Pinko wrote: Prove that for every positive integer $n \geq 2$ the following inequality holds: $e^{n-1}n!<n^{n+\frac{1}{2}}$ Use induction and the inequality $e < \left(1+\frac 1 n\right)^{n+1/2}$: $$e^n(n+1)!=(n+1)e\cdot e^{n-1}n! < (n+1)e n^{n+1/2} < (n+1)\left(1+\frac 1 n\right)^{n+1/2} n^{n+1/2}=(n+1)^{n+1+1/2}.$$