How to show that $$ \sum_{k=1}^{\infty}\left ( \frac{x}{k} -\operatorname{Log}\left ( 1+\frac{x}{k} \right )\right ) $$ converges uniformly for $x$ in compact subsets of $x\in \mathbb{C}$ with $\Re x>0$? Here $\operatorname{Log}$ denoted the principal logarithm.
If $x>0$, we we can use $1-1/x\leq \log x$ to show that the $k$'th term in the series is less than $x^2/k^2$ for alle $x>0$, and so it clearly converges uniformly on compact sets for $x>0$. But, what about the case $x\in \mathbb{C}$ with $\Re x>0$?
Initially, I was thinking about looking at the imaginary part of the series, as the real part is already done. But, how does the imaginary part of this expression look like?