Convergence of $\sum\limits_{n=1}^\infty \frac{n!}{n^n} \times (5x)^n$

Question

I have to check for which $x$ the series converges/diverges.

$\sum\limits_{n=1}^\infty\frac{n!}{n^n} \times (5x)^n$

I know that for $|x| < \frac{1}{5}e$ it converges and for $|x| > \frac{1}{5}e$ it diverges by using the ratio test $\frac{a_{n+1}}{a_n}$. However, this test does not tell me anything about $|x| = \frac{1}{5}e$.

How do I prove that the series diverges for $|x| = \frac{1}{5}e$ ? (Wolfram Alpha told me so)

I thought that it has something to do with $\frac{n!}{x^n}$ which would be $\frac{1}{e^x}$, but I just failed to find a proper proof.

Any help would be appreciated!

Answer 1

$$x=\frac e5\implies\;\text{we have the series}\;\;\sum_{n=1}^\infty\frac{n!5^ne^n}{n^n5^n}=\sum_{n=1}\frac{n!e^n}{n^n}$$

and now you can use Stirling's Approximation

$$n!\sim\frac{n^n}{e^n}\sqrt{2\pi n}$$

so our series behaves asimptotically (for large values of $\;n\;$) as the series

$$\frac{n^n}{e^n}\sqrt{2\pi n}\frac{e^n}{n^n}=\sqrt{2\pi n}$$

and thus clearly our series diverges.