# What is meant by a divergent sequence?

Aug 30, 2015

A divergent sequence is a sequence that fails to converge to a finite limit.

#### Explanation:

A sequence ${a}_{0} , {a}_{1} , {a}_{2} , \ldots \in \mathbb{R}$ is convergent when there is some $a \in \mathbb{R}$ such that ${a}_{n} \to a$ as $n \to \infty$.

If a sequence is not convergent, then it is called divergent.

The sequence ${a}_{n} = n$ is divergent. ${a}_{n} \to \infty$ as $n \to \infty$

The sequence ${a}_{n} = {\left(- 1\right)}^{n}$ is divergent - it alternates between $\pm 1$, so has no limit.

We can formally define convergence as follows:

The sequence ${a}_{0} , {a}_{1} , {a}_{2} , \ldots$ is convergent with limit $a \in \mathbb{R}$ if:

$\forall \epsilon > 0 \exists N \in \mathbb{Z} : \forall n \ge N , \left\mid {a}_{n} - a \right\mid < \epsilon$

So a sequence ${a}_{0} , {a}_{1} , {a}_{2} , \ldots$ is divergent if:

$\forall a \in \mathbb{R} \exists \epsilon > 0 : \forall N \in \mathbb{Z} , \exists n \ge N : \left\mid {a}_{n} - a \right\mid \ge \epsilon$

That is ${a}_{0} , {a}_{1} , {a}_{2} , \ldots$ fails to converge to any $a \in \mathbb{R}$.