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 1

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1、ABouncingBallDropaballfromHmetersaboveaflatsurface.Eachtimetheballhitsthesurfaceafterfallingadistanceh,itreboundsadistancerh,Lecture1whererispositivebutlessthan1.findthetotaldistancetheballtravelsupanddown.Itiseasytoseethatthetotaldistanceis23s=H++++222HrHrHr?Thequestionisthathowtocalculatethesumw

2、ithinfiniteterms.Itiswellknownthat,iftherehasfinitetermsSeriesofConstantTerms23ns=++H222HrHr+Hr++?2Hrnn2(Hr1)−r=+H1−rn2(Hr1l−im)r2HrlimsH=+n→∞=+H.Itisclearthatnn→∞11−rr−12RepeatingDecimalsInfiniteSeriesDefinitionThefollowingrepresentationExpresstherepeatingdecimal5.232323…astheratiooftwointegers.∞

3、aa12++++??anor∑ann=1Solutionwhichconsistsofallthetermsofasequence{a}andconnectedn5.232323...5=+23+++2323?successivelybytheplussign,iscalledaninfiniteseriesorsimplya23100(100)(100)series,andaniscalledthegeneraltermoftheseries.⎛⎞⎛⎞2Ifthetermsofthesequencearenumbersthentheseriesiscalleda23⎛⎞1⎛⎞1=+51⎜

4、⎟++⎜⎟+?seriesofconstantterms.100⎜⎟⎝⎠100⎝⎠100Ifthereislittlechanceofconfusion,itmaybecalledsimplyaseries.23⎛⎞123518Itisconvenienttousesigmanotationtowritetheseriesas=+55⎜⎟=+=.1000.99⎝⎠9999∞∞Ausefulshorthand∑an,∑ak,or∑anwhensummationfromn=1k=11to∞isunderstood34InfiniteSeriesInfiniteSeries∞1111Afinit

5、esumofrealnumbersalwaysproducesarealnumber,butan∑an=1+++++24816?n=1infinitesumofrealnumbersissomethingelseentirely.ThisiswhywePartialSumValueneedacarefuldefinitionofinfiniteseries.First:s1=121−Webeginbyaskinghowtoassignmeaningtoanexpressionlike111111Second:s=1+2−1+++++?22224816111Third:s=1++2−Thew

6、aytodosoisnottotrytoaddallthetermsatonce(wecannot)3424………butrathertoaddthetermsoneatatimefromthebeginningandlookforapatterninhowthese“partialsum”grow.1111nthsn=++++1?n−12−n−12422561PartialSumConvergenceandDivergenceDefinition(ConvergenceandDivergenceofaseries)∞DefinitionThesumofthefirstntermsofthe

7、seriesIfthesequenceofpartialsumsofaseries∑akhasalimitSasnk=1Saann=+++=12??a∑ank,(1,2,)=napproachinginfinite,wesaythattheseriesconvergestothesumS,k=1andwewriteiscalledthepartialsumoftheseries.∞aaa123+++++=??ank∑aS