Be $\psi_0(s)$ the polygamma function of order zero in $s \in C$. Do you think is correct to write $ \psi_{0}(s) + \psi_{0}(-s) = \sum_{n=1}^{\infty} \frac{1}{-s+n} + \sum_{n=1}^{\infty} \frac{1}{s+n} = lim_{n \rightarrow \infty} \psi_0(-s + n + 1) + \psi_0,(s + n + 1) -\psi_0 (1 - s) - \psi_0(s + 1) = -\psi_0 (1 - s) - \psi_0(s + 1)$
where partial sum is given by mathwolfram site rappresenting $\psi(s),\psi(-s)$ as series $\sum_{m=1}^n(1/(-s + m) + 1/(s + m)) = \Psi_0(-s + n + 1) + \Psi_0,(s + n + 1) -\Psi_0 (1 - s) - \Psi_0(s + 1) $
The correct approximation as $n\to\infty$ is $$\sum _{k=1}^{n } \frac{1}{k-s}+\sum _{k=1}^{n } \frac{1}{k+s}\sim \frac{1}{n}+2 \log n -\psi ^{(0)}(s+1)-\psi ^{(0)}(1-s)$$