Does the series $\sum n!/n^n$ converge or diverge?
Hint:
$$\frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{n^n}{n!}=\frac{1}{\left(1+\frac{1}{n}\right)^n}\xrightarrow[n\to\infty]{}\ldots$$
And yes: it converges.
$$\frac{n!}{n^n}=\frac{1}{n}\cdot\frac{2}{n}\cdot\frac{3}{n}\cdot ... \cdot \frac{n}{n}\leq \frac{1}{n}\cdot\frac{2}{n}\cdot1\cdot ... \cdot 1= \frac{2}{n^2}$$