+~I+...+~:)=xl"Lf3IEI+f32E2+...+fJ"En] t-e::~l[f3n+l+f3n+2!+...] andourassertion followsataglance,fortheexpression inthelastsquare bracket tendstof3n+lasx--++00,andthe(finitely numerous) E/Stend toO. Taking g(z)=+1__andreplacing F(x)byF(x)-aa,itfollowsaoz asaparticular case,provided onlythatao=f:0,that I 1 al1al'-aua.1 [t(,.) ,.....,a~-Go'x+-~3-X'+ Hencewe"may"dividebyasymptotic expansions withnon-vanishing constant terms;thiscompletes theproofoftheorem 1. Takingg(z)=eO,weobtain,without anyrestrictions. e~'(x),.....,ea,[1+~~+lal~+a.+...]. Inparticular, wemaywrite,by299,5, nI,.....,(~)n"'27rn[1+ri-,;+2sina-5~::O~+...J. Term~by-term integration anddifferentiation arealsovalidwith suitable provisos. Wehave I~ (aM) 542Chapter XIV.Euler'ssummation formula andasymptotic expansions. Theorem 3.IfF(x)'"ao+111+ll:+...andifF(x)iscontinuousx x fOTX;;::::xo,then '"tp(x)=f(F(t)-ao-Cl1)dt'"a.+2a'2+...+a71T;+....t x x nx x IfF(x)hasacontinuous derivative, andifF'(x)isknowntopossess anasymptotic expansion, thenthisexpansion maybeobtained by differentiating term-by-term, i.e. F'(x)'"_a1_2a2_ _ (n-})an:-:J_x2x3 • • • X'i •••• Proof. Sincet2(F(t)- ao-:1).-,..a2ast.-,..+00,theintegral whichdefinesthefunction tp(x)alwayscxistsforx>xo'Further, wemay F()a1 an+1 e:(t)>fid)} )sett-ao-t-•••-tn+1=tn+i(n=1,xe,wlcree(t.-,..0as t.-,..+00.Hence tp() _~_ _ a1l+1=f"'e:(r)dtxx• • •nx"t'rl+l· x Nowife(x)denotes themaximum valueofIe(t)1inx~t<+oo, then . c(x)E(x).-,..0alsoasxincreases; andsincethelastmtegral ::;;:11x'"after multiplication byxnitlikewise tendstoo. Nowifthederivative F'(x),whichiscontmuous forx>.ro' possesses anasymptotic expansion F'(x)'"bo+~+~~+..., wehave z z z F(x)=-fF'(t)dt+C 1=f(bo+t) dt+f(F'(t)-bo-t)dt-I-C] ~ Zo %0 '" =box+bllogx+C'J-f(F'(t)-bo-~)dt , 'J" whereCl'C'Jareconstants. Bywhatwehavejustproved, andbe­ causeafunction defines itsasymptotic expansion uniquely, itfollows thatbo=bl=0andthatbn= -(n-1)an-lforn:22. Theexpansion F(x)=e-rDsin(erD )'"0+~+~2+... exemplifies thefactthatF'(x)neednotpossess anasymptotic ex pansion, evenwhenF(x)does. §66.Specialcasesofasymptotic expansions. 543 Theorems 1-3laythefoundation forPoincare's veryfruitful applications ofasymptotic seriestothesolution ofdifferential equations 14. Adetailed account liesoutside theplanofthisbook,however, and wemustcontent ourselves bygivinganexample ofthisapplication ofasymptotic seriesinthefollowing section. §66.Specialcasesofasymptotic expansions. Theuseofasymptotic expansions raisestwomainquestiono;: firstthereisthequestion whether thefunction underconsideratIOn possesses anasymptotic expansion atall,andhowitistobefound inagivencase(theexpansion problem); ontheotherhand,there 1<; thequestIOn howthefunction, orrather,afunction, istobefound, whichisrepresented byagivenasymptotic expansion (thesummation problem). Inthecaseofbothquestions, theanswers available inthe presentstateofknowledge arenotcompletely satisfactory asyet,for although theyareverynumerous andinpartofremarkably wide rangp,theyaresomewhat isolated andlackmethodical andfunda­ mentalconnections. Thissection willtherefore consist ratherofa collection ofrepresentative examples thanofasatisfactory solution of thetwoproblems. A.Examples oftheexpansion problem. 1.Fromthetheoretical pointofview,theexpansion ofgiven303• functions wasthoroughly dealtwithinthenote301,3;butitisonly seldom thattherequired determinations oflimitscanallbecarried out.Themethod alsofailsIflimF(x)doesnotexist,i.e.Ifonlyan x-)ooo asymptotic expansion inthemoregeneral sensementioned in:l01,6 canbeconsidered. Itisonlywhenf(x)andg(x),thefunctions In­ volved,havebeenfoundthatwecanproceed asin301,3. 2.Wehavelearned thatasymptotic seriesveryfrequently arise fromEuler'ssummation formula: butthereitisnotsomucha caseofexpanding givenfunctions asthatbyspecialchoiceofthe functionf(x)inthesummation formula weareoftenledtovaluable expansions. 3.Aswehavealready emphasized, hitherto perhaps themost important application ofasymptotic expansions isPoincare's useof theminthetheoryofdifferential equations 14•Thesimplefundamental laAveryclearaccount ofthecontents ofPOlncare's paper,including all theessential points,isgivenbyE.Borelinhis"Lec;ons surlesseriesdiver· gentes" (v.266). 544ChapterXIV.Euler'ssummation formula andasymptotic expansions. ideaisthis:suppose weknowthatafunction y=F(x)satisfies a differential equation ofthentborder rp(x,y,y',...,y(nl)=0, whererpdenotes arational function ofthevariables involved. Now ifweknowthaty=F(x)anditsfirstnderivatives allpossessasymptotic representations, theexpansions fory',y",...,yen)follow, by302 Theorem 3,fromthefirstone, Ifwesubstitute theseexpansions inthedifferential equation, inaccord ancewith302,Theorem 1,wemustobtainanexpansion whichstands for0,allthecoefficients ofwhichmusttherefore vanish. From theequations obtained inthisway,together withtheinitialcondi­ tions,thecoefficients andhencetheexpression forF(x)areingeneral found. Thuse.g.thefunction y=F(x)=e'"je~'dt, s whichisdefined forx>0,hasforitsderivative ODf-t -Ill I y'=F'(x)=e,1ll Tdt-elll~=y-x' z thatis,itsatisfies thedifferential equation y'-y+..!.