Investigate the convergence of $ \sum_{n=1}^{\infty } (-1)^{n}\frac{n+2}{n(n+1)} $ I am supposed to investigate the convergence of $ \sum_{n=1}^{\infty } (-1)^{n}\frac{n+2}{n(n+1)} $. I'm unsure whether to use Leibniz' criterion or a comparison test and I really can't start. Thanks

You can directly use Leibniz' criterion, but you have to show the absolute value of the general term is non-increasing. Here, I suggest an (arguably) simpler way to see what's happening and prove convergence:
You have
$$
\frac{n+2}{n(n+1)} = \frac{1}{n+1} + \frac{2}{n(n+1)}
$$
and therefore
$$
\sum_{n=1}^\infty (-1)^n \frac{n+2}{n(n+1)} = \sum_{n=1}^\infty  \frac{(-1)^n}{n+1} + 2\sum_{n=1}^\infty \frac{(-1)^n}{n(n+1)}
$$
The first series on the right-hand-side converges conditionally by Leibniz's criterion, the second converges absolutely.
