locally uniform convergence
Imagine a sequence of functions slowly settling onto a limiting function, like a flickering image sharpening into focus. There are several strengths of 'settling down'. The weakest, pointwise convergence, only asks that at each individual point the values eventually get close — but the rate can be hopelessly different from point to point, so the limit can be ugly even when every term is smooth. The strongest, uniform convergence on the whole region, demands a single error bound that works everywhere at once. Locally uniform convergence sits in between and is exactly the right notion for holomorphic functions: the convergence is uniform on every compact (closed and bounded) piece of the region, even if it degrades as you approach the boundary or run off to infinity.
Precisely, a sequence f_n converges locally uniformly to f on an open region D if, for every compact subset K of D, the maximum of |f_n(z) - f(z)| over K tends to 0 as n -> infinity. The key relaxation compared with global uniform convergence is that you are never forced to control the behavior near the edge of D all at once; you only have to win on each fixed inner chunk. A clean test: a sequence converges locally uniformly exactly when it converges uniformly on every closed disk that sits strictly inside D. For example, the partial sums of the geometric series sum z^n converge locally uniformly to 1/(1 - z) on the open unit disk |z| < 1 — uniformly on each smaller disk |z| <= r with r < 1 — yet not uniformly on the full open disk, since the error blows up as |z| -> 1.
This is the convergence that respects holomorphy. Under locally uniform convergence a limit of holomorphic functions is again holomorphic, and (a small miracle with no real-variable analogue) the derivatives converge too — that is the Weierstrass convergence theorem. It is also the natural topology behind normal families and Montel's theorem. A common slip is to expect global uniform convergence; on a non-compact domain that is usually too much to ask, and demanding it would throw away most of the useful limits. Locally uniform is the honest, workable standard.
The partial sums S_N(z) = 1 + z + z^2 + ... + z^N converge to 1/(1 - z) locally uniformly on the open unit disk. On the disk |z| <= 1/2 the tail error is at most (1/2)^(N+1)/(1 - 1/2), which goes to 0 uniformly; on |z| <= 0.99 you still win, just more slowly. But no single N tames the whole open disk at once.
Uniform on each inner disk, yet never uniform on the whole open disk — the signature of locally uniform convergence.
Locally uniform convergence on an open set is equivalent to uniform convergence on every compact subset; the phrases 'normal convergence' and 'uniform on compacta' mean the same thing. Do not confuse it with global uniform convergence, which is strictly stronger.