In classical mathematical analysis, Abel's uniform convergence test, one of many things named after Niels Henrik Abel, is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts.
The test is as follows. Let {gn} be a uniformly bounded sequence of real-valued continuous functions on a set E such that gn+1(x) ≤ gn(x) for all x ∈ E and positive integers n, and let {ƒn} be a sequence of real-valued functions such that the series Σƒn(x) converges uniformly on E. Then Σƒn(x)gn(x) converges uniformly on E.