Power series: find the radius of convergence, R, of the series

Question

n=1 (8)^n*(x)^n / (n)^5

R=

find the interval, I, of convergence of the series ( use interval notation )

I=

Answer 1

∑₁^∞(8ⁿ·xⁿ)/n⁵.

To get radius of convergence we using limit ratio test: lim n→∞ of abs(u_n+1/un) = lim n→∞ of abs(8ⁿ⁺¹xⁿ⁺¹/(n+1)⁵·(n⁵/8ⁿxⁿ) = lim n→∞ of 8(n/(n+1))⁵abs(x) = 8abs(x) <1. From here abs(x) < 1/8, so

-1/8< x < 1/8 and radius of convergense R = 1/8