Monotonicity of $\frac{n}{\sqrt[n]{(n!)}}$ It is known that when $n\rightarrow\infty$ the sequence $$\frac{n}{\sqrt[n]{(n!)}}$$ has limit $e$ but I don't know how to prove it's monotonicity. After a short calculus using WolframAlpha I found that this sequence is actually increasing. I tried to compare $2$ consecutive members but I couldn't manage to show something. It is obvious that $$\frac{n^n}{n!}$$ is increasing but that don't help us much (I think)

Taking $\log$ gives $$\sum_{i=1}^n\frac{-\log \frac i n}{n}$$
This is a Riemann sum of the integral $\int_{0}^1(-\log x)\,dx$.
Since the function $-\log x$ is decreasing and convex, its Riemann sums have to increase monotonically as the partition gets finer, if the partitions are regular. (Proof here).

