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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? $$\lim\limits_{m \to \infty} \sum_{k=m}^\infty{a_k}=0$$

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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.

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