# Cauchy Criterion for Infinite Series

The Cauchy Criterion states that for each $$\epsilon\gt0$$ there exists a number N such that $$n\geq m\gt N$$ implies $$\left|\sum_{k=m}^n{a_k}\right|\lt\epsilon$$ Does $$n$$ have to be finite or if it can be taken to be infinity? If the Cauchy Criterion is satisfied does that imply the statement below? $$\lim\limits_{m \to \infty} \sum_{k=m}^\infty{a_k}=0$$

## 1 Answer

You say a series converges if each partial sum converges, i.e, if you define $$S_k=\sum_{1\leq i \leq k} a_i$$, if you consider the sequence $$(S_k)_{k=1}^{\infty}$$, it has to converge.

And as every convergent sequence is Cauchy, thus $$\forall n,m\geq k$$ (suitable $$k$$) you must have $$|\sum_{1\leq i \leq n} a_i-\sum_{1\leq i \leq m} a_i|=|\sum_{1\leq i \leq n-m} a_i| \leq \epsilon$$ (WLOG letting $$n>m$$ and suitable $$\epsilon>0$$) and just shift $$n\to n+m$$ to get why it works.

For your last part, i fo not understand why you would like to write that way. You are dealing with the sequence of the sums, not the entire sum itself.