examples of power series.pdf

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1、Examples of Power Series Leif Mejlbro Download free books at LeifMejlbroExamplesofPowerSeriesCalculus3c-3ExamplesofPowerSeries–Calculus3c-3©2008LeifMejlbro&VentusPublishingApSISBN978-87-7681-377-2Calculus3c-3ContentsContentsIntroduction51.Powerseries;radiusofconvergenceandsum62.Po

2、werseriesexpansionsoffunctions353.Cauchymultiplication454.Integralsdescribedbyseries485.Sumsofseries51Fast-trackyourcareerMastersinManagementStandoutfromthecrowdDesignedforgraduateswithlessthanoneyearoffull-timepostgraduateworkexperience,LondonBusinessSchool’sMastersinManagementwi

3、llexpandyourPleaseclicktheadvertthinkingandprovideyouwiththefoundationsforasuccessfulcareerinbusiness.Theprogrammeisdevelopedinconsultationwithrecruiterstoprovideyouwiththekeyskillsthattopemployersdemand.Through11monthsoffull-timestudy,youwillgainthebusinessknowledgeandcapabilitie

4、stoincreaseyourcareerchoicesandstandoutfromthecrowd.LondonBusinessSchoolApplicationsarenowopenforentryinSeptember2011.Regent’sParkLondonNW14SAUnitedKingdomFormoreinformationvisitwww.london.edu/mim/Tel+44(0)2070007573Emailmim@london.eduemailmim@london.eduorcall+44(0)2070007573www.l

5、ondon.edu/mim/4Calculus3c-3IntroductionIntroductionHerefollowsacollectionofgeneralexamplesofpowerseries.ThereaderisalsoreferredtoCalculus3b.Theimportanttechniqueofsolvinglineardifferentialequationswithpolynomialcoefficientsbymeansofpowerseriesispostponedtothenextbookinthisseries,Calc

6、ulus3c-4.ItshouldnolongerbenecessaryrigourouslytousetheADIC-model,describedinCalculus1candCalculus2c,becausewenowassumethatthereadercandothishimself.EvenifIhavetriedtobecarefulaboutthistext,itisimpossibletoavoiderrors,inparticularinthefirstedition.Itismyhopethatthereaderwillshowsom

7、eunderstandingofmysituation.LeifMejlbro14thMay20085Calculus3c-3Powerseries;radiusofconvergenceandsum1Powerseries;radiusofconvergenceandsumExample1.1Findtheradiusofconvergenceforthepowerseries,∞1nx.nnn=11Leta(x)=xn.Thenbythecriterionofrootsnnn

8、x

9、n

10、an(x)

11、=→0forn→∞,nandtheseriesisc

12、onvergentforeveryx∈R,hencetheinte