I'm sure this has been asked a million times, but it's hard to google for a particular series without knowing its name.
$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$
I know this converges absolutely to $\frac{\pi^2}{6}$ and I know that it is absolutely convergent so that the terms can be rearranged.
So the sum is equal to $-1 + \sum_{n=2}^\infty \frac{1}{n^2} - \frac{1}{(n+1)^2} = -1 + \sum_{n=2}^\infty \frac{2n + 1}{n^2(n+1)^2}$. Which got me nowhere. Is it a clever rearrangment we're looking for here, or is there another tool to be used?