TITLE: Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$
QUESTION [6 upvotes]: I am trying to calculate the following limit without Stirling's relation. 
\begin{equation}
\lim_{n \to \infty} \dfrac{n!}{n^n}
\end{equation}
I tried every trick I know but nothing works. Thank you very much.

REPLY [3 votes]: Consider the series $$\sum_{n=1}^\infty \frac{n!}{n^n} $$ of positive terms. The ratio of two consecutive terms is $$\frac{a_{n+1}}{a_n}=\frac{(n+1)!/(n+1)^{n+1}}{n!/n^n}= \left( \frac{n}{n+1} \right)^n=\left[ \left(1+\frac{1}{n} \right)^n \right]^{-1} $$ which tends to $e^{-1}<1$. It follows from the ratio test that the series converges, and by the necessary condition for convergence of series the limit is obtained. We have $$\lim_{n \to \infty} \frac{n!}{n^n}=0. $$