uniform convergence on compact subsets
When a series of functions converges, there is a gentle kind of convergence and a strong kind. Pointwise, the gentle kind, only promises that at each individual point the partial sums eventually get close to the limit — but how many terms you need may grow without bound as you move around. Uniform convergence, the strong kind, promises a single rate that works everywhere at once: pick a tolerance, and from some fixed N onward EVERY point is within that tolerance. For complex analysis the right version is uniform convergence on compact subsets: uniform on each closed bounded piece you carve out, even if not uniform on the whole region.
Here is why this is exactly the right notion for power series. A power series converges uniformly on every closed disk |z - z_0| <= r with r < R strictly inside its open disk of convergence — though not necessarily uniformly on the full open disk, because near the boundary the convergence slows down. 'Compact subset' in an open disk means a closed, bounded set that stays a positive distance away from the boundary, and any such set sits inside some smaller closed disk. So 'uniform on compacta' captures precisely the truth: as good as uniform everywhere you stay safely inside, with the only weakening reserved for the unreachable rim.
This is the workhorse behind the theory's best theorems. Uniform convergence is what lets you swap a limit with an integral or a derivative — so it justifies integrating and differentiating a power series term by term inside its disk. It also underlies Weierstrass's theorem that a locally uniform limit of holomorphic functions is again holomorphic, and the whole notion of normal families later on. Whenever you see a complex analyst confidently interchange a sum and an integral, locally uniform convergence is the licence being quietly invoked.
The geometric series sum z^n converges to 1/(1 - z) uniformly on each closed disk |z| <= r with r < 1: the tail past N is bounded by r^(N+1)/(1 - r), one number that controls every such z. But it is NOT uniform on the full open disk |z| < 1, because as z approaches the boundary the needed N runs off to infinity.
Uniform on every closed disk strictly inside, but the uniformity degrades as you let r approach the radius — exactly the compacta hedge.
Uniform convergence on compact subsets is strictly weaker than uniform convergence on the whole open disk — and that is a feature, not a bug. Demanding global uniformity would exclude almost every interesting power series, since most fail it right at the boundary.