Cauchy's convergence test

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" $inmathbb{N}$ such that

: $|a_{n+1}+a_{n+2}+cdots+a_{n+p}|$

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

$|s_m-s_n|$

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

: $|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+cdots+a_{n+p}|$

ee also

*Series (mathematics)
*Convergent series

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Cauchy's convergence test

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