Mathematics for the Liberal Arts

This problem is from ``Mathematics for the Liberal Arts'' (Wiley, 2012).

Leonhard Euler (1707-1783) proved the formula $${1\over 1^2} + {1\over 2^2} + {1\over 3^2} + \cdots =\frac{\pi^2}{6}.$$ Can you use Euler's formula to find the exact value of $$\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}-\frac{1}{6^2}+\cdots ?$$

My math page: https://sites.google.com/site/martinerickson/