Cauchy test

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)