Prove that $n^{(n+1)} > (n+1)^{n}$ when $n\geq 3$ Firstly, for $n=3$,
$$3^4 > 4^3$$.
Secondly, $(n+1)^{(n+2)} > (n+2)^{(n+1)}$
Now I'm stuck. 

The inequality at stake is equivalent to $n > \left( 1+\frac{1}{n}\right)^n$.
Taking log yields $\frac{\ln n}{n} >\ln \left(1+\frac{1}{n} \right)$.
This is true since the concavity of $\log$ implies the stronger $\frac{1}{n} >\ln \left(1+\frac{1}{n} \right)$.
Using this argument, your inequality can be refined to $(n+1)^n<e^1n^n$
