Cauchy Criterion (Series)

analysis

Definition

Cauchy Criterion (Series)

Let ${\sum }_{n\ge 0}{a}_n$ be an infinite series. The series is convergent iff:

$$\sum _{n\ge 0}{a}_n\text{ convergent}⟺\forall \epsilon >0:\exists N(\epsilon )\in \mathbb{N}:\forall m\ge n>N(\epsilon ):|\sum _{k=n}^m{a}_k|<\epsilon$$