TITLE: What test can be used to show $\sum \dfrac{n!}{n^n}$ converges?
QUESTION [15 upvotes]: Does $\sum \dfrac{n!}{n^n}$ converge? Intuitively I think it converges but I am not sure what test can be used to show that.

REPLY [9 votes]: $$\frac{n!}{n^n} = \frac{1}{n} . \frac{2}{n}  \dots \frac{n}{n} \leq \frac{1}{n} .\frac{2}{n} . 1 \times 1 \dots \times 1 = \frac{2}{n^2}. $$