Answer
Convergent
Work Step by Step
By Root Test, we have
$\lim_{n\to\infty}\sqrt[n]{\biggl| \biggl(\frac{n^2}{(n+1)^3}\biggr)^n\biggr|}=\lim_{n\to\infty}\biggl|\frac{n^2}{(n^3+3n^2+3n+1)}\biggr|=0 < 1$
Since the limit is $ < 1$, the series is convergent