Advanced Calculus fi..

nL: Chapter 6 Infinite Series 429have nonzero convergence radii and have equal sums wherever both series converge,then the series are identical; that is,fcn=Cn, n=0,1,2 ,....Proof. Let the convergence radii be r* and R*, respectively, where 0 < r* 5 R*.Then, by assumption,By the preceding theorem, one must then have co = Co = f (a), andThus the coefficients are equal for every value of n.COROLLARY If a power series has a nonzero convergence radius and has a sumthat is identically zero, then every coefficient of the series is zero.PROBLEMS1. a) Obtain the Maclaurin series1 1 xnlog-=x+-x2+...+-+...1 -x 2 n, -1