I'm trying to prove the sum
$H_x :=\lim_{n \to \infty} \sum_{k=1}^{n} \left( \frac{1}{k} -\frac{1}{x+k} \right)$
converging uniformly at $x\neq -1, -2, ...$.
Using the weirstrass m-test i proved it for $x\geq0$, but have trouble proving it on the intervals $... \cup (-3,-2) \cup (-2,-1)$.
So far I got
$$\lvert \frac{1}{k}-\frac{1}{x+k}\rvert=\lvert \frac{x}{k(x+k)} \rvert= \frac{-x}{k\lvert(x+k)\rvert}$$
at which point I'm not sure how to manipulate the denominator.