# Summing up all digamma roots

Calculus Level 5

$\Large \sum_{x \in \Psi} \frac{1}{x(x-2)}=\dfrac{1-\frac{\gamma^A}{B}-\frac{\pi^C}{D}}{1-\gamma}$

Let $$\Psi$$ is the set consisting of all roots of the digamma function.

Given that $$A,B,C$$ and $$D$$ are positive integers satisfying the equation above, find $$A+B+C+D$$.

Notation: $$\gamma$$ denotes the Euler-Mascheroni constant, $$\displaystyle\gamma = \lim_{n\to\infty} \left( - \ln(n) + \sum_{k=1}^n \dfrac1k \right) \approx 0.5772$$.

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