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### Digamma and Trigamma series

Posted: Sun Dec 15, 2019 10:51 pm
Let $\psi^{(0)}$ and $\psi^{(1)}$ denote the digamma and trigamma functions respectively. Prove that:

$\sum_{n=1}^{\infty} \left ( \psi^{(0)}(n) - \ln n + \frac{1}{2} \psi^{(1)}(n) \right ) = 1+ \frac{\gamma}{2} - \frac{\ln 2\pi}{2}$

where $\gamma$ denotes the Euler – Mascheroni constant.