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The Cauchy criterion for the convergence of a series: Given a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208401.png" /> with non-negative real terms, if there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208403.png" />, such that, for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208404.png" />, one has the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208405.png" />, which is equivalent to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208406.png" />, then the series is convergent. Conversely, if for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208407.png" /> one has the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208408.png" />, or even the weaker condition: There exists a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c0208409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084010.png" /> with terms satisfying the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084011.png" />, then the series is divergent.
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{{MSC|40A05}}
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{{TEX|done}}
  
In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084012.png" /> exists and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084013.png" />, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084014.png" /> is convergent; if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084015.png" />, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084016.png" /> is divergent. This was proved by A.L. Cauchy . In the case of a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084017.png" /> with terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084018.png" /> of arbitrary sign, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084019.png" /> implies that the series is divergent; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084020.png" />, the series is absolutely convergent.
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The term is used for several tests which can be used to determine whether a series of real numbers converges or diverges. These tests are sometimes called Cauchy criteria. However, the latter term is most commonly used for a characterization of convergent sequences in the Euclidean space (and in general in complete metric spaces, see [[Cauchy criteria]]).
 
 
The integral Cauchy test, or the Cauchy–MacLaurin integral criterion: Given a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084021.png" /> with non-negative real terms, if there exists a non-increasing non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084022.png" />, defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084023.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084025.png" /> then the series is convergent if and only if the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084026.png" /> is convergent. This test was first presented in a geometrical form by C. MacLaurin [[#References|[2]]], and later rediscovered by A.L. Cauchy [[#References|[3]]].
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Cauchy,  "Analyse algébrique" , Gauthier-Villars  (1821)  pp. 132–135  (German translation: Springer, 1885)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. MacLaurin,  "Treatise of fluxions" , '''1''' , Edinburgh  (1742)  pp. 289–290</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.L. Cauchy,  "Sur la convergence des séries" , ''Oeuvres complètes Ser. 2'' , '''7''' , Gauthier-Villars  (1889) pp. 267–279</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>
 
  
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====Cauchy criterion====
 +
A series $\sum a_i$ of real numbers converges if and only if for every $\varepsilon$ there is an $N$ such that
 +
\[
 +
\left|\sum_{i=m}^n a_i\right| < \varepsilon \qquad \forall m, n \geq N\, .
 +
\]
  
 +
====Root test====
 +
Let $\sum a_i$ be a series. If
 +
\[
 +
\limsup_{n\to\infty} \left|a_n\right|^{1/n} < 1
 +
\]
 +
then the series converges absolutely. If
 +
\[
 +
\limsup_{n\to \infty} \left|a_n\right|^{1/n} > 1
 +
\]
 +
then the series diverges.
  
====Comments====
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When
See also [[Cauchy criteria|Cauchy criteria]]. The following is also known as Cauchy's condensation test or Cauchy's convergence theorem (criterion): If the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084027.png" /> of a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084028.png" /> form a monotone decreasing sequence, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084029.png" /> and
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\[
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\limsup_{n\to \infty} |a_n|^{a/n} = 1
 +
\]
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it is possible that the series diverges, converges (but not absolutely) and converges absolutely. In particular, consider the series
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\begin{eqnarray}
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&\sum_{n=1}^\infty \frac{1}{n}\label{e:harmonic}\\
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&\sum_{n=1}^\infty (-1)^n \frac{1}{n}\label{e:harmonic_-}\\
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&\sum_{n=1}^\infty \frac{1}{n^2}\, .
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\end{eqnarray}
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In all these cases
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\[
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\lim_{n\to \infty} |a_n|^{1/n} =1\, .
 +
\]
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However the first series diverges, the second converges, but not absolutely, and the third converges absolutely.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020840/c02084030.png" /></td> </tr></table>
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====Cauchy-MacLaurin integral test====
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Let $f: [0, \infty[\to \mathbb R$ be a nonincreasing nonnegative function. Then the series $\sum f(n)$ converges if and only if the integral
 +
\[
 +
\int_0^\infty f(x)\, dx
 +
\]
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is finite.
  
