Convergence of the function series $\sum \frac{n!}{(nx)^n}$ for $x<0$
Do you know Stirling's approximation to the factorial?
It says that $$ n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n\text{ as }n\to\infty. $$ As such, $$ \frac{n!}{n^n}\sim\frac{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}{n^n}=\frac{\sqrt{2\pi n}}{e^n}. $$ So, you need $x^n\to0$ faster than $e^n\to\infty$, which is where the result comes from.