LLL Determine for which z the following series converge 2m 2

Last updated: 5/21/2023

LLL Determine for which z the following series converge 2m 2 2 e f a 2 1 c k 1 d e b k 0 R lim k d We apply the ratio test for am 1 m thus am m xam 1 00 m 0 m 0 lim M 2 1 m 1 m 1 om 1 lim n m 2m 1 m m z i 1m 1 m m m n 3 n z 3 2 2 2 i lim m x hence the radius of convergence is R 1 Therefore the series converges for all satisfying 2 i 1 and diverges for all z satisfying 2 i 1 Now consider z such that z i 1 then 1 2 m 1 m 1 Since 11 m converges the power series converges for such z also Hence the given series converges for all satisfying 12 il 1