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.
Given a set S and functions (or to any normed vector space), the series
A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒn(x) = xn/n on the open interval (−1,0), then the series Σfn(x) converges uniformly by comparison of the partial sums to those of Σ(−1)n, and the series Σ|fn(x)| converges absolutely at each point by the geometric series test, but Σ|fn(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.
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.
Pairwise Output Convergence in Selected Countries of East Asia and the Pacific: An Application of Stochastic Unit Root Test
Apr 01, 2011; I. Introduction The concept of convergence is defined in the literature as implying "forces accelerating the growth of nations...