TITLE: Is $\sum\limits_{n=1}^{\infty}\frac{n!}{n^n}$ convergent?
QUESTION [8 upvotes]: I can clearly see that $\dfrac{n!}{n^n}\to 0$ when $n\to\infty$. But how do I know if the sum $$\sum_{n=1}^{\infty}\frac{n!}{n^n}$$ is convergent or not? I know this might be basic, but thank you if anyone can help me.

REPLY [4 votes]: Hint: $n!/n^n = (n/n) ((n-1)/n) ((n-2)/n) \cdots (2/n) (1/n)$.  Each term in the product is less than or equal to one.  So the whole thing is at most $2/n^2$ (for $n \geq 2$).