Cauchy's convergence test

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Not to be confused with Cauchy condensation test.

The Cauchy convergence test is a method used to test infinite series for convergence. A series

\sum_{i=0}^\infty a_i

where the real or complex summand "a"i" is convergent if and only if for every \varepsilon>0 there is a natural number N such that

|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon

holds for all n > N and p ≥ 1.[1]

The test works because the space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are both complete. Then the series is convergent if and only if the partial sum

s_n:=\sum_{i=0}^n a_i

is a Cauchy sequence.

(a) The plot of a Cauchy sequence (x_n), shown in blue, as x_n versus n 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.

A sequence of real or complex numbers s_n is Cauchy if and only if s_n converges ( to some point a in R or C).[2] The formal definition states that for every \varepsilon>0 there is a number N, such that for all n, m > N holds

|s_m-s_n|<\varepsilon.

We will assume m > n and thus set p = m − n.

|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon.

Showing that a sequence is Cauchy is useful since we do not need to know the limit of the sequence in question. This is based on the properties of metric spaces, in which all such sequences converge to a limit. We need only show that its elements become arbitrarily close to each other after a finite progression in the sequence. There are computer applications of the Cauchy sequence, in which an iterative process may be set up to create such sequences.

The Cauchy Criterion[edit]

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 a_k , the infinite series

\sum_{k=1}^\infty a_k

converges if and only if for every \varepsilon>0 there is a number N, such that

m ≥ n ≥ N imply

|\sum_{k=n}^m a_k|<\varepsilon..[3]

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]

  1. ^ Abbott, Stephen (2001). Understanding Analysis, p.63. Springer, New York. ISBN 9781441928665
  2. ^ Wade, William (2010). An Introduction to Analysis. Upper Saddle River,NJ: Prentice Hall. p. 59. ISBN 9780132296380. 
  3. ^ Wade, William (2010). An Introduction to Analysis. Upper Saddle River, NJ: Prentice Hall. p. 188. ISBN 9780132296380. 
  4. ^ Encyclopedia of Mathematics. "Cauchy Criteria". European Mathematical Society. Retrieved 4 March 2014.