Does Convergence of Sum(z_j) and Sum((z_j)^2) Imply Convergence of Sum(|z_j|^2)?

[ilikegroupreps]

Homework Statement

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?

Homework Equations

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.

 

[HallsofIvy]

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

 

[ilikegroupreps]

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