# Cauchy test

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 , and later rediscovered by A.L. Cauchy .