-= 0x forx>O.Itmaybeproveddirectly -butwecannotgivethedetails here-thatthisequation hasonlyonesolution ysuchthatyandy' existforx>Xo~0andhaveanasymptotic representation. Ifwe accordingly set +a1+Qg+y'"aoz7...• wehavetheequationssothat ag= -al'"',a..+1= -n a..,.•.• whence itfollows that ao=O, Wethereforea1=1,a2=-1,...,a,,+l=(-l)"nl, .... findthat 1 1 2131F(x)'"x-7+7-Z4+ -.... §66.Specialcasesofasymptotic expansions. 545 co laThefunction e-ZF(x)=Je-'tdt.which4.Thefunction intheprevious example canbeasymptotically expanded byanother method, whichisfrequently applicable. Ifwe putt=u+a;Jwehave QO F(x)=J:~uxdu. o Here,byCauchy's observation (v.300,1).wecanput 11"u9 uti 1un+l_= +__+"'+(_1)"_+(_1)"+ {}--u+x xx9x· xn+1 xn+2' O<{}<1, forallpositive valuesofxandu.Itfollows that F(x)=~_.-!-+~_ +...+(_l)n~ +(_l)n+1{} (n+~xx9x' :r.n+1 1xn+2' 0<{}l<l. Thuswehaveagainfoundtheexpansion inthelastexample 15. 5.If.f(u)isafunction whichisdefined foru>0andispositive there,andiftheintegrals 00If(u)U"-ldu=(-1)"-1an o existforeveryintegral n2':1,wcsimilarly obtaintheasymptotic expan­ sion forthefunction <Xl F(x)=ff(u)__d u •u+xo Moreover, thepartialsumsofthisseriesrepresent F(x),exceptforan error,whichislessinabsolute valuethanthefirsttermneglected andis ofthesamesignasthelatter.Expansions ofthiskindhavebeeninvesti­ gatedespecially byTh.J.Stieltjes 16.(Forfurtherparticulars, seebelow. B,p.549.) r« becomes-fIdfJwiththeogfJ x 0 transformation e-t=v,isknownastheLogarithmic-integral function ofy=e-"'. 18Stieltjes, Th.J.:Recherches surlesfractions continues, Annales dela Fac.desSciences deToulouse, Vols.8and9,1894and1895. 546Chapter XIV.Euler'ssummation formula andasymptotIC expansions. 6.Certain methods requiring themoreadvanced resources ofthe .theoryoffunctions datebacktoLaplace, buthaverecently beenex­ tendedbyE.W.BarnesI1l,H.Burkhardt18,O.Perron19,andG.Faber20• Wecannotgointodetails,butmustcontentourselves withthefollowing remarks. Barnes givestheasymptotic expansions ofmanyintegral functions, e.g..2nl(=:0)'({}=l=0,-1,-2,...),andsimilarfunc· tions.Besides theexpansions wchavemetwith,O.PerrolIobtains as examples theasymptotic expansion intermsof11ofcertainintegrals whIch occurinthetheoryofKeplfTimz motion, suchas +:n: fen(t-EsintH A(n)=---r=-;costdt, -n +J< C(n)=Jen{t-sIntli dt. -n~o<e<1,naninteger), Fromourpointofviewitisnoteworthy thatintheseexamples the termsoftheexpansion donotproceed byintegral powers of~,butby fractional powers. ThustheexpansIOn ofC(n)isoftheform C(n)"'~~+++~+~-+....n/,n/,n/,n/, Thissuggests another extension ofthedefinition 301,6,which,however, weshallnotdiscuss. Numerous additional examples ofasymptotic expansions ofthis kind,inparticular thoseoftrigonometrical integrals occurring inphysical andastronomical investigations, aretobefoundinthearticlebyH.Burk­ hardt:'''Obertrigonometrische Reihen undIntegrale", intheEnzyklo· padiedermathematischen Wissenschaften, Vol.Il,1,pp.815-1354. 7.Anexpansion, whichwasfirstgivenbyL.Fe7cr21,andwas subsequently treatedindetailbyO.Perron 22,isofamorespecialized nature; itsobjectistodeduce anasymptotic representation forthe 'I:a-coefficients oftheexpansion inpowerseriesofeI-x,or,moregener· 17Barnes.E.W.:TheAsymptotic Expansions ofIntegral Functions defined byTaylor's Series, Pbi\.TransRoy.Soc.,A,206,pp.249-297. 1906. 18Burkhardt. H.:OberFunktionen graDerZablen, Sitzungsber. d.Bayr. Akad.d.Wlssenscb., pp.1-11.1914. 10Perron.0.:Oberdienaherungsweise Berecbnung vonFunktionen grofier Zablen, Sitzungsber. d.Bayr.Akad.d.Wissenscb., pp.191-219. 1917. 20Faber.G.:Abschatzung vonFunktionen groDer Zahlen, Sitzungsber. d.Bayr.Akad.d.Wissensch'Jpp.285-304. 1922. 11Fejtr.L.:inapaperinHungarian. 1909. 22Perron.0.:-aberdasinfinitare Verbalten dEfKoeffizienten einerge· wissenPotcnzreihe, Archivd.Math.u.Phys.(3),Vo!.22,pp.329-340. 1914. §66.Special casesofasymptotic expansions. ally,ofelX/(l-x)P, wheree>0andex>O.Weatoncefindthat at_X '"1 (X)keI-a:=.E r:t.=1+Cx+Cx2+... k~Ok!1 -x 1 2 wherethecoefficients Cnhavethevalues Cn=1:(11-1)exV v"-,,lv-IJ)!.547 ForthesePerronshowed inalaterwork 23thattheyhaveanasymptotic expansion oftheform 24 Cr-.../_~vae,-vr:t.n_ (1+a~-I-t2J+.Q!.+...). n2V1'(e7.-\/n3 V'IV11'V113 R.Finally, wedrawattention tothefactthattheasymptotic repre­ sentation ofcertainfunctions formsthesubjectofmanyprofound in­ vestigations intheanalytical theoryofnumbers. Infact,ourexamples 301,6,a,b,andd,belongtothisclass,forthefunctions expanded have ameaning onlyforintegral valuesofthevariable inthefirstinstance. Justtoindicate thenatureofsuchexpansions, wegiveafewmoreexamples, without proof: a)IfT(n)denotes thenumberofdivisorsofIt, whereCisEuler'sconstant 25.Regarding thenextterm26,practically allthatisknownisthatitislowerindegreethann-Ibutnotlowerthan n-1• b)Ifa(n)denotes thesumofthedivisors ofn, u(1)+u(2)+...+u(n)..2n r-.../12n-I-..• 23Perron,0.:ObcrdasVerhalten einerausgearteten hypergeometrischen Relhebeiunbcgrenzten Wachstum einesParameters. J.reineu.angew. Math. Vo!.151,pp.63-78. 1921. 2&Anelementary proofofthefarlesscomplete result logcnr-.../2v~n isgivenbyK.Knoppand1.Schur: Elementarer Beweiseinigerasymptotischer Formeln deraddltiven Zahlentheone, Math.Zcitschr., Vo!.24,p.559.1925. 