(a) The plot of a Cauchy sequence shown in blue, as versus . If the space containing the sequence is complete, the "ultimate destination" of this sequence (that is, the limit) exists.
(b) A sequence that is not Cauchy. The elements of the sequence fail to get arbitrarily close to each other as the sequence progresses.
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.
Proofedit
We can use the results about convergence of the sequence of partial sums of the infinite series and apply them to the convergence of the infinite series itself. The Cauchy Criterion test is one such application. For any real sequence , the above results on convergence imply that the infinite series
converges if and only if for every there is a number N, such that m ≥ n ≥ N imply
Probably the most interesting part of this theorem is that the Cauchy condition implies the existence of the limit: this is indeed related to the completeness of the real line. The Cauchy criterion can be generalized to a variety of situations, which can all be loosely summarized as "a vanishing oscillation condition is equivalent to convergence".[4]
cauchy, convergence, test, confused, with, cauchy, condensation, test, cauchy, convergence, test, method, used, test, infinite, series, convergence, relies, bounding, sums, terms, series, this, convergence, criterion, named, after, augustin, louis, cauchy, pub. Not to be confused with Cauchy condensation test The Cauchy convergence test is a method used to test infinite series for convergence It relies on bounding sums of terms in the series This convergence criterion is named after Augustin Louis Cauchy who published it in his textbook Cours d Analyse 1821 1 Contents 1 Statement 2 Explanation 3 Proof 4 ReferencesStatement editA series i 0 a i displaystyle sum i 0 infty a i nbsp is convergent if and only if for every e gt 0 displaystyle varepsilon gt 0 nbsp there is a natural number N displaystyle N nbsp such that a n 1 a n 2 a n p lt e displaystyle a n 1 a n 2 cdots a n p lt varepsilon nbsp holds for all n gt N displaystyle n gt N nbsp and all p 1 displaystyle p geq 1 nbsp 2 Explanation edit nbsp a The plot of a Cauchy sequence x n displaystyle x n nbsp shown in blue as x n displaystyle x n nbsp versus n displaystyle n nbsp If the space containing the sequence is complete the ultimate destination of this sequence that is the limit exists nbsp b A sequence that is not Cauchy The elements of the sequence fail to get arbitrarily close to each other as the sequence progresses This section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed February 2022 Learn how and when to remove this message The test works because the space R displaystyle mathbb R nbsp of real numbers and the space C displaystyle mathbb C nbsp of complex numbers with the metric given by the absolute value are both complete From here the series is convergent if and only if the partial sums s n i 0 n a i displaystyle s n sum i 0 n a i nbsp are a Cauchy sequence Cauchy s convergence test can only be used in complete metric spaces such as R displaystyle mathbb R nbsp and C displaystyle mathbb C nbsp 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 Proof editWe can use the results about convergence of the sequence of partial sums of the infinite series and apply them to the convergence of the infinite series itself The Cauchy Criterion test is one such application For any real sequence a k displaystyle a k nbsp the above results on convergence imply that the infinite series k 1 a k displaystyle sum k 1 infty a k nbsp converges if and only if for every e gt 0 displaystyle varepsilon gt 0 nbsp there is a number N such that m n N imply s m s n k n 1 m a k lt e displaystyle s m s n left sum k n 1 m a k right lt varepsilon nbsp 3 188 Probably the most interesting part of this theorem is that the Cauchy condition implies the existence of the limit this is indeed related to the completeness of the real line The Cauchy criterion can be generalized to a variety of situations which can all be loosely summarized as a vanishing oscillation condition is equivalent to convergence 4 This article incorporates material from Cauchy criterion for convergence on PlanetMath which is licensed under the Creative Commons Attribution Share Alike License References edit Allegranza Mauro Answer to Origin of Cauchy convergence test History of Science and Mathematics StackExchange Retrieved 10 September 2021 Abbott Stephen 2001 Understanding analysis Undergraduate Texts in Mathematics New York NY Springer Verlag p 63 ISBN 978 0 387 21506 8 Wade William 2010 An Introduction to Analysis Upper Saddle River NJ Prentice Hall ISBN 9780132296380 Kudryavtsev Lev D De Lellis Camillo Artemisfowl3rd 2013 Cauchy criteria In Rehmann Ulf ed Encyclopedia of Mathematics Springer European Mathematical Society a href Template Cite encyclopedia html title Template Cite encyclopedia cite encyclopedia a CS1 maint numeric names authors list link Retrieved from https en wikipedia org w index php title Cauchy 27s convergence test amp oldid 1225242114, wikipedia, wiki, book, books, library,