How to prove this inequality with factorials: $n!>n^{\frac {n}{2}}$

Question

This is what I am trying to solve:

$n!>n^{\frac {n}{2}}$

I tried with induction and somehow it doesn`t work this way, I tried with logarithms and also didn´t find a way. Is there an elementary(if possible) way to prove this?

Or is it the case that for some large $n_0$ this inequality reverses sign?

Answer 1

For $n$ even, you get $n! = (n/2)! \cdot B,$ where $B$ is the product of the $(n/2)$ numbers from $1 +(n/2)$ to $n.$ So, $$ B > (n/2)^{(n/2)} = \frac{n^{n/2}}{2^{n/2}}. $$ Then you need $$(n/2)! > 2^{n/2} $$ which is not difficult by induction.

Mild revisions for odd $n$

Answer 2

$(n!)^2=\{1.2.3......n\}\{n.(n-1)......3.2.1\}=\{(1.n)(2.(n-1))........(n.1)\} \geq n.n.n.....n (\text{n}\space times) =n^n \implies n! \geq (n)^{n/2}$.