Uniform convergence
A series of functions converges uniformly on a set when one tail bound works for every point of the set at once.
A series of functions \sum f_n(z) converges uniformly on a set S if, for every \varepsilon>0, some single index N makes the tail past N smaller than \varepsilon at every point of S simultaneously -- one N that works everywhere on S at once, rather than a different N needed at each point. Uniform convergence is what licenses term-by-term operations -- integrating a series along a contour, or differentiating it, by working on each term separately -- and a power series converges uniformly on every closed disk strictly inside its radius of convergence.
Examples
\sum z^n converges uniformly on the closed disk |z|\le 0.5, but not uniformly on the full open disk |z|<1, where terms near the boundary shrink arbitrarily slowly.
Uniform convergence on a contour C lets \int_C \sum f_n(z)\,dz be computed as \sum \int_C f_n(z)\,dz, integrating term by term.
See also