are [[Equiconvergent series|equiconvergent series]], i.e. both converge or both diverge (cf. [[#References|[a1]]], [[#References|[a2]]]).
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====Cauchy condensation test====
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Let $\{a_n\}$ be a monotone vanishing sequence of nonnegative real numbers. Then $\sum_n a_n$ converges if and only if the following series converges
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\[
 +
\sum_{n=0}^\infty 2^n a_{2^n}\, .
 +
\]
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.H. Hardy,  "A course of pure mathematics" , Cambridge Univ. Press (1975)</TD></TR></table>
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{|
 +
|-
 +
|valign="top"|{{Ref|Ca1}}|| A.L. Cauchy,  "Analyse algébrique" , Gauthier-Villars  (1821)  (German translation: Springer, 1885)
 +
|-
 +
|valign="top"|{{Ref|Ca2}}|| A.L. Cauchy,  "Sur la convergence des séries" , ''Oeuvres complètes Ser. 2'' , '''7''' , Gauthier-Villars  (1889)  pp. 267–279
 +
|-
 +
|valign="top"|{{Ref|Ha}}||    G.H. Hardy,  "A course of pure mathematics" , Cambridge Univ. Press (1975)
 +
|-
 +
|valign="top"|{{Ref|Kn}}||  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer   (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)
 +
|-
 +
|valign="top"|{{Ref|ML}}|| C. MacLaurin,  "Treatise of fluxions" , '''1''' , Edinburgh  (1742)  pp. 289–290
 +
|-
 +
|valign="top"|{{Ref|Ni}}||  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' ,  MIR  (1977) (Translated from Russian)
 +
|-
 +
|}

Latest revision as of 20:30, 8 December 2013

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

The term is used for several tests which can be used to determine whether a series of real numbers converges or diverges. These tests are sometimes called Cauchy criteria. However, the latter term is most commonly used for a characterization of convergent sequences in the Euclidean space (and in general in complete metric spaces, see Cauchy criteria).

Cauchy criterion

A series $\sum a_i$ of real numbers converges if and only if for every $\varepsilon$ there is an $N$ such that \[ \left|\sum_{i=m}^n a_i\right| < \varepsilon \qquad \forall m, n \geq N\, . \]

Root test

Let $\sum a_i$ be a series. If \[ \limsup_{n\to\infty} \left|a_n\right|^{1/n} < 1 \] then the series converges absolutely. If \[ \limsup_{n\to \infty} \left|a_n\right|^{1/n} > 1 \] then the series diverges.

When \[ \limsup_{n\to \infty} |a_n|^{a/n} = 1 \] it is possible that the series diverges, converges (but not absolutely) and converges absolutely. In particular, consider the series \begin{eqnarray} &\sum_{n=1}^\infty \frac{1}{n}\label{e:harmonic}\\ &\sum_{n=1}^\infty (-1)^n \frac{1}{n}\label{e:harmonic_-}\\ &\sum_{n=1}^\infty \frac{1}{n^2}\, . \end{eqnarray} In all these cases \[ \lim_{n\to \infty} |a_n|^{1/n} =1\, . \] However the first series diverges, the second converges, but not absolutely, and the third converges absolutely.

Cauchy-MacLaurin integral test

Let $f: [0, \infty[\to \mathbb R$ be a nonincreasing nonnegative function. Then the series $\sum f(n)$ converges if and only if the integral \[ \int_0^\infty f(x)\, dx \] is finite.

Cauchy condensation test

Let $\{a_n\}$ be a monotone vanishing sequence of nonnegative real numbers. Then $\sum_n a_n$ converges if and only if the following series converges \[ \sum_{n=0}^\infty 2^n a_{2^n}\, . \]

References

[Ca1] A.L. Cauchy, "Analyse algébrique" , Gauthier-Villars (1821) (German translation: Springer, 1885)
[Ca2] A.L. Cauchy, "Sur la convergence des séries" , Oeuvres complètes Ser. 2 , 7 , Gauthier-Villars (1889) pp. 267–279
[Ha] G.H. Hardy, "A course of pure mathematics" , Cambridge Univ. Press (1975)
[Kn] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
[ML] C. MacLaurin, "Treatise of fluxions" , 1 , Edinburgh (1742) pp. 289–290
[Ni] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
How to Cite This Entry:
Cauchy test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_test&oldid=15773
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article