Convergence of the series $\sum_{n=1}^{\infty}\frac{(1+nx)^n}{n!}$

Question

I'm trying to find the radius of convergence of the following series. But I'm not having success in finding it.

Convergence of the series : $$\sum_{n=1}^{\infty}\frac{(1+nx)^n}{n!}$$

Thank you.

Answer 1

By the root test, you require $$ \lim_{n\to \infty}\frac{|1+nx|}{(n!)^{1/n}}<1 $$ but by Stirling's, $$ n!\sim n^{n+1/2}e^{-n}\implies (n!)^{1/n}\sim n^{1+1/2n}e^{-1}\sim n/e $$ and so your requirement is $$ \lim_{n\to \infty}\frac{|1+nx|}{n}e<1\implies |x|<1/e $$