25Lejeune-Dirichlet, P.G.:OberdieBestimmung dermittleren Wertein derZahlentheorie (1849),Werke, Vol.I1,pp.49-66. 26Hardy, G.H.:OnDinchlet's Divisor Problem, Proc.Lond.Math.Soc. (2),15,pp.1-11). 1915. 548ChapterXIV.Euler'ssummation formula andasymptotic expansions. c)Ifcp(n)denotesthenumber ofnumbers lessthannandprime toit, 'jl(1)+cp(2)+...+cp_(!'2,....,_3n+.... n ,,2 cl)IfTT(n)denotesthenumberofprimesnotgreaterthann, n TT(n),....,--+....logn Inalltheseandinmanysimilarcases,itisnotknownwhether a complete asymptotic expansion exists.Hencetherelationwhichwchave writtendownonlymeansthatthedifference oftherightandlefthand sidesisofsmallerorder,asregardsn,thanthelasttermontherighthand side. e)Ifp(n)denotesthenumberofdifferent waysinwhichnmaybe partitioned intoasumof(equalorunequal) positive integers 27, 1"!v'~np(n)""---&+....4n~/3 Inthisparticularly difficultcaseG.H.HardyandS.RamarlUjan 28suc­ ceededbymeansofveryprofound investigations incontinuing the expansion totermsoftheorder-i1•yn B.Examples ofthesummation problem. 304. Herewehavetodealwiththeconverse question, thatoffindinga functionF(x)whoseasymptotic expansion a+~+~+ ...oxx2 isanassigned, eVf~rywhere divergent series 21.Theanswers·to thisques­ tionarestillmoreisolatedandlackingingenerality thanthoseoftheprevioui> division. WhenthefunctionF(x)isfound,ithassomeclaimtoberegarded asthe"sum"ofthedivergent seriesE:::inthesenseof§59,sinceit becomes moreandmorecloselyrelatedtothepartialsumsoftheseries astheirindexincreases. Thisisthecaseonlytoaverylimitedextent, however, since,aswehavealreadyemphasized, thefunctionF(x)isnot 17E.g.P(4)=5,since4admitsofthefivepartitions: 4,3+I,2+2, 2+1+1,and1+1+1+1. 18Hardy,G.H.,andS.Ramanu]an: Asymptotic Formulae inCombinatory Analysis, Proc.Lond.Math.Soc.(2),Vo!'17,pp.75-115. 1917.Seealso Rademacher, H.:AConvergent SeriesforthePartItion Function. Proc.Nat. Acad.ScLU.S.A.,Vo!'23,pp.78--84. 19:J7. 12Whentheseriesconverges, therequired function isdefinedbytheseries itself. §66.Specialcasesofasymptotic expansions. 549 defineduniquely bytheseries.Thusthequestion howfarF(x)behaves likethe"sum"oftheseriescanonlybeinvestigated ineachparticular caseaposteriori. 1.Themostimportant advance inthisdirection wasmadebyStieltjes 30. Wesawabove(v.A,5),thatafunction givenintheform rf) F(x)=j/(uLdux+uo 'f"IGn. h'hpossesses theasymptotic expansion ~x'"InWIC 00(-l)n-lan=JJ(II)un-1du, o(n=I,2,3,...). Conversely, ifwearegiventhe~pansion Ean,withcoefficients QIJa2,xn aa,...,andifwecandiscover apositive functionf(u)definedinu>0, forwhichtheintegral in(*)hasthegivenvaluesall-a2,aa,.••,for '1=I,2,3,•••,thenthefunction rf) F(x)=jIJu:> dUx+uo willbeasolution ofthegivensummation problem, byA.5.Theproblem offinding, givenamafunctionf(u)whichs:'tisfiesthesetofequations (*) isnowcalledStieltjes' problem ofmoments. Stieltjes givesthenecessary andsufficient conditions forittobecapableofsolution, and,inparticular, fortheexistence ofjustonesolution, withverygeneralassumptions. In particular, iff(u),andhenceF(x),isuniquely determined bytheproblem ofmoments -suchaseriesEaniscalledaStieltjes seriesforshort-wexn aremorejustified inclaimingF(x)asasumofthedivergent seriesE~".,forxn instance asitsS-sum. Lackofspaceprevents usfromentering intocloserdetailsofthese verycomprehensive investigations. Anaccount whichincludes every­ thingessential isgivenbyE.Borelinhis"Le~ons surlesseriesdivergentes", whichwehaverepeatedly referred to.Asanexample, suppose wearegiven theseries (t)1l!2!3!x-x2+x3-x'+- ".. 80Lac.cit.(footnotes 12,16),andalsoinhismemoir, Surlareduction en fraction continue d'uneseneprocedant suivant lespuissances descendantes d'une variable, Annales delaFac.Scienc,Touloulle Vo!.3,H.1-17. 1889, 550Chapter XIV.Euler'ssummation formula andasymptotic expansions. Thestatement oftheproblem ofmoments is 00 ff(u)un-1du=(n-I)!, tJn=1,2, whichobviously possesses thesolutionf(u)=e-U•Inthiscasewecan provewithout difficulty thattheaboveistheonlysolution. Hencein fOOe-U F(x)=x+l.ldu o wehavenotonlyfoundafunction whoseasymptotic expansion isthe givenseries,but,inthesenseof§59,wecanregardF(x)astheS-sum 31 ofthe(everywhere) divergent series(t). 2.Theappealtothetheoryofdifferential equations isjustasuseful inthesummation problem asintheexpansion problem (v.A,3).Fre­ quently wecanwritedownthedifferential equation whichisformally satisfied byagivenseriesandamongthesolutions theremaybeafunction whoseasymptotic expansion istheoriginal series. Asarule,however, matters arenotasdescribed above,norasinA.3,butthedifferential equation itselfistheprimary problem. Itisonlywhenthisequation can besolvedformally bymeansofanasymptotic series,aswasindicated inA,3,andprovided wesucceed insumming theseriesdirectly thatwe canhopetoobtainasolutionofthedifferential equation inthisway.Other­ wisewemusttrytodeducetheproperties ofthesolution fromtheasymp­ toticexpansion. Poincare's researches 32,whichwereextended later, especially byA.KneseTandJ.Horn 33,dealwiththisproblem, whichlies outsidethescopeofthisbook. 