Want to test the series for convergence/divergence $\sum_n \frac{\sqrt{n+1} - 1}{(n+2)^3 - 1} $ Want to test the series $$\frac{\sqrt{2} - 1}{3^3 - 1} + \frac{\sqrt{3} - 1}{4^3 - 1} + \frac{\sqrt{4} - 1}{5^3 - 1} + \cdots$$ for convergence/divergence.
My attempt is
$U_n =\frac{\sqrt{n+1} - 1}{(n+2)^3 - 1} $ Now I wish to find $V_n $ which is less than (or greater than $U_n $) such that its convergence/divergence is known  so that I can find the behaviour of $U_n $

Hint: $\sqrt{n+1}-1\sim\sqrt n$ and $(n+2)^3-1\sim n^3$ (when $n\to +\infty$). Since the series have non-negative terms, we just have to deal with the convergence of $\sum\limits_{n\geq 1}\frac{\sqrt n}{n^3}$.
