北京邮电大学国际学院高等数学(下)幻灯片讲义(无穷级数)lecture 2new

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1、SeriesofFunctionsDefinition(SeriesofFunctions&ConvergenceDomain)∞Lecture2Aseriesoftheform,whereisafunctionofina∑uxn()uxn()xn=1set,iscalledaA⊆Rseriesoffunctionsdefinedintheset.A∞Iftheseriesofconstanttermsconverges,thenwesaythe∑uxn()0∞n=1seriesconver∑uxn()gg,esatxx=0,andiscalleda

2、x0convergencen=1pointoftheseriesoffunctions;otherwise,iscalledaxdivergence0PowerSeriespoint.Thesetconsistingofallconvergencepointsiscalledtheconvergencedomainoftheseriesandthesetconsistingofalldivergencepointsiscalledthedivergencedomainoftheseries.12SeriesofFunctionsSeriesofFun

3、ctionsnDefinition(SumfunctionandConvergence)DefinitionThequantityiscalledtheSnk()xu=∑()xpartialsum∞∞k=1Ifistheconvergencedomainoftheseries,then,D∑uxn()∀∈x0Doftheseries,while∑uxn()∞n=1n=1theserieshasasum.Hencethesumoftheseries∑uxn()0S()x0∞∞n=1R()xSxSx=()−=()∑uxxD(),∈∑ux(),,isact

4、uallyafunctionofdefinedin,calledtheS()xxDnnknkn=+1n=1sumfunctionanddenotedbyiscalledaremainderoftheseries.∞∑uxSxxDn()=∈(),Itiseasytoseethatthenecessaryandsufficientconditionforaseries∞n=1∞∑ux()tobeeverywhereconvergentinisthatDnInthiscase,theseriesissaidtoexhibit∑uxn()everywhere

5、n=1n=1convergenceorpointwiseconvergencetoin.Generally,itS()xDlimRnn()xS=0orlim()x=∈S(),xxDnn→∞→∞iscalledsimplyconvergencetoin.S()xD34PowerSeriesTheGeometricSeriesDefinition(PowerSeries)ThegeometricseriesAnexpressionoftheform∞nn2∞∑xx=1+++++xx??nn2∑cxnn=++ccxcx012++??cx+n=0n=01is

6、apowerseriescenteredat.Itconvergestoontheintervalx=0isapowerseriescenteredatx=0.Anexpressionoftheform1−x−11<

7、ecenterastheseries,orconvergesonlyatthecenterisapowerseriescenteredatx=x.Thetermisthecxx()−nnth0n0itself.term;thenumberisthex0center.561ApplyingtheDefinitionApplyingtheDefinitionThepowerseries2(2xx−−)(2)2⎛⎞1nnnso=1(−+−??+⎜⎟−−xx2)+<,0<4112⎛⎞1nx24⎝⎠21−−+−++(xx2)(2)??⎜⎟−−+(x2)(1)2

8、4⎝⎠2Series(1)generatesusefulpolynomialapproximationsof