Series of points in a bounded sector of a complex half-plane The question is: consider an infinite sequence of points which lie in a bounded sector of the complex plane, whose angular width is strictly less than pi (that is, it's an open sector of a half-plane). Prove that the sum of the sequence and the sum of the moduli (norm or abs. val.) of its terms either both converge or both diverge.
Attempt at a solution: Complex absolutely convergent series also converge, which gives us half the implications we want, which isn't terribly helpful. I'm not really sure how to use the boundedness of the arguments, but I noticed that for every z in the sector, -z is not in the sector, meaning there is no 'alternating series' type effect that could cause convergence here, but I'm not sure how to explain that in complex terms. Also, the argument of the partial sums tends towards a limit. This made me think I could use something like a complex form of the limit comparison test, (which might then tend toward an element of the unit circle?), but that seems to only apply to positive real sequences, which would be more useful if the sequence was constrained to a quadrant rather than a half-plane. 
 A: If $\sum_n z_n$ converges, then so does $\sum_n \text{Re}(\alpha z_n)$ for any $\alpha$.  In particular we can take $\alpha_1$ and $\alpha_2$ that are not collinear, such that all $\text{Re}(\alpha_i z_n) \ge 0$. Now there is $c > 0$ 
such that for every $z$, 
$$c |z| \le \left|\text{Re}(\alpha_1 z)\right| + \left|\text{Re}(\alpha_2 z)\right|$$ 
and then the Comparison Test implies that $\sum |z_n|$ converges.
EDIT: To explain further: for any nonzero complex number $\alpha$, 
$\text{Re}(\alpha z) \ge 0$ for $z$ in a half-plane through the origin (it is $0$ for real multiples of $i \bar{\alpha}$).  The sector is the intersection of two of these half-planes.  
You can get $c$ as the minimum value of $\left|\text{Re}(\alpha_1 z)\right| + \left|\text{Re}(\alpha_2 z)\right|$ on the unit circle, using the fact that a continuous function on a compact set attains its minimum.
Now the point is that since $\text{Re}(\alpha_1 z) \ge 0$ and $\text{Re}(\alpha_2 z) \ge 0$, we don't need the absolute values:
$$c |z_n| \le \text{Re}(\alpha_1 z_n) +  \text{Re}(\alpha_1 z_n) $$
and then, knowing that $\sum_n \text{Re}(\alpha_1 z_n)$ and $\sum_n \text{Re}(\alpha_1 z_n)$ converge, the Comparison Test tells us that $\sum_n |z_n|$ converges.
