Cauchy criterion for series
= Cauchy criterion for series
{c}
{title2=$\forall\varepsilon>0\ \exists N\ \forall m>n\geq N:\ |\sum_{k=n+1}^m a_k|<\varepsilon$}
A <series> over the <real numbers> or <complex numbers> converges if and only if its tails satisfy: for every $\varepsilon>0$, some $N$ makes $|\sum_{k=n+1}^m a_k|<\varepsilon$ whenever $m>n\geq N$. This says exactly that its <partial sums> form a <Cauchy sequence>; sufficiency uses <completeness>.