Hyperbolic Cotangents And Sums Of Squares

Calculus Level 5

It can be shown that \[ \sum_{n=1}^\infty \frac{\coth^2 \pi n}{n^2} \; = \; \frac{P}{Q}G + \frac{R}{S}\pi^T \] where \[G \;=\; \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^2} \;=\; 0.915965594\ldots \] is Catalan's constant, and where \(P,Q,R,S\) and \(T\) are all positive integers with \(P,Q\) and \(R,S\) coprime pairs. Find \(P+Q+R+S+T\).


Inspiration.

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