Question

(2 pts) Consider the power series Find the radius of convergence R. If it is infinite, type "infinity" or "inf Answer: Rinf What is the interval of convergence? Use interval notation for an interval. Type the number, if it is only a single point. Type NONE if there it is the empty set. Answer: (-inf inf)

I'm struggling with two simpler calc questions, and I would appreciate it if someone could walk me through the steps on these to show me where I'm going wrong. Thank you.

Answer 1

Sigma n = 1 infinity n^a (x-1)^n let an = x^n therfore radius of convergence, rho = 1/limn right arrow nsquarerootof module a_n = 1/limnright arrow infinity n = 0 . interval of convergence is right arrow [1 - 0, 1 + 0] = {1} \r\n therfore the power services is convergent only for x = 1 an = 2.4.6.....(2n)/1.3.5.....(2n - 1) therfore an + 1/a_n = 2.4.6....(2n + 2)/1.3.5....(2n - 1) times 1.3.5....(2n - 1)/2.4.6....2n 2n + 2/2n + 1 = (1 + 1 1 + 1/2n) therfore lim nright arrow infinity a_n + 1/a_n = lim xright arrow infinity 1 + 1 1 + 1/2n = 1. therfore radius of convergence , rho = 1/limx right arrow infinity (an + 1/an) = 1. Interval of convergence is (0 - 1, 0 + 1) = (- 1, 1)