Difference between revisions of "Cauchy test"
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− | The Cauchy criterion for the convergence of a series: Given a series $\sum_{n=1}^{\infty}u_n$ with non-negative real terms, if there exists a number $q$, $0\leq q<1$, such that, for all sufficiently large $n$, one has the inequality | + | The Cauchy criterion for the convergence of a series: Given a series $\sum_{n=1}^{\infty}u_n$ with non-negative real terms, if there exists a number $q$, $0\leq q<1$, such that, for all sufficiently large $n$, one has the inequality $(u_n)^{1/n}\leq q$, 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. |
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. | 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. |
Revision as of 10:19, 16 January 2013
The Cauchy criterion for the convergence of a series: Given a series $\sum_{n=1}^{\infty}u_n$ with non-negative real terms, if there exists a number $q$, $0\leq q<1$, such that, for all sufficiently large $n$, one has the inequality $(u_n)^{1/n}\leq q$, which is equivalent to the condition , then the series is convergent. Conversely, if for all sufficiently large
one has the inequality
, or even the weaker condition: There exists a subsequence
,
with terms satisfying the inequality
, then the series is divergent.
In particular, if exists and is
, then the series
is convergent; if it is
, then the series
is divergent. This was proved by A.L. Cauchy . In the case of a series
with terms
of arbitrary sign, the condition
implies that the series is divergent; if
, the series is absolutely convergent.
The integral Cauchy test, or the Cauchy–MacLaurin integral criterion: Given a series with non-negative real terms, if there exists a non-increasing non-negative function
, defined for
, such that
,
then the series is convergent if and only if the integral
is convergent. This test was first presented in a geometrical form by C. MacLaurin [2], and later rediscovered by A.L. Cauchy [3].
References
[1] | A.L. Cauchy, "Analyse algébrique" , Gauthier-Villars (1821) pp. 132–135 (German translation: Springer, 1885) |
[2] | C. MacLaurin, "Treatise of fluxions" , 1 , Edinburgh (1742) pp. 289–290 |
[3] | A.L. Cauchy, "Sur la convergence des séries" , Oeuvres complètes Ser. 2 , 7 , Gauthier-Villars (1889) pp. 267–279 |
[4] | S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) |
Comments
See also Cauchy criteria. The following is also known as Cauchy's condensation test or Cauchy's convergence theorem (criterion): If the terms of a series
form a monotone decreasing sequence, then
and
![]() |
are equiconvergent series, i.e. both converge or both diverge (cf. [a1], [a2]).
References
[a1] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) |
[a2] | G.H. Hardy, "A course of pure mathematics" , Cambridge Univ. Press (1975) |
Cauchy test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_test&oldid=29316