Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$ 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.
 A: Use Stirling's approximation:
$    n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n $
and you'll get
$$
\lim_{n \rightarrow \infty} \frac{n}{(n!)^{1/n}}
=\lim_{n \rightarrow \infty} \frac{n}{(\sqrt{2 \pi n} \left(\frac{n}{e}\right)^n)^{1/n}}
=\lim_{n \rightarrow \infty} \frac{n}{({2 \pi n})^{1/2n} \left(\frac{n}{e}\right)}
=\lim_{n \rightarrow \infty} \frac{e}{({2 \pi n})^{1/2n} }=e,
$$
because $\lim_{n\to \infty} ({2 \pi n})^{1/2n}= \lim_{n\to \infty} n^{1/n}=1$.
A: 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 
$$
\lim_{n\to\infty}\frac{n}{(n!)^{1/n}}=e.
$$
A: Alternatively, you could use the fact that for a sequence $(a_n)$ of positive terms, if $\lim\limits_{n\rightarrow\infty} {a_{n+1}\over a_n}$ exists, then so does $\lim\limits_{n\rightarrow\infty}\root n\of{a_n}$ and the two limits are equal.
For your problem, consider $a_n={n^n\over n!}$. Then 
$${a_{n+1}\over a_n}={(n+1)^{n+1}\over (n+1)!}\cdot {n!\over n^n}= {1\over n+1}\cdot\Bigl({n+1\over n}\Bigr)^n\cdot(n+1)=\Bigl(1+{1\over n}\Bigr)^n
\ \ \buildrel{n\rightarrow\infty}\over\longrightarrow\ \  e.
$$
Thus $\lim\limits_{n\rightarrow\infty}{a_{n+1}\over a_n}=e$. As $\root n\of {a_n}={n\over(n!)^{1/n}}$, we have  $\lim\limits_{n\rightarrow\infty}{n\over(n!)^{1/n}}=e$ as well.
