Evaluate if series with exponential diverges or converges

Question

The task is to evaluate for what values of $a \in \Bbb R_+$ does the series $$\sum_{n=1}^\infty \frac{a^n \times n!}{n^n}$$ converge. I've already checked with the ratio test that it converges for $ a < e $ and diverges for $a > e$, but I can't seem to find the answer what happens for $a = e$: $$\sum_{n=1}^\infty \frac{e^n \times n!}{n^n}$$

What test should I apply?

Answer 1

By using the Stirling formula, as $n \to \infty$, $$ n!\sim \sqrt{2\pi n}\frac{n^{n}}{e^{n}} $$ one gets that, as $n \to \infty$,

$$ \frac{e^n \times n!}{n^n}\sim \sqrt{2\pi n} \to +\infty \neq0 $$

and the initial series is divergent.