$ 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+.-1<x<1)-Y1-xn=0tn80t"1(n+1)2"xWUn+1limim(n+ 1)2"(n + 2)2n+1t"2n->0unonThen it converges when x<2801>when 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 .