TITLE: Is there a standard way to compute $\lim\limits_{n\to\infty}(\frac{n!}{n^n})^{1/n}$?
QUESTION [8 upvotes]: I'm computing the radii of convergence for some complex power series. For one I need to compute
$$\lim_{n\to\infty}\left(\frac{n!}{n^n}\right)^{1/n}.$$
I know the answer is $\frac{1}{e}$, so the radius is $e$. But how could you compute this by hand? I tried taking the logarithms and raising $e$ by this logarithm, but it didn't lead me to the correct limit. (This is just practice, not homework.)

REPLY [2 votes]: Stirling's formula for $n!$ works wonders here, but I guess that may not qualify as "computing by hand"?
Just for the sake of completeness, Stirling's formula states that
$$n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$
or:
$$\displaystyle\lim_{n\to\infty} \frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n} = 1$$
Substituting the RHS from the first equation and taking the limit as $n \to \infty$ pretty much yields the solution instantly.