Test The Convergence

the expression above is simply equal to:

$\sum_{n = 2}^{\infty} \cfrac{(n-1)^2}{(n^2)(2n-1)} = \sum_{n = 2}^{\infty} \cfrac{1}{(2n- 1)} - \sum_{n = 2}^{\infty} \cfrac{1}{(n^2)} \\ = \sum_{n=2}^N \cfrac{1}{n} - \sum_{n=2}^{\left \lfloor \cfrac{N}{2} \right \rfloor} \cfrac{1}{2n} - \sum_{n = 2}^{\infty} \cfrac{1}{(n^2)}$

But as $\displaystyle \sum_{n = 2}^{\infty} \cfrac{1}{(n)}$ diverge, hence, $\displaystyle \sum_{n = 2}^{\infty} \cfrac{(n-1)^2}{(n^2)(n+1)}$ also diverge.