Studying convergence/divergence of the series $\sum\limits_n\frac{(a+nx)^n}{n!}$

Question

Study the convergence/divergence of $$\sum_{n=1}^{\infty}\frac{(a+nx)^n}{n!}$$

So far, I've managed to break it down to: $$U_n=\frac{(a+nx)^n}{n!}$$

But I can't judge a way how to proceed. Please help.

Answer 1

Hint: $a$ is unimportant (why?). Now try to use the limit comparison test and Stirling's Approximation.