Factorial-over-power series
= Factorial-over-power series
For $a_n=n!/n^n$,
$$
\frac{a_{n+1}}{a_n}=\left(\frac n{n+1}\right)^n\to e^{-1}.
$$
Thus $\sum n!/n^n$ converges and $n/(n!)^{1/n}\to e$.
= Factorial-over-power series
For $a_n=n!/n^n$,
$$
\frac{a_{n+1}}{a_n}=\left(\frac n{n+1}\right)^n\to e^{-1}.
$$
Thus $\sum n!/n^n$ converges and $n/(n!)^{1/n}\to e$.