## growth of function given by infinite series

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