# Radius of convergence for $\sum (xn)^n/n!$ without Stirling's approximation.

This problem is from a calculus competition.

What is the radius of convergence of $\sum_{n=1}^{\infty} \frac{(xn)^n}{n!}$?

I know the answer is 1/e, but the solution uses Stirling's approximation which I am not familiar with. Is there any way to do this using the ratio test?

• How about the ratio test? – zhw. Jun 28 '17 at 1:43