[Cauchy Criterion for Series (2.7.2)] The series \(\sum_{k=1}^{\infty} a_k\) converges if and only if, given \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that whenever \(n > m \geq N\) it follows that $$ \begin{align*} | a_{m+1} + a_{m+2} + ... + a_{n} | < \epsilon. \end{align*} $$

For the absolute value function definition and other properties see here.
For the definitions of sequences and what it means to for a sequence to converge, see this, for subsequences see this.
For the definitions of series, and what it means to for a series to converge, see this.
For the “show the limit” template and an example, see this.

Formal Proof

For the forward direction, suppose that \(\sum_{k=1}^{\infty} a_k\) converges, we will prove that given \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that

$$ \begin{align*} | a_{m+1} + a_{m+2} + ... + a_{n} | < \epsilon \quad \text{whenever $n > m \geq N$} \end{align*} $$

To so do, we’re given that \(\sum_{k=1}^{\infty} a_k\) converges, this means that the sequence of partial sums \((s_m)\) where \(s_m\) is

$$ \begin{align*} s_m = a_1 + a_2 + ... + a_m, \end{align*} $$

converges. Since the sequence of partial sums converges. Since it converges then by the Cauchy criterion for sequences, this sequence is also Cauchy. Now, let \(\epsilon > 0\). Because the sequence of partial sums is Cauchy, then by definition we can find \(N \in \mathbb{N}\) such that if \(n > m \geq N\), we have \(|S_n - S_m| < \epsilon\). Expand \(|S_n - S_m|\) to see that

$$ \begin{align*} |S_n - S_m| &= |(a_1 + a_2 + ... + a_m + ... + a_n) - (a_1 + a_2 + ... + a_m)| \\ &= |a_{m+1} + ... + a_n|. \end{align*} $$

Which is what we wanted to show. For the backward direction, Suppose that there exists an \(N \in \mathbb{N}\) such that whenever \(n > m \geq N\) it follows that

$$ \begin{align*} | a_{m+1} + a_{m+2} + ... + a_{n} | < \epsilon. \end{align*} $$

This term \(| a_{m+1} + a_{m+2} + ... + a_{n} |\) is equivalent to \(|S_n - S_m|\) and so \(|S_n - S_m| \leq \epsilon\). This means that the sequence of partial sums \((S_m)\) is a Cauchy sequence by definition. Therefore, by the Cauchy criterion for sequences, \((S_m)\) converges as we wanted to show. \(\blacksquare\)
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