Question

1. Consider the power sets Consider the w -1) 2 5"n 1 (+1)". Determine the interval of convence 2. Given that = " for IE(-1,1), write each of the following as a power series centered at To 0 and indicate the open interval on which the power series converge 4 + 2. (4 + 2 )

Answer 1

00 n =0 (1²4) gennal feom ga power series Cm (-a)" for the giun dentes cuising (947)" (n = (ighsn (121) >comparting with general a = -1 ) form To find the racias a conurgence, R use the natio test by ratio test racios y comurgence R is given bi R = tem n> 0 n/ Cnti . for the green senes (ann no Th²H) (ight 5 (141) (htt1 )

R- lim (1) x 50 x ((1+1) + 1) / insa (nu) (-1)"" s(14) - m o vie son camps 41) | (n*+)9%2C1)'e mht!) - im In C+080) - n°(*) (*) no - lim h00 1 Km (14) (145) | a ... on ten val o convergence shich is given as (a-R, atk) (= (-1-1, -1+1) = (-2,0) (14)

20 bgium take no (xe (-1,1)) OL 4+2x 4( 1+ 2x) 4 (172) given lala 1 <1 .. The guen denies can be written as. tx which is oqual to ☆ 1- n=0 n=d mm data). I drile) de o dre 4+28) (4+2x)2 - -2 (4+2x) 2

but we already showed n = 0 n =0 + cy)" nakane) n=0 20 2x4 n=0 an or - 1 ) A. n. 36-1) & n=0 an on 2 (-1)+n.gen-1) n = 0 -1 " (+3)