In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.
Motivation.
A convergent series of numbers can often be reordered in such a way that the new series diverges. This is not possible for series of nonnegative numbers, however, so the notion of absolute-convergence precludes this phenomenon. When dealing with uniformly convergent series of functions, the same phenomenon occurs: the series can potentially be reordered into a non-uniformly convergent series, or a series which does not even converge pointwise. This is impossible for series of nonnegative functions, so the notion of uniform absolute-convergence can be used to rule out these possibilities.
Definition.
Given a set "X" and functions formula_1 (or to any normed vector space), the series
formula_2
is called uniformly absolutely-convergent if the series of nonnegative functions 
formula_3
is uniformly convergent.
Distinctions.
A series can be uniformly convergent "and" absolutely convergent without being uniformly absolutely-convergent. For example, if "ƒ""n"("x") = "x""n"/"n" on the open interval (−1,0), then the series Σ"f""n"("x") converges uniformly by comparison of the partial sums to those of Σ(−1)"n"/"n", and the series Σ|"f""n"("x")| converges absolutely "at each point" by the geometric series test, but Σ|"f""n"("x")| does not converge uniformly. Intuitively, this is because the absolute-convergence gets slower and slower as "x" approaches −1, where convergence holds but absolute convergence fails.
Generalizations.
If a series of functions is uniformly absolutely-convergent on some neighborhood of each point of a topological space, it is locally uniformly absolutely-convergent. If a series is uniformly absolutely-convergent on all compact subsets of a topological space, it is compactly (uniformly) absolutely-convergent. If the topological space is locally compact, these notions are equivalent.