TITLE: Proving $\frac{1}{2}(n+1)<\frac{n^{n+\frac{1}{2}}}{e^{n-1}}$ without induction
QUESTION [5 upvotes]: I want to show that $\displaystyle\frac{1}{2}(n+1)<\frac{n^{n+\frac{1}{2}}}{e^{n-1}}$. But except induction, I do not know how I could prove this?

REPLY [3 votes]: If $n\ge3$, $\frac12(n+1)<n$ and $n\ge\mathrm{e}$ hence $n^{n-1}\ge\mathrm{e}^{n-1}$ and it is enough to prove that $n<n^{3/2}$, which is obviously true. Then you can check manually the cases $n=1$ (for which the strict inequality is false, by the way) and $n=2$.