TITLE: Does the series $\sum n!/n^n$ converge or diverge?
QUESTION [13 upvotes]: so I used the root test, but i'm not quite sure if i'm allowed to. I think im performing the operations correctly,a and i keep ending up with $(1)^{\infty}$. So really my question is am i performing the operations wrong or do i have to use a different test?

REPLY [18 votes]: $$\frac{n!}{n^n}=\frac{1}{n}\cdot\frac{2}{n}\cdot\frac{3}{n}\cdot ... \cdot \frac{n}{n}\leq 
\frac{1}{n}\cdot\frac{2}{n}\cdot1\cdot ... \cdot 1= \frac{2}{n^2}$$