Uniform Convergence

uniform convergence, in analysis, property involving the convergence of a sequence of continuous functions—f₁(x), f₂(x), f₃(x),…—to a function f(x) for all x in some interval (a, b). In particular, for any positive number ε > 0 there exists a positive integer N for which |f_n(x) − f(x)| ≤ ε for all n ≥ N and all x in (a, b). In pointwise convergence, N depends on both the closeness of ε and the particular point x.

An infinite series f₁(x) +  f₂(x) +  f₃(x) + ⋯ converges uniformly on an interval if the sequence of partial sums converges uniformly on the interval.

Many mathematical tests for uniform convergence have been devised. Among the most widely used are a variant of Abel’s test, devised by Norwegian mathematician Niels Henrik Abel (1802–29), and the Weierstrass M-test, devised by German mathematician Karl Weierstrass (1815–97).