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Here, one can see that the sequence is converging to the limit 0 as n increases. In the real numbers , a number L {\displaystyle L} is the limit of the sequence ( x n ) {\displaystyle (x_{n})} , if the numbers in the sequence become closer and closer to L {\displaystyle L} , and not to any other number.
A series is convergent (or converges) if and only if the sequence (,,, … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit ; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices , one gets partial sums that become closer and closer to a given number.
It states that for a converging sequence the sequence of the arithmetic means of its first members converges against the same limit as the original sequence, that is () with implies (+ +) / . [ 1 ] [ 2 ] The theorem was found by Cauchy in 1821, [ 1 ] subsequently a number of related and generalized results were published, in particular by Otto ...
In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given any real number r > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance r/2 of s, and any two terms of the original sequence are within distance r/2 of ...
Absolute convergence implies Cauchy convergence of the sequence of partial sums (by the triangle inequality), which in turn implies absolute convergence of some grouping (not reordering). The sequence of partial sums obtained by grouping is a subsequence of the partial sums of the original series.
Cauchy's convergence test can only be used in complete metric spaces (such as and ), which are spaces where all Cauchy sequences converge. This is because we need only show that its elements become arbitrarily close to each other after a finite progression in the sequence to prove the series converges.
Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are called conditionally convergent series. Those that converge to the same value regardless of rearrangement are called unconditionally convergent series.
Firstly, we will acknowledge that a sequence () (in or ) has a convergent subsequence if and only if there exists a countable set where is the index set of the sequence such that () converges. Let ( x n ) {\displaystyle (x_{n})} be any bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} and denote its index set by I {\displaystyle I} .