The Cauchy convergence test is a method used to test infinite series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 for convergence. A series


with real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 or complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 summands a_i is convergent if and only if for every there is a natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

 N such that


holds for all and .

The test works because the space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are both complete, so that the series is convergent if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 the partial sum



is a Cauchy sequence

Cauchy sequence

In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...

: for every there is a number N, such that for all n, m > N holds



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