3.InStieltJes' processthecoefficients anwererecovered, sotospeak, fromthegivenseries13~~,byreplacing anbyn_lX 00f(-l)n-lf(u)u n-1du o 31Thusforx=1weobtainthevalue JOOdu •=e-U- -=O'5!J63471+u .•• o 00 fortheS-sumofthedivergent seriesE(-l)nn!Thisserieshadalready been n~O studied byEu/er(whoobtained thesamevalueforItssum),Lacroix, andLaguerre. Laguerre's workformedthestarting-point ofStieltjes' investigations. 82Poincare, lac.Clt.(footnote 13). 83Acomprehensive account isgivenbyJ.Horn:Gewohnliche Differential­ gleichungen. 2nded.,Leipzig 1927. §66.Specialcasesofasymptotic expansions. 551 andhencetheseriesby Jf(u)G-;.+::- +...)duo u InplaceoftheseriesExntherenowappears theverysimplegeometrical series,multiplied through bythefactorf(u).Thesolution oftheproblem ofmoments isnecessary inordertodetermine f(u),andthisisusually noteasy.Wecan,however, maketheprocess moreelasticbyputting EGrJ=EC(~)-!.:\.n 11enxn' andchoosing thefactors Cnfirstlysothattheproblem ofmoments linkupinthis IfthefunctIOn '"F(x)=Je-u(/J(;)du o isasolution ofthegivensummation problem 34.Herewecannotdiscuss thedetailsoftheassumptions underwhichthismethod leadstothedesired00 Cn=If(u)undu o issoluble, andsecondly sothatthepowerseries E'!E(~)n enx represents aknownfunction. Thuswecanforinstance waywithBaret'ssummation process, byputting Cn=1l!. Efill('!)n=Ea~('!)n=lP('!) enx ".X IX canberegarded asknown,then 3.Theconnection withBorel's summation process canbeestablished as follows. Thefunction ~xR Y=e-xEsnI' 11~On. whichwasintroduced in§59,7forthedefimtion ofBaret's process, hasforits derivative '" ny'=e-a:Ean+1x_. n~O n! '"nThusifwesetEa-tn=lP(t),asinthetextabove,wchavey'=e-:rlP'(x),sothat n.'Otl! x y=ao+Ie-trp'(t)dt. o HenceiftheB-sumof~anexists,itisgivenby 00 S=aD+Ie-trp'(t)dt, o -anexpression which,withsuitable assumptions, canbetransformed intofe-tiP(t)dt o byintegration byparts.Thiscorresponds precisely tothevalueofF(I)deduced IDthetext. 552Chapter XIV.Euler'ssummation formulaandasymptotic expansions. result. Weshallconclude withafewexamples ofthismethodofsum­ mation: inthese,ofcourse,thequestion whether thefunction foundis reallyrepresented bytheseriesmustremainunsettled, sincewehavenot provedanygeneraltheorems. Itcan,however, easilybeverifiedaposteriori. a)Fortheseries Il!2!3!x-x'+x'-x'+-...· whichwehavealreadydiscussed in1.wehave an=(-I)n-l(n-I)!,sothatep(;)=log(1+~), andaccordingly co F(x)=Je-Ulog(1+~)du. o Byintegration byparts,itiseasilyshownthatthefunction ISidentical withthatdiscussed in1. b)Ifwearegiventheasymptotic series I1·3 1·3·5...(2n-1)1 -2.x+2f~~X2-+...+(-l)n---If';~x''- ---+.... wehaveep(~)=(1+;)-.,sothat co co-Je-U.I-J /2d F(x)=vx-==-du =2ea:vXe-t.vu+x _o '\'x Thisprovides, further, theasymptotic expansion co G(z)=Je-t"dt=~e-z"G-2~3+..)- z forwhatisknownasGauss'serror-function, whichisofspecialimportance inthecalculus ofprobabilities. c)Ifwearegiventhesomewhat moregeneralseries withex>0,wehaverp(;)=(1+~)-"'.sothat 00 001 1 Je-U 1Ji--2dtF(x)=x'"---du=-x'"ea:e-tt'"•(u+x)" eto x'". Exercises onChapterXIV. d)Fortheseries 12141---+---+ ...xxax5 wehave(/>(~)=tan-1(~),sothatlS53 ~ ~ F(x)=fe-utan-1(~)du=xfx/-;·u'duo o 0 Ifthisisregarded astheS-sumofthegivendivergent series,weobtain e.g.thevalue a'l fe-U S=ITu'du=0'6214... o forthesumoftheseries 1-2!+41-61+- .... Exercises onChapter XIV. 217.Generalize thisresultandprovethefollowing statements: If__~__ICl C2 ,+Cnn _c", eX+ 1-+I!x+2!x+... TiTx+...-e symbohcally, wehave,inthefirstinstance, C=(I+2B)n+l-(2B)nH= _ ~(2n+I-=--!)Bn+! n n+l n+l and (C+I)n+cn=0forn~I, sothat Co=I,1 2'C.=0,1Ca=4'c.=0,1C.= -2' Usingthesenumbers, wehave(againsymbolically) IP-2P+ 3P-+...+(-l)nnP=~{(-l)n-l(C+ 1 + n)P-CP}. 218.Generalize theresultofExercise 217anddeduce afonnula forthesum f(l)-f(2)+f(3)-+...+(-I)n-If(n), wheref(x)denotes apolynomial. 219.Following 296and298,deduce afonnula for fo-fl+f.-+...+(-I)nfn. 220.a)Following 299,3,andusingEuler'ssummation fonnula, deriv~the powersenesexpansion for-f--.smx b)InEuler'ssummation fonnula, put f(x)=xlogx,x'logx,x"logex,x(logx)',••. andinvestigate therelations soobtained. (Cf.Exercise 224.) 554Chapter XIV.Euler'ssummation formula andasymptotic expansions. 22].Ruler'ssummation formula 298canofcoursebeusedequally ",-ell fortheevaluatIOn ofmtegrals asfortheevaluation ofsums.ShowinthISwaythat jI_~lt'dt=0·62U••• u (v.Ex.223below). I I222.a)Thesum1+2+:f+•..+-11hasthevalue 7·4H5470 forn--1000, an<Ithevalue 14·39272tJ for11-~1000000. ProvethIS,firstassumlOg thatCisknown,andthenwithout assuming 3knowledge ofC. b)Provethatn!hasthevalue 104561>73•2·8242••• for11-~105,andthevalue 105565708 . 82639.•• forn=106• c)Provethatr(x+-1-)hasthevalue 102566.1·2723•.• forx=10',andthevalue 105565705. S2639••• forx=106• d)WIthout assummg aknowledge ofthevalueof,,',evaluate 1 I I10--+-11"+...+;;2forn=10" antifindthelimitofthissumasn40OCJ.(Weobtain0·104IGG83..••0·10516633.••). e)Usmgd),showthat,,'-6-=1·64493406 ••. f)Showthat 00IE-.=1·20205690 •••• ,,~ln andthat 001E~-=2·61237.••on=lnlJ" 1 1 1 g)Provethat1+~lf+V'3c+...