# Infinite Poly-summation

Calculus Level 5

$\large S = \sum_{k=1}^\infty \left ( \dfrac1{k^2} {\psi^{(1)} \left( \frac{k+1}2\right) } \right)$

Let $$\psi^{(1)} (\cdot)$$ denote the Trigamma function, $$\displaystyle \psi^{(1)}(z) = \sum_{n=0}^\infty \dfrac1{(z+n)^2}$$.

If $$S$$ can be expressed as $$\dfrac{\pi^A}B$$, where $$A$$ and $$B$$ are integers, find $$A+B$$.

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