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In a topological abelian group, convergence of a series is defined as convergence of the sequence of partial sums. An important concept when considering series is unconditional convergence, which guarantees that the limit of the series is invariant under permutations of the summands.
The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick ...
That the series formed by calculating the expected value of the outcome's distance from a particular value may converge to 0; That the variance of the random variable describing the next event grows smaller and smaller. These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been ...
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence () =, with {, +}, the series = converges. If X {\displaystyle X} is a Banach space , every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general.
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 Σf 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 ...
Divergent series (2 C, 15 P) L. Limits (mathematics) (1 C, 18 P) ... Modes of convergence; Modes of convergence (annotated index) Mosco convergence; N. Normal ...
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (,,, …) defines a series S that is denoted = + + + = =. The n th partial sum S n is the sum of the first n terms of the sequence; that is,