limits - Is there a standard way to compute $limlimits_{ntoinfty}(frac{n!}{n^n})^{1/n}$?

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.)

Answer

There are two formulas to compute radius of convergence of the series $\sum\limits_{n=1}^\infty{c_n}z^n$
$$
\frac{1}{R}=\lim\limits_{n\to\infty}|c_n|^{1/n}=\lim\limits_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|.
$$
Use the second one.