Question 10
Prove that, for every integer $n \geq 2$, we have:
$$
\sum_{k=2}^{n} \frac{1}{\sqrt[k]{(2 k)!}} \geq \frac{n-1}{2 n+2}
$$
N.B. If $a\gt 0$, we denote by $\sqrt[k]{a}$ the unique real number $b\gt 0$ such that $b^{k}=a$.
$$
\sum_{k=2}^{n} \frac{1}{\sqrt[k]{(2 k)!}} \geq \frac{n-1}{2 n+2}
$$
N.B. If $a\gt 0$, we denote by $\sqrt[k]{a}$ the unique real number $b\gt 0$ such that $b^{k}=a$.