Hello!

Suppose that $\displaystyle \{z_n\}$ is a sequence of complex numbers satisfying $\displaystyle Re\{z_n\}\geq0$ and that the two series $\displaystyle \sum_{n=1}^\infty z_n$ and $\displaystyle \sum_{n=1}^\infty z_n^2$ are convergent. Prove that the series $\displaystyle \sum_{n=1}^\infty |z_n|^2$ is also convergent.

Well, assuming $\displaystyle z_n=x_n+iy_n$, we can deduce from the given that the following series converge:

$\displaystyle \sum_{n=1}^\infty x_n$, $\displaystyle \sum_{n=1}^\infty |x_n|$,

$\displaystyle \sum_{n=1}^\infty y_n$,

$\displaystyle \sum_{n=1}^\infty x_n^2-y_n^2$,

$\displaystyle \sum_{n=1}^\infty 2x_n y_n$.

The sequence $\displaystyle \sum_{n=1}^\infty |z_n|^2$ can be written as:

$\displaystyle \sum_{n=1}^\infty x_n^2+y_n^2$ or $\displaystyle \sum_{n=1}^\infty |x_n^2-y_n^2+2ix_ny_n|$

I am stuck here for now. Any ideas?

Thanx

Mohammad