PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (526)
Corrections (13)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
Cauchy criterion for convergence (Theorem)

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

$\displaystyle \Vert a_{n+1}+a_{n+2}+\cdots+a_{n+p}\Vert<\varepsilon$
holds for all $ n>N$ and $ p\geq1$.

Proof:

First define
$\displaystyle s_n:=\sum_{i=0}^n a_i.$
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:
$\displaystyle \Vert s_m-s_n\Vert<\varepsilon.$
We can assume $ m>n$ and thus set $ m=n+p$. The series is convergent iff
$\displaystyle \Vert s_{n+p}-s_n\Vert=\Vert a_{n+1}+a_{n+2}+\cdots+a_{n+p}\Vert<\varepsilon.$



"Cauchy criterion for convergence" is owned by mathwizard.
(view preamble | get metadata)

View style:

Log in to rate this entry.
(view current ratings)

Cross-references: Cauchy sequence, converges, complete, number, iff, Banach space, series
There are 3 references to this entry.

This is version 11 of Cauchy criterion for convergence, born on 2003-01-16, modified 2007-12-15.
Object id is 3894, canonical name is CauchyCriterionForConvergence.
Accessed 12438 times total.

Classification:
AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
Pending Errata and Addenda
None.
[ View all 8 ]
Discussion
Style: Expand: Order:
forum policy
Unproven by fernsanz on 2007-07-05 14:23:48
The entry appears as unproven although it is not.
[ reply | up ]
Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)