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
$ 12.3 Power SeriesII.Power Series2. Convergence Set1.The convergenceset of a power seriesis not null80(-1)"x" x = O is always a convergence point.E.G.n=02. The convergence set of a power series may be the real set .sinnxsinnx1ZE.G.<n=0
§12.3 Power Series II. Power Series 2. Convergence Set 1. The convergence set of a power series is not null. n n n x = − 0 ( 1) x = 0 is always a convergence point. 2. The convergence set of a power series may be the real set . =0 2 sin n n nx 2 2 sin 1 n n nx E.G. E.G
$ 12.3 Power SeriesIl.Power Series2. Convergence Set:Wx"=1+x+x2+.-10unonThen it converges when xwhen x = 2,diverges.(n+ 1)108(-1)"Nwhen x = -2,converges.(n+1)0
§12.3 Power Series II. Power Series 2. Convergence Set = + + ++ + = n n n x x x x 2 0 1 ( 1 1) 1 1 − − = x x n=0 ( + 1)2 n n n x n n n u u 1 lim + → n n n n n x n n x ( 1)2 ( 2)2 lim 1 1 + + = + + → 2 x = Then it converges when |x|<2 diverges. ( 1) 1 when 2, 0 = + = n n x converges. ( 1) ( 1) when 2, 0 = + − = − n n n x
$ 12.3 Power SeriesI.Power Series2. Convergence SetTheoreml80The convergence setfora,x" is always an interval of one ofn=0the following three types :(i) The single point x=0.(i) The whole real line.(iii) An interval(-R,R), plus possibly one or both end pointsIn (i), (ii) and (iii), the series is said to have radius ofconvergence 0,R,and c0
§12.3 Power Series II. Power Series 2. Convergence Set the following three types: The convergence set for is always an interval of one of 0 n= n an x Theorem1 (i) The single point x=0. (ii) The whole real line. (iii) An interval (-R,R), plus possibly one or both end points. In (i), (ii) and (iii), the series is said to have radius of convergence 0,R,and .
S 12.3 Power SeriesIL.Power Series2.Convergence SetTheoreml80+The convergencesetfora,x" is always an interval of one ofn=0the following three types :The theorem means that if a power series converges at x,it converges absolutely for all x such that x < x .on the other hand, if a power series diverges at x,, itdiverges absolutely for all x such that x ≥ |x2 l
§12.3 Power Series 2. Convergence Set the following three types: The convergence set for is always an interval of one of 0 n= n an x Theorem1 on the other hand, if a power series diverges at , it diverges absolutely for all x such that 2 x . x x2 The theorem means that if a power series converges at , it converges absolutely for all x such that 1 x . x x1 II. Power Series
$ 12.3 Power SeriesIl.Power Series2. Convergence SetHow to find the radius R ?82anxn+1n+1limimn->0n8n-80uan=01when px 1, that is,the seriesdivergesp- is the radius.SO.p
§12.3 Power Series II. Power Series n n n a x =0 2. Convergence Set How to find the radius R ? n n n u u 1 lim + → n n n n n a x a x 1 1 lim + + → = x x a a n n n = = + → 1 lim , the seriesconverges. 1 when 1,that is, x x , the seriesdiverges. 1 when 1,that is, x x is the radius. 1 so,