In the sum, group the terms with index k and ľk (that's what the hint about reordering the sum is getting at!), to make it . In that form, it's easier to see what happens as |Im(z)| goes to infinity.
This is all spelt out in Ahlfors' Complex Analysis (Section 5.2 in the edition that I have). If you want to get to grips with complex analysis, Ahlfors is really an essential possession.