Definition:Oscillating Sequence

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

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

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

That is, let:
 * $x_n \not\to \infty$ 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 Sequences may be Bounded.