What kind of algebraic manipulations are required to show the following?
$ \frac2{1^2}-\frac2{2^2}+\frac2{3^2}-\frac2{4^2}+\cdots=\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots. $
The LHS has alternating signs and is twice the RHS, and the series converges to $\pi^2/6$.