## You are here

HomeCauchy criterion for convergence

## Primary tabs

# Cauchy criterion for convergence

A series $\sum_{{i=0}}^{\infty}a_{i}$ in a Banach space $(V,\|\cdot\|)$ is convergent iff for every $\varepsilon>0$ there is a number $N\in\mathbb{N}$ such that

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

holds for all $n>N$ and $p\geq 1$.

# Proof:

First define

$s_{n}:=\sum_{{i=0}}^{n}a_{i}.$ |

Now, since $V$ is complete, $(s_{n})$ converges if and only if it is a Cauchy sequence, so if for every $\varepsilon>0$ there is a number $N$, such that for all $n,m>N$ holds:

$\|s_{m}-s_{n}\|<\varepsilon.$ |

We can assume $m>n$ and thus set $m=n+p$. The series is convergent iff

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

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

40A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

## Info

## Attached Articles

## Corrections

the the by yark ✓

use \cdots by Mathprof ✓

contains own proof by rspuzio ✓

Not quite by rm50 ✓

Contains own proof by rm50 ✓

use \cdots by Mathprof ✓

contains own proof by rspuzio ✓

Not quite by rm50 ✓

Contains own proof by rm50 ✓

## Versions

(v14) by mathwizard 2013-03-22

## Comments

## Unproven

The entry appears as unproven although it is not.