How to find the value $\sum_{n\geq2}^{\infty}(-1)^{n+1}\frac{n}{n^2-1}$ converges to? How to find the value this sum converges to?$$\sum_{n\geq2}^{\infty}(-1)^{n+1}\frac{n}{n^2-1} $$
I've tried writing it like this
$$\sum_{n\geq2}^{\infty}(-1)^{n+1}·n·\Bigg(\frac{1/2}{n-1}-\frac{1/2}{n+1}\Bigg) $$
and writing a few terms, but they won't cancel and I ended up with no ideas, any hint?
I haven't learnt integration nor differentiation FYI.

Note that$$(-1)^{n+1}\frac n{n^2-1}=\frac12\times\frac{(-1)^{n+1}}{n-1}+\frac12\times\frac{(-1)^{n+1}}{n+1}.$$But$$\sum_{n=2}^\infty\frac{(-1)^{n+1}}{n-1}=\sum_{n=1}^\infty\frac{(-1)^n}n=-\log(2)$$and$$\sum_{n=2}^\infty\frac{(-1)^{n+1}}{n+1}=\sum_{n=3}^\infty\frac{(-1)^n}n=\left(\sum_{n=1}^\infty\frac{(-1)^n}n\right)+1-\frac12=-\log(2)+\frac12.$$
