How does one show that the series $$\sum_{k = 1}^\infty \left\{\frac{s}{k} - \log\left(1 + \frac{s}{k}\right)\right\}, \quad s \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$$ is uniformly convergent? Here, $\log z$ is the principal value of the logarithm (with the branch cut along the negative real axis).
Any help would be appreciated.
Edit: It has been pointed out in the comments that this series is only locally uniformly convergent.