absolute and conditional convergence

Imagine adding up an infinite list of complex numbers, each a little arrow in the plane. Some arrows point one way, some another, and they may partly cancel. There are two strengths of convergence you can ask for. The strong one ignores direction: replace every term by its length |z_n| and ask whether the sum of those lengths is finite. If sum |z_n| converges, we say the original series converges ABSOLUTELY. The weak one keeps the directions: maybe the lengths add up to infinity, but thanks to cancellation the actual arrows still settle to a finite total. If sum z_n converges but sum |z_n| does not, we say it converges CONDITIONALLY.

Absolute convergence is the gold standard, and here is the key fact: absolute convergence implies ordinary convergence. The reason is that sum |z_n| is a series of non-negative real numbers, so all the comfortable real-variable tests (comparison, ratio, root) apply to it, and once you control the lengths you control the sum. Better still, an absolutely convergent series may be rearranged in any order and regrouped freely without changing its value — it behaves like a finite sum. Conditional convergence is fragile by contrast: rearrange a conditionally convergent series and you can make it sum to anything you like (Riemann's rearrangement theorem, on the real part and imaginary part separately).

Why care in complex analysis? Because the whole power-series machine runs on absolute convergence. Inside its disk of convergence a power series sum a_n (z - z_0)^n converges absolutely, and that is precisely what lets us differentiate and integrate it term by term, multiply two such series together, and rearrange at will. When we later prove a holomorphic function equals its Taylor series, the safety of those manipulations rests on this absolute (indeed uniform) convergence.