Show that $\Sigma_{n=1}^\infty |z_n| $ converges.

Question

Assume $z_k = |z_k|e^{i\alpha_k}$ are complex numbers

and that exists $0<\alpha<\pi/2$ s.t $\forall k -\alpha < \alpha_k<\alpha$

assume that $\Sigma_{n=1}^\infty z_n $ converges.

we need to show that $\Sigma_{n=1}^\infty |z_n| $ converges.

I kind of have no idea how to approach this one, its from a test I attended and I'm going for a second go at it and I want to understand the first one, first.

Answer 1

Hint: compare the real part of $z_n$ to $|z_n|$.

Answer 2

Hint: Two parts:

1) $\sum z_n$ converges iff $\sum Re(z_n)$ and $\sum Im(z_n)$ converge

2) $Re(z_n) = |z_n|\cos(\alpha_n) \geq |z_n|\cos(\alpha)$

Can you do the rest?