Discuss the monotonicity of $\sqrt[n]{n!}$. It seems that $\sqrt[n]{n!}$ is increasing because it turns to $+\infty$ as $n\to+\infty$. How to prove it?

Hint:
$$
\begin{align}
\frac{n!^{n+1}}{(n+1)!^n}
&=\frac{n!}{(n+1)^n}\\
&=\frac1{n+1}\cdot\frac2{n+1}\cdot\frac3{n+1}\cdots\frac{n}{n+1}
\end{align}
$$

Another Hint: Notice that $n!^{\frac1n}$ is the geometric mean of first $n$ natural numbers. $(n+1)!^{\frac1{n+1}}$ is the geometric mean of the same set of natural numbers and one larger natural number.
