Trigamma series

Let $\psi^{(1)}$ denote the trigamma function. Prove that

$\sum_{n=1}^{\infty} \left (\psi^{(1)} (n) - \frac{1}{n} \right )=1$

Solution

$\begin{align*} \sum_{n=1}^{\infty} \left ( \psi^{(1)}(n) - \frac{1}{n} \right ) &= \lim_{N \rightarrow +\infty}\sum_{n=1}^{N} \left ( \psi^{(1)}(n) - \frac{1}{n} \right ) \\ &=\lim_{N \rightarrow +\infty} \left ( \log N + 1 + \gamma + \mathcal{O}\left( \frac{1}{N} \right) - \mathcal{H}_n \right ) \\ &=\lim_{N \rightarrow +\infty} \left ( \log N - \mathcal{H}_n + \mathcal{O}\left(\frac{1}{N}\right) \right ) + 1 + \gamma\\ &=-\gamma + 1 + \gamma \\ &= 1 \end{align*}$