# Complex Analysis - Series

1. Feb 15, 2013

### ilikegroupreps

1. The problem statement, all variables and given/known data
Assume that z_j is a sequence where j indexes from 1 to infinity are in the complex numbers such that the real part of z > 0. Is it true or false that if sum(z_j) and sum_((z_j)^2) both converge then sum(|z^j|^2) also converges?

2. Relevant equations

3. The attempt at a solution
I tried breaking up z into real and imaginary parts and looking at what has to converge. For example, if z_j=x_j+iy_j, then the sum of x_j and sum of y_j both converge. Not really sure where else to go.

2. Feb 15, 2013

### HallsofIvy

Staff Emeritus
What are the real and imaginary parts of $z_j^2$?

3. Feb 15, 2013

### ilikegroupreps

so the real part is x^2-y^2 and the imaginary part is 2xy