Cauchy's convergence test

From Wikipedia, the free encyclopedia

The Cauchy convergence test is a method used to test infinite series for convergence. A series

\sum_{i=0}^\infty a_i

is convergent if and only if for every \varepsilon>0 there is a number N\in\mathbb{N} such that

|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon

holds for all n > N and p \geq 1.

The test works because the series is convergent if and only if the partial sum

s_n:=\sum_{i=0}^n a_i

is a Cauchy sequence: for every \varepsilon>0 there is a number N, such that for all n, m > N holds

|s_m-s_n|<\varepsilon.

We can assume m > n and thus set p = m - n. The series is convergent if and only if

|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon.

[edit] See also

This article incorporates material from Cauchy criterion for convergence on PlanetMath, which is licensed under the GFDL.