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Digamma series

Let \psi^{(0)} denote the digamma series. Evaluate the series

    \[\mathcal{S} = \sum_{n=1}^{\infty} \left( \psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n} \right)\]

Solution

The series telescopes;

    \begin{align*} \mathcal{S} &=\sum_{n=1}^{\infty} \bigg(\psi^{(0)} \left(\frac{1+2n-1}{2} \right) -\psi^{(0)} \left( \frac{2n-1}{2} \right) - \\ &\quad \quad \quad -\frac{1}{2n-1}+\psi^{(0)} \left( \frac{1+2n}{2}\right)-  \psi^{(0)}\left(\frac{2n}{2} \right)-\frac{1}{2n} \bigg)\\ &=\sum_{n=1}^{\infty} \bigg( \psi^{(0)}\left( n \right)-\psi^{(0)} \left( n-\frac{1}{2} \right) -\frac{1}{2n-1}+ \\ &\quad \quad \quad +\psi^{(0)} \left( n+\frac{1}{2} \right)-\psi^{(0)} \left( n \right) -\frac{1}{2n} \bigg) \\ &=\sum_{n=1}^{\infty} \bigg( \psi^{(0)} \left(n+\frac{1}{2}\right) -\psi^{(0)} \left(n-\frac{1}{2}\right)- \\ &\quad \quad \quad -\frac{1}{2n-1}-\frac{1}{2n} \bigg)\\ &=\sum_{n=1}^{\infty} \left( \frac{2}{2n-1}-\frac{1}{2n-1}-\frac{1}{2n} \right) \\ &=\sum_{n=1}^{\infty} \left( \frac{1}{2n-1}-\frac{1}{2n} \right)\\ &=\ln 2 \end{align*}

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