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Exercise 1. Find the radius of convergence of each of the following power series. 1. Š«=", . (Hint: Think of writing w = 2², and write this first as a power series in w.)

Answer 1

lesing rat tot, o homem Votzen 121 - 121 lim nn 1 : 121:02 Inco = Izl = the series converges if 121<1. Therefore the radius of convergence is R = L. 2) Ž 2 h = Ě wh ² = w using root test, f = lim ~ win no 4" Y = IWI 4. = the series converges if Iwl< 4 ie 12²1<4 ie 121² <4 ie 121<2 Therefore the radius of convergence is R = 2 0 2 1 2” - using root test, r= I'm 1n! 2²1 lim (!) 121 – Jaco , since lim (n!)""= - the radius of convergence is R=0 n=0 n700 V co noo

Using root test, r= lima 1 nos 121 700 V 13-41 = 121 - - = the the series converges if radius of convergence 12144. is R=4.