Hello,

Suppose $\displaystyle 0<\alpha<\frac{\pi}{2}$ and $\displaystyle \{z_n\}$ is a sequence of complex numbers satisfying $\displaystyle -\alpha<arg(z_n)<\alpha$. Prove that the two infinite series $\displaystyle \sum_{n=1}^\infty {z_n}$ and $\displaystyle \sum_{n=1}^\infty |{z_n}|$ are both convergent or both divergent.

First, I'm trying to show that if the first converges, then the second also converges. Setting $\displaystyle z_n=x_n+iy_n$, I know that all $\displaystyle x_n$ are positive and hence $\displaystyle \sum_{n=1}^\infty |x_n|$ converges. But for $\displaystyle y_n$, this is still unclear for me.

Thank you

Mohammad