I apologize but I have a bit open-ended question... Imagine you have a sequence of real numbers that go to infinity very fast (let's say \sum_{n=1}^\infty \frac{1}{|z_j|^\varepsilon} < \infty for any \varepsilon)

Then the series f(z)=\sum_{j=1}^\infty \frac{1}{z-z_j} defines an analytic function on the whole complex plain except at z_j's where it has poles. I'm concerned with its behavior at infinity in the upper half plane. Is it possible to show that

\lim_{y \to +\infty} f(iy) =0

It looks so obvious but I'm not sure how to show it. Am I being dumb? The second question is -- can we determine the leading term behavior? Is it always O(\frac1y)?

Any ideas will be appreciated