# Definition:Oscillating Sequence

## Definition

Let $S$ be one of the standard number fields $\Q$, $\R$ or $\C$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $S$.

Let $\left \langle {x_n} \right \rangle$ be divergent.

Suppose $\left \langle {x_n} \right \rangle$ is not divergent to $\infty$.

That is, let:

$\neg x_n \to \infty$ as $n \to \infty$.

Then $\left \langle {x_n} \right \rangle$ is said to oscillate.

## Also see

An example is the sequence $\left \langle {x_n} \right \rangle$ where $x_n = \left({-1}\right)^n$ as demonstrated in Divergent Sequence may be Bounded.