Question 92
Prove that for all positive integers $n \ge 2$ we have
$$
\frac{1}{2} + \sqrt{\frac{1}{2}} + \sqrt[3]{\frac{2}{3}} + \dots + \sqrt[n]{\frac{n-1}{n}} \lt \frac{n^2}{n+1}.
$$
$$
\frac{1}{2} + \sqrt{\frac{1}{2}} + \sqrt[3]{\frac{2}{3}} + \dots + \sqrt[n]{\frac{n-1}{n}} \lt \frac{n^2}{n+1}.
$$