+--.y-;;hasthevalue 1998·54014 ••• forn~106• 223.Taking§66,B.3dasamodel,findtheS-sumofthefollowing series, forfixedp=I,2,3,...: -n a)};(-I)n(pn)!. nooQ00 b)E(-I)n(pn+I)! •.... n--O 00 c)};(-Ir(p"+p-I)!. n-O (Cf.Ex,221.) Exercises onChapter XIV. 224.Provethefollowing relationship~, statedbyGlclllher: n2i..."I_In' 11•2~. 33•••••nn'""A.n~~t12e • whereAhasthefollowmg value: A=23'~...~expD(-~C+~S2-~S3+ -...)]~~1·282~27I..•, '"1whereCisEuler'sconstant andSkdenotes thesumE(2-1)1..'(Cf.Exercise 220,b.) ,,-0n+ 225.ForthefunctIOn 00(I)n F(x)=E--- flcO(x+n) obtaintheasymptotic expansion ooa 1 1 1 F(x)'--'nEx~""2x+4X2-8x'+ 2n-1andprovethatthecoefficient a..hasthevalue-,1--Bnfor1162. 556 Bibliography. Bibliography. (Thisincludes somefundamental papers, comprehensive accounts, and textbooks.) 1.Newton. 1.'Deanalysiperaequationes numero terminorum infinitas. Lon· don1711(written in1669). 2.Wallis.John:Treatise ofalgebra bothhistorical andpractical, withsome additional treatises. London 1685. 3.Bernoulli. James:Propositiones arithmeticae deseriebus infinitis earumque summa finita,withfouradditiom;. Basle1689-1704. 4.Eulef'.L.:Introductio inanalysin infinitorum. Lausanne 1748. 5.Eul,f',L.:Institutiones calculidifferentialis cumejususuinanalysi Intini· torumacdoctrina serierum. Berlin1755. 6.Eulef'.L.:Institutiones calculiintegralis. St.Petersburg 1768-69. 7Gauss,K.F.:Disquisitiones generales circaseriemintinitam l+a·pz+ a(a+l)·p(P+l)#+etc. 1." 1.2.,,(,,+1) Gottingen 1812 8.Cauchy, A.L.:Coursd'analyse del'ecolepolytechnique. PartI.Analyse algebrique. Paris1821. mm(m-l)9.Abel.N.H.:Untersuchungen UberdieReihe1+-x+---#+....1 1·2 Journal fUrdiereineundangewandte Mathematik, Vol.1,pp.311 to339.1826. 10.duBois-Reymond, P.:EineneueTheorie derKonvergenz undDivergenz vonReihen mitpositiven Gliedern. Journal (lirdiercineundange­ wandte l\fathematik, Vol.76,pp61-91. 1873. 11.Pringsheim, A.:AlIgemeine Theorie derDivergenz undKonvergenz von Reihen mitpositiven Gliedern. Mathematische Annalen, Vol.35. pp.297-394. 1890. 12.Pringsheim, A.:Irrationalzahlen undKonvergenz unendlicher Prozesse. Enzyklopiidie dermathematischen Wissenschaften. Vol.I,I,3 Leipzig 1899. 13.Borel,E.:Lec;:ons surlesseriesatermes positifs. Paris1902. 14.Runge,C.:Theorie undPraxisderReihen. Leipzig 1904. 15.Sto11l.0.,andA.Gmeinef': Einleitung indieFunktionentheorie. Leipzig 1905 16.Prin~sheim, A.andJ.Molk:Algorithmes illimites denombres reels.En· cyclopedie desSciences Mathematiques, Vol.J,1,4.Leipzig 1907. 17.Bromwlch. T.J.l'A.:Anintroduction tothetheory ofinfinite series London 1908:2nded.1926. 18.Pringsheim. A.andG.Faber:Algebraische Analysis. Enzyklopadie der mathematischen Wissenschaften, Vol.11,C.1.Leipzig 1909. 19.Fabry,E.:Thcorie desseriesatermesconstants. Paris1910. 20.Pringsheim, A.,G.Faber,andJ.Molk: Analyse algebrique. Encyclopedie desSciences Mathcmatiques, Vol.11,2.7.LeipziglOll. 21.Stolz,0.•andA.Gmeiner: Theoretische Arithmetik, Vol.11.2ndedition. Leipzig 1915. 22.Pringsheim, A.:Vorlesungen uberZahlen- undFunktionenlehre, Vol.I,2 and3.Leipzig, 1916and1921.2nd(unaltered) ed.1923. NameandSubject Index. Thereferences aretopages. Cahen,E.,290,441. Cajori,F.,322. Cantor, G.,I,26,33,68,355. Cantor,M.,12. 667Abel,N11.,122,127,2Il,281,290 seq.,299,313,314,321,424seqq., 459,467,556. Abel-Dim theorem, 290. Abel'sconvergence test,314. -lImittheorem, 177,349. -partialsummatIOn, 313,397. -senes,122,281,292. -theorem, extension of,406. AbSCissa ofconvergence, 44l. Absolute convergence ofsenes, 136 seq".,396. -ofproducts, 222. Absolute value,7,390. Adams,J.C.,183,256. Addition, 5,30, 32. -termbyterm,48,70,134. Addition theorem fortheexponential function, 191. -forthebInomIal coefficients, 209. -forthetrigonometrical functIOns, 199,415. Aggregate, closed,7. -ordered, 5. d'Alembert, J.,458, 459. BAlmost all",65. Alterations, fimtenumber of,forse­ quences, 47,70,95. -forseries,130,476. AlternatIng senes,131,250,263seq., 316,518. Ames,L.D.,244. AmplItude, 3UO. Analytic functions, 401seqq. -senesof,429. Andersen, A.P.,488. Approach withinanangle,404. Approximation, 65,231. Archimedes, 7,104. Area,169. Anthmetic, fundamental lawsof,5. -means, 72,460. Arrangement bysquares, bydiagonals, 90. Arzeta,S.,344. Associative law,5,6. -forseries,132.Asymptotically equal,68. -proportional, 68,247. Asymptotic series(expansion, repre- sentation), 518seq.,535seqq. Averaged companson, 464-66. Axiom, Cantor-Dedekind, 26,33. Axioms ofanthmetlc, 5. Bachmann, F.,2. Barnes,E.W.,546. Bernoulli, JamesandJohn,18,65,184, 2:38,244,457,52:1seq.,556. Bernoulli's inequality, 18. Bernoulli, Nicolaus, 324. Bernoulli's numbers, 183,203--4, 237, 479. -polynomials, 523,534seqq. Bertrand, J.,282. Bieberbach, L.,478. Bmaryfraction, 39. BInomial senes,127,190, 208-H, 423-8. -theorem, 50,190. Bacher,M.,350. Bohr,H.,492. duBois-Reymond, P.,68,87,96,301, 304,305, 353, 355, 379, 556. duBois-Reymond's test,315,348. Bolzano, B.,87,91,394. Bolzano- Weierstrass theorem, 91,394. Bonnet,0.,282. Boormann, J.M.,195. Borel,E.,320,471seqq.,477,543,549, 551,556. Bound, 16,158. -upper,lower,96,159. Bounded functions, 158. -sequences, 16,44,80. Breaking offdecimals, 249. Briggs,H.,58,257. Bromwich, l'A.,477,556. Brouncker, W.,104. Burkhardt, H.,353,375,546. l)58 Index. Cantor-Dedekind