Definition:Oscillating Sequence

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

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

Let $$\left \langle {x_n} \right \rangle$$ diverge neither to $+\infty$ nor to $-\infty$.

That is, let neither:
 * $$x_n \to +\infty$$ as $$n \to \infty$$, nor
 * $$x_n \to -\infty$$ as $$n \to \infty$$.

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

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