Cauchy criterion for series

Statement

Let $(a_n)$ be a real sequence. The series $\sum a_n$ converges if and only if the following holds for each $\varepsilon > 0$. There is a natural number $N$ such that $$\left| \sum_{k=m+1}^{n} a_k \right| < \varepsilon \quad \text{for all natural numbers } n > m \ge N.$$

Remarks

The sum in the condition is $s_n - s_m$, a difference of partial sums. So the condition says that $(s_n)$ is a Cauchy sequence, and the result is the Cauchy criterion for sequences. With $n = m + 1$ it gives the term test. It proves that absolutely convergent series converge.

Explanations

L1 human

A series converges exactly when its tails become small. Each block of consecutive terms far out in the series must have a small sum.