TITLE: evaluation of limit $\lim_{n\rightarrow \infty}\left(\frac{n!}{n^n}\right)^{\frac{1}{n}}$
QUESTION [9 upvotes]: Calculate the value of the limit
$$
\lim_{n\rightarrow \infty}\left(\frac{n!}{n^n}\right)^{1/n}
$$

Can we solve this without using a Riemann sum method? If so, how?

REPLY [6 votes]: We will use the result 

$$ \lim_{n \to \infty} a_n^{1/n} = \lim_{n\to \infty}\frac{a_{n+1}}{a_n} .$$

Let
$$ a_n=\frac{n!}{n^n} \implies \frac{a_{n+1}}{a_n}=\frac{(n+1)!n^n}{(n+1)^{n+1}n!} = \left(\frac{n}{n+1}\right)^n    $$
$$ \implies \lim_{n\to \infty}\frac{a_{n+1}}{a_n}= \lim_{n\to \infty }\frac{1}{(1+1/n)^n} =e^{-1}. $$