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 \[ \frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}-\frac{1}{6^2}+\cdots ? \]

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