Why is this identity true? $$\sum_{k=1}^{\infty} \frac{\sin(k\pi/2)}{k^2} = G$$ where $G$ is Catalan's constant.

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    $\begingroup$ Hint: What is the definition of the Catalan constant, and what does the sequence of numbers $\sin(k\pi / 2)$ look like? $\endgroup$ – John Barber Dec 29 '15 at 20:49

The series sums to $$\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\cdots$$ which is just the definition of Catalan's constant.


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