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1、9.6 Power SeriesProblem IntroductionWhat is power series?What are the characteristics and properties of power series?Definition of Power SeriesDefinition:Power SeriesA power series in has the form 20120.nnna xaa xa x=+We call the set on which a power series converges its convergence set.Example 1For
2、 what s does the power seriesconverge and what is the sum?Assume that .230nnaxaaxaxax=+It is a geometric series.It converges for ,and has sum given by(),111aS xxx=0a 11x()S xExample 2Find the convergence set for .0!nnxn=()1!lim1!nnnxnnx+=+lim1nxn=+0=By the Absolute Ratio Test,the series converges fo
3、r all.Theorem A:Abel THdiverges at ,0nnna x=0 xx=(2)If0|xxthendiverge.all x0|xxthenall x converge.converges at ，00(0)xx x=(1)If0nnna x=converge on the whole number line,there must be a fully determined positive R,which has the following properties:Deduction of Abel THthe power series converges absol
4、utely;If|,xRIf|,xRthe power series diverges;the power seriesIf or,xRxR=may converges or diverges.The power seriesnnnxa =1converges at x=0,and doesnt The convergence set for a power series is always an interval of one of the following three types:(i)The single point .(ii)An interval ,plus possibly so
5、me end points.(iii)The whole real line.In(i),(ii),and(iii),the series is said to have radius ofconvergence 0,and,respectively.nna x0 x=(,)R RTheorem BDefinition of Geometric ExplainOx Geometric explain:R RConvergent areaDivergent areaDivergent area(1)Positive R is called radius of convergence of the
6、 power series.Domain of convergence of the power series is called interval of convergence.),RR,(RR.,RR),(RR Definition of Geometric ExplainProvisions:Question:How to find the radius of convergence of the power series?(2)The power series converges only at x=00,R=interval of convergence:0;x=(3)The pow
7、er series converges for all x,R=+interval of convergence:(,).+Theorem C(1)If 0,1;R=(2)If 0,=;R=+(3)If,=+0.R=Let 0nnna x=If the power series is,and0.na 1limnnnaa+=()or limnnna=A power series converges absolutely on the interior of its interval of convergence.0nnna x=Example 3Find radius of convergenc
8、e and the convergence set of the following power series:121(2)(1)()2nnnnxn=1(1)2nnnxn=Example 31(1)2nnnxn=112(1)lim12nnnnn+=2R=12=1limnnnaa+=lim2(1)nnn=+1R=If 2,x=If 2,x=1(1),nnn=11,nn=Alternating series,converges.Harmonic series,diverges.So,the convergent set is 2,2).Example 3121(2)(1)()2nnnnxn=212
9、1|=xt(0,1)xIf 0,x=11nn=If 1,x=1(1)nnn=Diverges.Converges.So,the convergence set isR=12=1Let,2tx=12(1)nnnntn=Other ways?1limnnnaa+=1lim2nnn+(0,1.That is,Converges.That is,Converges210ln(1).(21)!nnnxxn+=+Example 4it is a power series1()lim()nnnuxux+=Get rid of the first term,2321|(21)!lim(23)!|nnnxnnx
10、+=+2|lim(22)(23)nxnn=+So,Get rid of the first term,the series converges everywhere.0=As the domain of definition of ln is so,the convergence set of the given series is0,x(0,).+Ratio TestFind the convergence set for Example 5Find the convergence set for 210(1)31nnnnx+=This series is a power series wi
11、th missing terms.Let 2,yx=that is,10(1).31nnnny+=As 1131lim 131nynnR+=3If y=3,that is,103(1),31nnnn+=and3lim 31nnnso,this series diverges.It doesnt satisfy the conditions of Theorem B.10,=Example 5Find the convergence set for210(1)31nnnnx+=The convergence set of the given series y 0 isSo,the converg
12、ence set of the given series is33.xRadius of convergence:3.R=That is,03.y203,xSummaryDefinition of Power SeriesAbel TH and DeductionDefinition of Geometric explainQuestions and AnswersFind the convergence set for()()2011nnxn=+()()()()12211lim21nnnxnnx+=+()()221lim12nnxn+=+By the Absolute Ratio Test,It converges if ,that is,;1x=It diverges if .It converges at and .|1|1x|1|1x02x0 x=2x=0,2Power Series