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?

It is enough to show that $\frac{1}{n}\log(n!)=\frac{1}{n}\sum_{k=1}^{n}\log(k)$ is increasing.
Since $\log(x)$ is concave on $\mathbb{R}^+$, this is a simple consequence of Karamata's inequality:
$$ \tfrac{1}{n}\cdot\log(1)+\tfrac{1}{n}\cdot\log(2)+\ldots+\tfrac{1}{n}\cdot\log(n)+0\cdot \log(n+1)\\ \leq \tfrac{1}{n+1}\cdot\log(1)+\tfrac{1}{n+1}\cdot\log(2)+\ldots+\tfrac{1}{n+1}\log(n+1).$$

As an alternative approach, $n!^{n+1}\leq (n+1)!^n$ is equivalent to $n!\leq (n+1)^n$ which is trivial.