axiom,26,33. Carmichael, R.D.,477. Catalan, E.,247. Cauchy,A.L.,19,72,87,96,104,113, 117, 136, 138, 146, 147,148, 154, 186,190,219, 294, 408, 459, 534, 545,556. Cauchy's convergence theorem, 120. -doubleseriestheorem, 143. -inequality, 408. -limit theorem, 72. -product, 147,179,488, IH2. Cauchy-Toepllt::. limittheorem, 74,391. Centreofapowerseries,157. Cesaro,E.,292,318,322,466. Chapman, S.,477. Characteristic ofalogarithm, 58. Circleofconvergence, 402. Circular functions, 59:seealsoTrIgo­ nometrical functlOns. Closedaggregate, 7. -expressions forsumsofseries,232 to240. -interval, 20,162. Commutative law,5,6. -forproducts, 227. -forseries,138. Comparison testsofthefirstandsecond kmds,113seq.,274seq. Completeness ofthesystem ofreal numbers, 34. Complex numbers: seeNumbers. Condensation test,Cauchy's, 120,297. CondItionally convergent, 139,226seq. Conditions P,464. Contmued fractions, 105. Contmuity, 161-2,171,174,404. -ofpowerseries,174, 177. -ofthestraight line,26. -uniform, 162. Convergence, 64,78seq. -absolute, 136seq.,222,396seq.,435. -conditional, unconditional, 139,227. -ofproducts, 218,222. -ofseries,101. -uniform, 326seq.,381,428seq. Convergence, abSCIssa of,441. -circleof,402. -criteria of:seeConvergence tests, alsoMaincriterion. -general remarks ontheoryof,298to 305. -half-plane of,441. -interval of,153,327. -radiusof,151seqq. -rapidity of,251,262,279,332. -regionof,153.Convergence, systematization oftheon of,305to311. . -testsforFourier series,301,364-72. -forsequences, 78-8fl, -forseries,110-20, 124,282-90. -forseriesofcomplex terms,396--401. -forseriesofmonotonely dlmmishmg terms,120-4,294-6. -forseriesofpositive terms,116,117. -foruniform convergences, 332-8. Convergent sequences: seeSequences. Cosine, 199seq.,384,414scq. Cotangent, 202seq.,417seq. Curvesofapproximation, 329, 330. Decimal fractions, 116:seeRadixfrac- tions. -section, 24,51. Dedekind, R.,I,20,33,41. -section, 41. Dedekind's test,315,348. Dense, 12. Diagonals, arrangement by,90. Difference, 31,243. Difference-sequence, 87. Dlfferentiability, 163. -ofapowerscries,174-5. -righthand,lefthand,163. Differentiation, 163-4. -logarithmic, 382. -tcrmbyterm,175,342. Dini,a.,227,282, 290, 293,:nI, 344. Dini'srule,367-8,371. Dirichlet, G.Lejeune-, 138,329,347, 356,375,547. Dinchlet's integral, 356seq.,359. -rule,365,371. Dirichlet serics,317,441scq. Dirichlet's test,315, 347. DIsjunctive criterion, 118,308,309. DIstributive law,6,135,146seq. Divergence, 65,101,160,3!H. -definite, 66,101,160,391. -indefinite, 67,101,160. -proper, 67. Divergent sequences, 457seqq. -series,457seqq. Division, 6,32. -ofpowerseries,180seqq. -termbyterm,48,71. Divisors, number of,446,451, 5-17. -sumof,451,547. Doetsch, G.,478. Double series,theorem on,430. -analogue forproducts, 437-8. Duhamel,J.M.C.,285. Index. 559 e,82,194-8. -calculation of,251. Elsenstezn, G.,180. Elliot,E.B.,314. e:-neighbourhood, 20. Equahty, 2H. Equivalence theorem ofKnopp and Schnee,481. Ermakoff's test,296seqq.,3ll. Error,65. -evaluatIOn of:seeEvaluation ofre- mamders. Euclid, 7,14,20,69. Eudoxus, postulate of,11,27,34. -theorem of,7. Buler,L.,I,82,104,182,193,204, 211,22H,238, 243, 244,262,353, 375, 384, 385, 413,4111,4:3ll,445, 457seqq.,46Hseq.,5U7,518,535-6, 556. Buler's constant, 22fi,22S,271,522, 527seq.,536,53H,547,555. -<p-functlon, 451,548. -formulae, 35:3,415,518,536. -numbers, 239. --transformatIOn ofsenes, 244-6, 262-5,4G9,507. Evaluation, numerical, 247-60. --ofl',2111. -ofloganthms, 198,254--7. -of1t,252-4. -ofremainders, 250,525,531-5. - -moreaccurate, 259. -ofroots,257-8. -oftrigonometncal functions, 258-9. EvenfunctIOns, 173. Everywhere convergent, 15:3­ Exhaustion, method of,69. ExpanSIOn ofelementary functIOns in partial fractions, 205-8, 239,377 seqq.,419. -ofinfimteproducts, 437. -problem forasymptotic senes,543 seqq. Exponential function andsenes, 148, 191-8,411-4. Expressions forrealnumbers, 230. -forsumsofseries,230-73. -forsumsofseries,closed,232-40. Extension, 11,34. Faher,G.,546,Mi6. Fahry,E.,267,556. Faculty series,446seq. Fatzius, N.,244. Fejer,L.,493,496,546. Fejer'sintegral, 4!J4.Fejer'stheorem, 49:3. Fibonacci's sequence, 14,270,452. Fimtenumber, 15, 16. --ofalterations: seeAlterations. Fourier,j.R.,352, 375. -coeffiCients, constants, 354,361,362. -senes,350seqq.,492seqq. -Riemann's theorem on,363. Frobenius, G.,184,490. Frullani, 375. Fullymonotone, 263,264,305. Function, Hi8,403. -interval ofdefinition, limit,oscilla· tion,upperandlower bound~ ot, 158-9. Functions, analytic, 401seq. -arbitrary, 351-2. -cyc1ometrical, 213-5,421seq. -elementary, 189seq. -elementary analytiC, 410seq. -even,odd,173. -Integral, 408,411. -ofacomplex variable, 403seq. -ofarealvanable, 158seq. -rational, 189seq.,410seq. -regular, 408. -sequences of,326seq.,429. -trigonometncal, 198seq.,258,414 seq. Fundamental lawofnatural numbers, 6-7. -ofIntegers, 7. -lawsofanthmetic, 5,32. -oforder,5,29. Gamma-function, 225-6, 385,440, 5:30. Gapsmthesystemofrational numbers, 3seqq. Gauss,K.F.,I,113, 177, 288, 289, 552, 55fl. Geometric series:seeSeries. Glhbs'phenomenon, 