# 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 $$P+Q+R+S+T$$.

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