EULER AND HIS WORK ON INFINITE SERIES

BULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume44,Number4,October2007,Pages515–539S0273-0979(07)01175-5ArticleelectronicallypublishedonJune26,2007EULERANDHISWORKONINFINITESERIESV.S.VARADARAJANForthe300thanniversaryofLeonhardEuler’sbirthTableofcontents1.Introduction2.Zetavalues3.Divergentseries4.Summationformula5.Concludingremarks1.IntroductionLeonhardEulerisoneofthegreatestandmostastoundingiconsinthehistoryofscience.Hiswork,datingbacktotheearlyeighteenthcentury,isstillwithus,verymuchaliveandgeneratingintenseinterest.LikeShakespeareandMozart,hehasremainedfreshandcaptivatingbecauseofhispersonalityaswellashisideasandachievementsinmathematics.Thereasonsforthisphenomenonlieinhisuniversality,hisuniqueness,andtheimmenseoutputheleftbehindinpapers,correspondence,diaries,andothermemorabilia.OperaOmnia[E],hiscollectedworksandcorrespondence,isstillintheprocessofcompletion,closetoeightyvolumesand31,000+pagesandcounting.Avolumeofbriefsummariesofhislettersrunstoseveralhundredpages.Itishardtocomprehendtheprodigiousenergyandcreativityofthismanwhofueledsuchamonumentaloutput.Evenmoreremarkable,andinstarkcontrasttomenlikeNewtonandGauss,isthesunnyandequabletemperamentthatinformedallofhiswork,hiscorrespondence,andhisinteractionswithotherpeople,bothcommonandscientific.Itwasoftensaidofhimthathedidmathematicsasotherpeoplebreathed,effortlesslyandcontinuously.Itwasalsosaid(byLaplace)thatallmathematicianswerehisstudents.Itisappropriateinthis,thetercentennialyearofhisbirth,torevisithimandsurveyhiswork,itsoffshoots,andtheremarkablevitalityofhisthemeswhicharestillflourishing,andtoimmerseourselvesonceagainintheuniverseofideasthathehascreated.Thisisnotataskforasingleindividual,andappropriatelyenough,anumberofmathematiciansareattemptingtodothisandpresentapictureofhisworkanditsmodernresonancestothegeneralmathematicalcommunity.Tobehonest,suchaprojectisHimalayaninitsscope,anditisimpossibletodofulljusticetoit.InthefollowingpagesIshalltrytomakeaverysmallcontributiontothisproject,discussinginasketchymannerEuler’sworkoninfiniteseriesanditsmodernoutgrowths.MyaimistoacquaintthegenericmathematicianwithReceivedbytheeditorsApril20,2007and,inrevisedform,April23,2007.2000MathematicsSubjectClassification.Primary01A50,40G10,11M99.c
2007AmericanMathematicalSocietyRevertstopublicdomain28yearsfrompublication515516V.S.VARADARAJANsomeEulerianthemesandpointoutthatsomeofthemarestillawaitingcompleteunderstanding.Aboveall,itisthefreedomandimaginationwithwhichEuleroperatesthataremostcompelling,andIwouldhopethattheremarksbelowhavecapturedatleastsomeofit.ForatributetothisfacetofEuler’swork,see[C].TheliteratureonEuler,bothpersonalandmathematical,ishuge.Thereferencesgivenattheendarejustafractionofwhatisrelevantandareinnowayintendedtobecomplete.However,manyofthepointsexaminedinthisarticlearetreatedatmuchgreaterlengthinmybook[V],whichcontainsmoredetailedreferences.Afterthebookcameout,ProfessorPierreDeligne,oftheInstituteforAdvancedStudy,Princeton,wrotetomesomelettersinwhichhediscussedhisviewsonsomeofthethemestreatedinmybook.IhavetakenthelibertyofincludingheresomeofhiscommentsthathaveenrichedmyunderstandingofEuler’swork,especiallyoninfiniteseries.IwishtothankProfessorDeligneforhisgenerosityinsharinghisideaswithmeandforgivingmepermissiontodiscussthemhere.IalsowishtothankProfessorTrondDigernesoftheUniversityofTrondheim,Norway,forhelpingmewithelectroniccomputationsconcerningsomecontinuedfractionsthatcomeupinEuler’sworkonsummingthefactorial-likeseries.2.ZetavaluesEulermustberegardedasthefirstmasterofthetheoryofinfiniteseries.Hecreateditandwasbyfaritsgreatestmaster.PerhapsonlyJacobiandRamanujanmayberegardedasbeingevenclose.BeforeEulerenteredthemathematicalscene,infiniteserieshadbeenconsideredbymanymathematicians,goingbacktoveryearlytimes.Howevertherewasnosystematictheory;peoplehadonlyveryinformalideasaboutconvergenceanddivergence.Alsomostoftheseriesconsideredhadonlypositiveterms.Archimedesusedthegeometricseries43=1+14+142+143+...incomputing,bywhathecalledthemethodofexhaustion,theareacutoffbyasecantfromaparabola.Leibniz,Gregory,andNewtonhadalsoconsideredvariousspecialseries,amongwhichtheLeibnizevaluation,π4=1−13+15−17+...,wasamoststrikingone.Inthefourteenthcenturypeoplediscussedtheharmonicseries1+12+13+...,andPietroMengoli(1625–1686)seemstohaveposedtheproblemoffindingthesumoftheseries1+122+132+....Thisproblemgeneratedintenseinterest,andtheBernoullibrothers,JohannandJakob,especiallytheformer,appeartohavemadeeffortstofindthesum.ItcametobeknownastheBaselproblem.Butalleffortstosolveithadprovenuseless,andevenanaccuratenumericalevaluationwasextremelydifficultbecauseoftheslowdecayoftheterms.Indeed,since1n−1n+1=1n(n+1)1n21n(n−1)=1n−1−1nEULERANDHISWORKONINFINITESERIES517wehave1N+1∞
n=N+11n21N,sothattocomputedirectlythesumwithanaccuracyofsixdecimalplaceswouldrequiretakingintoaccountatleastamillionterms.Euler’sfirstattackontheBaselproblemalreadyrevealedhowfaraheadofeveryoneelsehewas.Sincethetermsoftheseriesdecreasedveryslowly,Eulerrealizedthathehadtotransformtheseriesintoarapidlyconvergentonet