# Cauchy test

(Redirected from Cauchy criterion)

2010 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).

## Contents

#### 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 criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cauchy_criterion&oldid=30862