Abel's Test - Abel's Uniform Convergence Test

Abel's Uniform Convergence Test

Abel's uniform convergence test 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 {g_n} be a uniformly bounded sequence of real-valued continuous functions on a set E such that g_n+1(x) ≤ g_n(x) for all x ∈ E and positive integers n, and let {ƒ_n} be a sequence of real-valued functions such that the series Σƒ_n(x) converges uniformly on E. Then Σƒ_n(x)g_n(x) converges uniformly on E.