Search results
Results From The WOW.Com Content Network
In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, see Modes of convergence (annotated index)
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 ...
The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the ...
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 ...
While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.