Cauchy Criterion (Series)

analysis

Definition

Cauchy Criterion (Series)

An infinite series ${\sum }_{n=0}^{\infty }{a}_n$ is convergent if and only if for every $\epsilon >0$, there exists an index $N(\epsilon )$ such that for all $n>m>N(\epsilon )$:

$$\left|\sum _{k=m+1}^n{a}_k\right|<\epsilon$$

This is equivalent to stating that the sequence of partial sums $s_n={\sum }_{k=0}^n{a}_k$ is a Cauchy sequence.

Visual Representation

Geometrically, this criterion means that the difference between any two partial sums $s_n$ and $s_m$ becomes arbitrarily small once the indices exceed $N$.

Significance

The Cauchy criterion for series is used to derive many other convergence tests, such as the Majorant Criterion and the Minority Criterion, as well as the Quotient Criterion and the Root Criterion. It shows that for a series to converge, the terms $a_n$ must necessarily form a zero sequence, although this condition alone is not sufficient.