12ChapterInfiniteSeriesSec.3Power Series$ 10.6$ 10.7
§10.6 §10.7 Chapter 12 Infinite Series Sec.3 Power Series
S 12.3 Power SeriesI.SeriesofFunctions1.Definition:Supposeu, (x),u,(x), ...,u, (x), ...are functions defined8on I R, then u,(x) = u,(x)+u,(x)+...+u,(x)+..n=1is called infinite series of functions defined on domainl.80sinxsin2xsin3xsinnxZe.g.194nn=1Two questions:1. For what x, does the power series converge?2. To what function does it converges?
§12.3 Power Series I. Series of Functions 1.Definition: Suppose u1 (x),u2 (x), ,un (x), are functions defined on I R, then = + ++ + = ( ) ( ) ( ) ( ) 1 2 1 u x u x u x un x n n is called infinite series of functions defined on domainI. = + + + = 9 sin3 4 sin2 1 sin sin 1 2 x x x n nx n e.g. Two questions: 1. For what x, does the power series converge? 2. To what function does it converges?
$ 12.3PowerSeriesLSeriesofFunctions2. The Convergence Point and Convergence Set8If Xo E I such that seriesZu,(xo) converges ,n=18Ziun(x),Then x, is called the convergence point ofn=18Zu,(x) converges itsWe call the set on whichn=lconvergence set.Similar to the definition of divergence point and divergenceset
§12.3 Power Series 2. The Convergence Point and Convergence Set If x I 0 such that series =1 0 ( ) n un x converges , Then 0 x is called the convergence point of ( ) 1 u x n n = , We call the set on which ( ) 1 u x n n = converges its convergence set. Similar to the definition of divergence point and divergence set. I. Series of Functions
S 12.3 Power SeriesLSeriesof Functions3.Function of Sum:For one x in convergence set,s(x)=u(x)+u2(x)+...+un(x)+..If the partial sum is s, (x),lim s,(x) = s(x)n-oThe remainder function is,rn(x) = s(x)- s,(x)lim r,(x) = 0Obviously,n→8
§12.3 Power Series I. Series of Functions lim s (x) s(x) n n = → The remainder function is, r (x) s(x) s (x) n = − n s(x) = u1 (x) + u2 (x) ++ un (x) + If the partial sum is s (x), n For one x in convergence set, 3.Function of Sum: lim ( ) = 0 → r x n n Obviously
$12.3Power SeriesIL.PowerSeries1.Definition:80Za,x" =a, +ax+a,x? +...+a,x" +..n=08Za,(x- )"= , + a(x-- xo)+ ,(x-xo)*..n=0a, is the coefficient of the power series
§12.3 Power Series II. Power Series 1.Definition: is the coefficient of the power series. an = + + ++ + = n n n n an x a a x a x a x 2 0 1 2 0 − = + − ++ − + = n n n n an (x x ) a a (x x ) a (x x ) 0 0 1 0 0 0