Need help calculating convergence, series.

I have an exercise where I'm supposed to find convergence if any and if there are show how it's... constituted. The series:

I'd like help when it comes to ways of solving it. I solve it as a real problem, getting the zeros on the real axis. Then I use these to find the zeros in the img-plane.

= =

For what z is the fraction smaller than 1?

I get - treating z as a real variable, so centre on x-axis should be

Now to the imaginary part.

which gives

I get - which goes nicely with the real values.

Centre =

Radius =

First on all, is this right? And is there a more solid way of doing this?