TITLE: Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$
QUESTION [6 upvotes]: I need to check if
$$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is appreciated. I had tried taking log and manipulating the sequence but I could not prove monotonicity this way.

REPLY [16 votes]: What you have is actually an indefinite integral in disguise. Let's
first consider the reciprocal of what you have:
\begin{eqnarray*}
\lim_{n\to\infty}\frac{(n!)^{1/n}}{n} & = & e^{{\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\ln\left(\frac{k}{n}\right)}}\\
 & = & e^{{\displaystyle \int_{0}^{1}\ln xdx}}\\
 & = & e^{-1}.
\end{eqnarray*}
Thus we get that 
$$
\lim_{n\to\infty}\frac{n}{(n!)^{1/n}}=e.
$$