It relies on bounding sums of terms in the series.
This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analyse 1821.
of complex numbers (with the metric given by the absolute value) are both complete.
From here, the series is convergent if and only if the partial sums are a Cauchy sequence.
Cauchy's convergence test can only be used in complete metric spaces (such as
This is because we need only show that its elements become arbitrarily close to each other after a finite progression in the sequence to prove the series converges.
We can use the results about convergence of the sequence of partial sums of the infinite series and apply them to the convergence of the infinite series itself.
there is a number N, such that m ≥ n ≥ N imply Probably the most interesting part of this theorem is that the Cauchy condition implies the existence of the limit: this is indeed related to the completeness of the real line.
The Cauchy criterion can be generalized to a variety of situations, which can all be loosely summarized as "a vanishing oscillation condition is equivalent to convergence".