Search results
Results from the WOW.Com Content Network
The sequence of partial sums obtained by grouping is a subsequence of the partial sums of the original series. The convergence of each absolutely convergent series is an equivalent condition for a normed vector space to be Banach (i.e.: complete).
Convergent evolution—the repeated evolution of similar traits in multiple lineages which all ancestrally lack the trait—is rife in nature, as illustrated by the examples below. The ultimate cause of convergence is usually a similar evolutionary biome , as similar environments will select for similar traits in any species occupying the same ...
In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. These moduli are often employed in the study of computable analysis and constructive mathematics.
It can be shown that a real-valued sequence is Cauchy if and only if it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, (, | |), is a complete metric space. In a general metric space, however, a Cauchy sequence need not converge.
The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is the Cauchy criterion for convergence of sequences : a sequence of real numbers is convergent if and only if it is a Cauchy sequence.
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. Below are some of the more common and typical examples. This article is intended as an introduction aimed to help practitioners explore appropriate techniques.
In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).