# Cauchy's convergence test

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.

## Statement

A series

$\sum_{i=0}^\infty a_i$ 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]

## Explanation

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.

## Proof

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 above results on convergence imply that 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

$|s_m-s_n|=|\sum_{k=n}^m a_k|<\varepsilon.$.[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]