# How can I tell if a sequence converges?

Nov 5, 2014

A sequence $\left\{{a}_{n}\right\}$ converges if ${\lim}_{n \to \infty} {a}_{n}$ exists.

A useful result is that, if the terms of the sequence get arbitrarily close as $n$ gets bigger, the sequence is convergent. This is called the Cauchy criterion and the sequence is called a Cauchy sequence.

Written in a more mathematical notation, a sequence is a Cauchy sequence if and only if:

$\forall \epsilon > 0 \exists N : n , m > N \implies | {a}_{n} - {a}_{m} | < \epsilon$

where $\epsilon \in \mathbb{R}$ and $n , m , N \in \mathbb{N}$.

I hope that this was helpful.