380,196. Glaisher,j.W.L.,180,555. Gmeiner,j.A.,399,556. Goldbach, 458. Goniometry, 415. Grandi, G.,133. Graphical representation, 8,15,20,390 seq. Gregory,j.,65,214. Gronwall, T.H.,380. Hadamard, j.,154,299,301, 314. Hagen,j.,182. Hahn,H.,2,305. Half-plane ofconvergence, ·141. MO Index. Hanstedt, B.,lRO. Hardy, G.H.,318,322,407,444,477 seq.,486, 487, 547, 548. Harmonic series:seeSeries. Hausdorff, F.,478. Hermann, j.,131. History ofinfinite series,104. llobson,E.W.,350. HOlder,0.,465,490. Holmboe, 459. Horn,j.,550. Hypergeometric series,289. Identically equal,15. Identity theorem forpowerseries,172. Improper integral: seeIntegral. Induction, lawof,6. Inequalities, 7. Inequality ofnests,29. Infinite number, 15. -series: seeSeries. Infinitely small,19. Innermost point,23,39!. Integrability inRtemann's sense,166. Integral, 165seq. -improper, lU9-70. -logarithmic, 545. Integral test,294. Integration byparts,169. -termbyterm,176,341. Interval, 20. -ofconvergence, 153,327. -ofdefinition, 158. Intervals, nestof,21,394. Inverse-sine function, 215,421seq. Inverse-tangent functIOn, 214,422seq. Isomorphous, 10. jacobi,C.G.j.,439. jacobsthal, E.,244,263. jemen,j.L.W.V.,74,76,441. jones,W.,253. jordan, C.,16. Karamata, j.,501,504. Keplerian motion, 546. Kneser,A.,550. Knopp,K.,2,75,241, 244, 247, 267, 350,404,448,467,477,481,487,507, 547. Kogbetliantz, E.,488. Kowalewski, G.,2. Kronecker, L.,theorem of,129,485. -complement totheorem of,150. Kummer, E.E.,241,247,260,311. Kummer's transformation ofsenes.247. 260.Lacroix, S.F.,5liO. Lagrange, j.L.,298. Laguerre, E.,550. Lambert,j.H.,448,451. -series,448seq. Landau,E.,2,4,11,444,446,452,484. Laplace, P.S.,546. Lasker,E.,490. Lawofformation, 15,37. -ofinduction, 6. -ofmonotony, 6. Lawsofarithmetic, 5,32. -oforder,5,29. Lebesgue, H.,168,350,::153. Ledert, 247. Lefthandcontinuity, 161. -differentiability, 163. -limit,159. Legendre, A.M.,375,520. Leibniz, G.W.,I,103,131,193,244,457. -equation of,214. -ruleof,131,316. Length, 169. LeRoy,E.,473. Levy,P.,398. Limit,64,462. -ontheleft,right,159. -upper,lower,92-3. Limitofafunction, 159,403-4. -ofasequence, 64. -ofaseries,101. Limitable, 462. Limitation processes, 403-77. -generalformof,474. Limiting curve,330. Limiting pointofasequence, 89,394. -greatest, least,92-3. Limittheorems: seeAbel,Cauchy. Toe- plitz. Lipschitz, R.,368, 371. Littlewood, j.E.,407,478,501. Loewy,A.,2,4. Logarithmic differentiation, 382. -scales,278seqq. -series,211seq.,419seq. -tests,281-4. Logarithms, 57-9,211seq.,420. -calculation of,24,198,254-7. Lyra,G.,487. Machin,j.,253. Maclaurin, C.,521. Maincriterion ofconvergence, first, forsequences, 80. -forseries,110. -second. forsequences. 84,87,393, 395. Index. 561 Maincriterion ofconvergence, second, forseries,126-7. -third,forsequences, 97. Malmsten, C.j.,316. Mangoldt, H.v.,2,350. Mantissa, 58. Markoff, A.,241,242,265. Markoff's transformation ofseries,242 to244,265seq. Mascheroni's constant: seeEuler'scon­ stant. Meanvaluetheorem ofthedifferential calculus, first,164. -oftheintegral calculus, first,168. -second, 169. Measurable, 169. Mercator, N.,1O'/'. Mertens, F.,321, 398. Method ofbisection, 39. Mittag-Leffler, G.,1. MiJblus' coefficients, 446,451. Modulus, 8,390. Molk,j.,556. Moments, Stieltjes' problem of,549. Monotone, 17,44,162-3. -fully,263-4,305. Monotony, p-fold,263-4. -lawof,5,6. deMorgan, A.,281. Motions ofx,HlO. Multiplication, 6,31,50. -ofmfinite series,146seq.,320seq. -ofpowersenes,179. -termbyterm,70,135. Napier,j.,58. Natural numbers: seeNumbers. Nestofmtervals, 21,394. -ofsquares, 394. Neumann, C.,17. Newton, I.,I,104,193,211,457, 556. Non-absolutely convergent, 136,396-7, 435. N6rlund, N.E.,521. Nullsequences, 17,45seq.,60-3,72, 74. Number axis,8. -concept, 9. -corpus, 7. Number system, 9. -extension of,Il,34. Numbers: seealsoBernouilli's numbers, Euler'snumbers. -complex, 388seq. -Irrational, 23seq. -natural, 4.Numbers, prime,14,445seq.,451,548. -rational; 3seqq. -real,33seqq. Numerical evaluations, 79,232-73, espe­ cially247-60. Oddfunctions, 173. Ohm,M.,184,320. Oldenburg, 211. Olivier,L.,124. Open,20. Ordered, 5,29. Ordered aggregate, 5. Orstrand, C.E.van,187. Oscillating series,101-3. Oscillation, 159. Pairoftests,308. Partialfractions, expansion ofelemen­ taryfunctions in,205-8, 239,377 seqq.,419. Partialproducts, 105,224. Partialsummation, Abel's,313,397. Partialsums,99,224. Partitions, number of,548. Passage tothelimittermbyterm, 338seqq.: seealsoAddition, Sub­ traction, Multiplication, DiVISIOn, Differentiation, Integration. Peano,G.,11. Periodstrip,413seq.,416-8. Periodic functions, 200,413seq. Permanence condition, 463. Perron,0.,105,475, 478, 546. 7t',200,230. -evaluation of,252-4. -senesfor,214,215. Poincare, H.,520,536,543,550. Poisson, S.D.,521. Poncelet,j.V.,244. Portionofaseries,127. Postulate ofcompleteness, 34. Postulate ofEudoxus, Il,27,34. Power series, 151seqq.,17lseqq., 401seqq. Powers, 49-50,53seq.,423. Primenumbers, 14,445seq.,451,548. Primitive period, 201. Prmcipal criterion: seeMaincriterion. Principal value,420,421-6. Pringsheim, A.,2,4,86,96,175,221, 291,298,300,301,309,320,39U, 490,556. Problem ofmoments, 559seq. Problems AandB,78,105,230seqq. Products, 31. -infinite, 104,218-29. 