Prove that $n^n\left(\frac{n+1}{2}\right)^{2n}\geq (n!)^3$ for natural number $n$

Question

Prove that

$$n^n\left(\frac{n+1}{2}\right)^{2n}\geq (n!)^3$$ for a natural number $n$.

what i try

i have use AM GM inequality

$$\frac{1^3+2^3+3^3+\cdots +n^3}{n}\geq ((n!))^{\frac{1}{3n}}$$

how i prove question inequality help me please

Answer 1

First of all, by expanding the terms we obtain$$n^n\bigg(\frac{n+1}{2}\bigg)^{2n}{={1\over n^n}\cdot \bigg(\frac{n(n+1)}{2}\bigg)^{2n}\\={1\over n^n}\cdot (1+2^3+\cdots +n^3)^n.}$$By substitution and taking the $n$-th root, we need to prove that $${1+2^3+\cdots+n^3\over n}\ge \sqrt[n]{n!^3}=\sqrt[n]{1^3\cdot 2^3 \cdots n^3},$$which is straight-forward by AM-GM.

Answer 2

You can also use Stirling approximation and Taylor expansions to show that $$a_n=n \log (n)+2 n \log \left(\frac{n+1}{2}\right) - 3 \log(n!) >0$$ This would give $$a_n=\left(2-\frac{3}{2} \log \left({2 \pi }\right)\right)+ (3-\log (4))\,n-\frac{3}{2} \log \left({n}\right)-\frac{5}{4 n}+O\left(\frac{1}{n^2}\right)$$ which is positive $\forall\, n \geq 2$.