Cauchy's Condition for Convergent Series

# Cauchy's Condition for Convergent Series

One very important result regarding convergent series of real numbers is called Cauchy's condition which we state and prove below.

Theorem 1 (Cauchy's Condition for Convergent Series): The series $\sum_{n=1}^{\infty} a_n$ converges if and only if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $\displaystyle{\mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid < \epsilon}$ for each $p = 1, 2, ...$. |

**Proof:**$\Rightarrow$ Suppose that $\sum_{n=1}^{\infty} a_n$ converges to $s$. Then the sequence of partial sums for this series, $(s_n(x))_{n=1}^{\infty}$ converges to $s$. But every convergent sequence is Cauchy, and so for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then:

\begin{align} \quad \mid s_m - s_n \mid < \epsilon \end{align}

- So so $n$ and let $m = n + p$ for $p = 1, 2, ...$. Then $m, n \geq N$ and:

\begin{align} \quad \mid s_{n +p} - s_n \mid = \biggr \lvert \sum_{k=1}^{n+p} a_k - \sum_{k=1}^{n} a_k \biggr \rvert = \mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid < \epsilon \end{align}

- $\Leftarrow$ Suppose that for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $\displaystyle{ \mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid < \epsilon}$ for each $p = 1, 2, ...$. Choose this same $N$, and let $n, m \geq N$ where $m(p) = n + p$ for $p = 1, 2, ...$. Then for all $m, n \geq N$ we see that:

\begin{align} \quad \mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid = \mid s_m - s_n \mid < \epsilon \end{align}

- So the sequence of partial sums $(s_n(x))_{n=1}^{\infty}$ is Cauchy. But this is a series of real numbers, so the corresponding sequence of partial sums are real numbers and $(\mathbb{R}, d)$ is a complete metric space so $(s_n(x))_{n=1}^{\infty}$ converges in $\mathbb{R}$ which implies that the series $\displaystyle{\sum_{k=1}^{\infty} a_n}$ converges. $\blacksquare$