562 Index. Products witharbitrary terms,221seq. -withcomplex terms,4:34seq. -withpositIve terms,218seq. -withvariable terms,380seq.,436seq. Pythagoras, 12. Quotient, 31. -ofpowerseries,182. Raabe,J.L.,285. Rademaeher, H.,318,548. RadIan, 59. Radiusofconvergence, 151. Radixfractions, 37seq. -breaking off,249. -recurring, 39. Raff,H.,475. Ramunujan, S.,/348. Rangeofaction,463. -ofsummation, 398. Rapidity ofconvergence: seeConver- gence. Ratiotest,116-7,277. Rational functions, 189seq.,410seq. -numbers: seeNumbers. Rational-valued nests,28. Realnumbers: sceNumbers. Rearrangement, 47,138. -inextended sense,142. -ofproducts, 227. -ofsequences, 47,70. -ofseries,136seqq.,318seqq.,398. -theorem, main,143, 181. - -application of,236-40. -Riemann's, 318seq. Reciprocal, 31. Regular functions, 408. Reiff,R.,104,133,457seq. Remainders, evaluation of,250, 259, 526,531-5. Representation ofrealnumbers ona straight hne,33. Representative point,33. Reversible functions, 163,184. Reversion theorem forpower series, 184,405. Riemann, B.,166,318,319,363. Riemann's rearrangement theorem, 318 to320. Riemann's theorem onFourier series, 363. Riemann's '-function, 345,444-6,491-2, 5:n,538. Riesz,M.,444, 477. Righthandcontinuity, HJl. -differentiability, 163. -limit, 159.Rogosinski, W.,350. Roots,50seqq. -calculatIOn of,257-8. Roottest,116-7. Runge,C.,556. Saalsehlltz, L.,184. Saehse,A.,:!G:l. Scales,logarithmic, 278seqq. Seherk,W.,239. Sehlollllleh, 0.,121,287,320. Schn'lldt, Herm.,212. Schnee,W.,481. Schr6ter, H.,20t. Sehur,l.,207,481,547. Section, 40,99. Seidel,Ph.L.v.,3:34. Semi-convergent, 520,536. Sequences, 14,43seqq. -bounded, IU. -complex, :l88seqq. -convergent, 64-78. -divergent, 65. -infimte, 15. -null,17seqq.,45seq.,00-3,72,74. -offunctions, 327seqq.,42!l. -ofpomts, 15. -ofportIOns, 127. -ratIOnal, 14. -real,15,43seq. Senes,alternatmg, 131,250,263seq., :H6,518. -asymptotIC, 535seqq. -binomial, 127, 190, 208seqq.,423 seqq. -Diriehlet, 317,441seqq. -divergent, 457~eqq. -exponential, 148,191,411. -faculty, 446seq. -fortrigonometrical functions, 19!? seq.,414seq. -Fourier, 350seqq.,492seqq. -geometric, 111,179,189,472,508. -harmonic, 81,112,115-7,150,2:37, 238. -hypergeometric, 289. -mfinite, !l8seqq. -mfinitesequence of,142. -Lambert, 448seq. -logarithmic, 211seq.,419seq. -ofanalytic functions, 428. -ofarbItrary terms, 126seqq.,312 seqq. -ofcomplex terms,388seqq. -ofpositive terms,110seqq.,274seqq. -ofpositive, monotone decreasing terms,120seqq.,294seq. Index. 563 Senesofvariable terms,152seq.,:12n seq.,42Sseq. -transformation of,240seqq., 2no seqq. -tngonometrical, 350seq. Sierpttiski, W.,320. SImilar systems ofnumbers, 10. Sine,199seq.,384,414seq. Sineproduct, 384. Squares, arrangement by,90. Stelnitz, E.,3H8. Stieltjes, Th.J.,238,302,321,530,545, 549seqq.,55-1. Stieltjes' moment problem, 549. -series,5!9. Stirllnl!,J.,240,448,530. $tzrlmg's formula, 529seq.,li38,li41. Stokes,G.G.,334. ,Sto1::::,0.,4,39,7n,87,311,407,550. StripsofconditIOnal convergen-:t:. 444. Sub-sequences, 46,H2. Sub-series, 116, 141. Subsidiary value,421. SubtractIOn, 5,31. -termbyterm,48,71,135. Sum,30. -ofdiVIsors, 451. .-ofaseries,101seq.,402. SummabIllty, boundary of,492. Summable, 4li2. -ab~olutely, 5J3. -uniformly, 490. Summation byarzthmetic means, 400 seqq. -ofDirichlet ~enes,4li4,491. -ofPourier series,41i4,492seqq. Summation formula, Ruler's, 518seqq. Summation, indexof,99. -rangeof,398. Summation problem forasymptotic senes,543,548seqq. SwnmatlOn processes, 4li4-76. -commutability of,509. Sumsofcolumns, ofrows,144. Sylvester,J.J.,180. Symbolic equations, 183,52:3,526. Tangent, 202seq.,417seq. Tauber,A.,48o,500. Tauberian theorems, 486,500. Taylor,B.,175. Taylor's series,175-6. Termbytermpassage tothelimit: seePassage. Termsofaproduct, 219. -ofaseries,99.Testsofconvergence: sceConvergence tests. Theory ofconvergence, general re- markson,298-30!). -systematization of,305-11. Tztchmarsh, E.C.,444. Toeplitz, 0.,74,474,489-90. Toeplitz' bmittheorem, 74,3!)I. Tonelli,L.,350. Transformation ofseries, 240seqq., 200seq. Trigonometrical functions, 198-208, 414-9. -calculation of,258-9. Trigonometrical series.350seq. Ultimate behaviour ofasequence, 16, 47,95,lOa. Unconditionally convergent, 139,2?6. Umform contmuity, 162. -convergence ofproducts, 381. - -ofseries,326seq.,428seq. - -ofDzrichlet series,442. - -offacultyseries,446-7. - -ofFourier series,35/l-6. - -ofLambert series,44H. - -ofpowerseries,332seqq. -convergence, testsof,344seq.,381. -summabIlity, 496. Umformly bounded, 337. Uniqueness ofthesystemofrealnum- bers,33seqq. Uniqueness, theorem of,35,17.t. Umt,10. Unitcircle,402. Valueofaseries,101,460. Vteta,P.,218. Vivallti, G.,344. Voss,A.,322. Wallis,J.,20,39,219,556. Wallis'product, 384,529. lVeierstrass, K.,I,91,:J34,345,379, 394, 398, 408, 430. Weierstrass' approxImation, 497. -testforcomplex series,398seq. -testofuniform convergence, 345. -theorem ondouble series,430seqq. Wiener,N.,451. Wirtinger, W.,521seq. Zero,10. ~-functlon, Riemann's, 345,444-6, 491-2,531,538. Zygmund, A.,350.