Proof of an inequality involving $(N-1)!$

Question

How is it possible to prove the following inequality? $$\displaystyle\prod_{k=1}^N\left(\dfrac{k^3+k+1}{k^2(k+1)}\right)^\dfrac{1}{N}\ge\left(\frac{1}{N+1}\right)^\dfrac{1}{N}-\left(\dfrac{1}{N^2(N-1)!}\right)^\dfrac{1}{N}$$ I found the equal holds for $N\to\infty$, but I'm unable to verify analytically the inequality even if numerically it holds. Thanks.

Answer 1

It is true even in the stronger form: $$\prod_{k=1}^{N}\frac{k^3+k+1}{k^2(k+1)}>\frac{1}{N+1},\tag{1}$$ since: $$\frac{k^3+k+1}{k^2(k+1)}>\frac{k}{k+1},\tag{2}$$ so: $$\prod_{k=1}^{N}\frac{k^3+k+1}{k^2(k+1)}>\prod_{k=1}^{N}\frac{k}{k+1}=\frac{1}{N+1}.$$