Suppose the series $\sum_{n=1}^\infty|a_n|^{-1}$ converges. Can it be shown that $\sum_{n=1}^\infty |x - a_n|^{-1}$ converges uniformly for $x$ in any bounded set that excludes all $a_n$?
Trying the obvious $||x| - |a_n|| \leq |x - a_n| \leq |x| + |a_n|$ doesn't seem to be much help.
The key is bounded set.
Suppose $X$ is a bounded set excluding all $a_n$s. Let $L= \sup_{x \in X} |x|$. Recall that necessarily $|a_n|^{-1} \to 0$, thus $|a_n| \to \infty$. This means that we can consider, for $n$ large enough, $$|a_n|> 2L$$ Now, $$|x-a_n| \ge |a_n| - |x| > |a_n|-L = |a_n| \cdot \left( 1- \frac{L}{|a_n|} \right) > \frac{1}{2}|a_n|$$ from this the result follows.