Cauchy's convergence test

Explanation

The theorem states that the values near the beginning of an infinite series can be ignored for determining convergence. Even if the initial values are extremely large, the convergence of the series depends more on the ultimate behavior of ${\displaystyle a_n}$ as ${\displaystyle n}$ approaches infinity.

The test works because the space ${\displaystyle \mathbb{R}}$ of real numbers and the space ${\displaystyle \mathbb{C}}$ 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

${\displaystyle s_n:=\sum _{i=0}^n{a}_i}$

are a Cauchy sequence.

Cauchy's convergence test can only be used in complete metric spaces (such as ${\displaystyle \mathbb{R}}$ and ${\displaystyle \mathbb{C}}$), 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

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 ${\displaystyle a_k}$, the above results on convergence imply that the infinite series

${\displaystyle \sum _{k=1}^{\mathrm{\infty }}{a}_k}$

converges if and only if for every ${\displaystyle \epsilon >0}$ there is a number N, such that m ≥ n ≥ N imply

${\displaystyle |s_m-s_n|=\left|\sum _{k=n+1}^m{a}_k\right|<\epsilon .}$

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".

This article incorporates material from Cauchy criterion for convergence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.