18 - Cauchy Convergence Criterion for Series

Cauchy Convergence Criterion for Series

Let $(a_k)$ be a sequence, and write

$$s_n=\sum _{k=1}^n{a}_k$$

Then ${\sum }_{k=1}^{\infty }{a}_k$ converges if and only if

$$\forall ϵ>0\exists N\in \mathbb{N}\text{ such that }\forall n>m\ge N:|s_n-s_m|=\left|\sum _{k=m+1}^n{a}_k\right|<ϵ$$

Proof

Follows from Caughy Convergence Criterion