Power series: find the interval of convergence of the series

Question

n=1 8^nx^n/n^5

Radius = 1/8

please help me find the interval of convergence of the series

Answer 1

Use the Ratio Test

a_n = (8ⁿxⁿ) / n⁵

l a_n+1 / a_n l = l (8ⁿ⁺¹xⁿ⁺¹) / (n+1)⁵ [n⁵ / (8ⁿxⁿ) l = 8[(n/(n+1)]⁵ l x l

Taking the limit of the above expression as n goes to infinity, we get 8(1) l x l

By the Ratio Test, the series converges if 8 l x l < 1. So, l x l < 1/8. Therefore, the series converges for -1/8 < x < 1/8. At the endpoints of the interval, the limit is 1 So, the Ratio Test is inconclusive. At x = 1/8, the series is the p-series with p = 5, so the series converges there also. When x = -1/8, we get a convergent alternating series. So the interval of convergence is -1/8 ≤ x ≤ 1/8. The radius of convergence, 1/8, is half the length of the interval of convergence.