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'Cauchy criterion for convergence'
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| Title of object: |
Cauchy criterion for convergence |
| Canonical Name: |
CauchyCriterionForConvergence |
| Type: |
Theorem |
| Created on: |
2003-01-16 05:12:35 |
| Modified on: |
2004-02-17 03:47:02 |
| Classification: |
msc:40A05 |
Revision comment (for changes between this and next version):
| Changes for correction #7412 ('the the'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
A series $\sum_{i=0}^\infty a_i$ is \PMlinkid{convergent}{2311} iff for every $\varepsilon>0$ there is a number $N\in\mathbb{N}$ such that
$$|a_{n+1}+a_{n+2}+\ldots+a_{n+p}|<\varepsilon$$
holds for all $n>N$ and $p\geq1$.
\subsection*{Proof:}
First define
$$s_n:=\sum_{i=0}^n a_i.$$
Now $(s_n)$ 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 the series is \PMlinkescapetext{convergent} iff
$$|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+\ldots+a_{n+p}|<\varepsilon.$$ |
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