Definition:Unbounded Divergent Sequence

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

Then $$\left \langle {x_n} \right \rangle$$ tends to $$+\infty$$ or diverges to $$+\infty$$ iff:
 * $$\forall H > 0: \exists N: \forall n > N: x_n > H$$.

That is, whatever (positive) number you pick, for sufficiently large $$n$$, $$x_n$$ will exceed $$H$$.

Similarly, $$\left \langle {x_n} \right \rangle$$ tends to $$-\infty$$ or diverges to $$-\infty$$ iff:
 * $$\forall H > 0: \exists N: \forall n > N: x_n < -H$$.

We write:
 * $$x_n \to +\infty$$ as $$n \to \infty$$; or:
 * $$x_n \to -\infty$$ as $$n \to \infty$$.

If we are not concerned about whether it is $$+\infty$$ or $$-\infty$$ that $$\left \langle {x_n} \right \rangle$$ diverges to, we can say:

$$\left \langle {x_n} \right \rangle$$ tends to $$\infty$$ or diverges to $$\infty$$ iff:
 * $$\forall H > 0: \exists N: \forall n > N: \left|{x_n}\right| < H$$.

Complex Sequence
As the Complex Numbers Can Not be Ordered, there is no concept of $$-\infty$$ in discussions relating to $$\C$$.

So we can use only the following definition:

Let $$\left \langle {z_n} \right \rangle$$ be a sequence in $\C$.

Then $$\left \langle {z_n} \right \rangle$$ tends to $$\infty$$ or diverges to $$\infty$$ iff:
 * $$\forall H > 0: \exists N: \forall n > N: \left|{z_n}\right| > H$$

where $$\left|{z_n}\right|$$ is the modulus of $$z_n$$.

We write:
 * $$x_n \to \infty$$ as $$n \to \infty$$.

Note
Compare the definition for divergent sequence.