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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