# Definition:Unbounded Divergent Sequence

## Definition

### Real Sequence

Let $\sequence {x_n}$ be a sequence in $\R$.

### Divergent to Positive Infinity

$\sequence {x_n}$ diverges to $+\infty$ if and only if:

$\forall H \in \R_{>0}: \exists N: \forall n > N: x_n > H$

That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will exceed $H$.

We write:

$x_n \to +\infty$ as $n \to \infty$

or:

$\displaystyle \lim_{n \mathop \to \infty} x_n \to +\infty$

### Divergent to Negative Infinity

$\sequence {x_n}$ diverges to $-\infty$ if and only if:

$\forall H \in \R_{>0}: \exists N: \forall n > N: x_n < -H$

That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will be less than $-H$.

We write:

$x_n \to -\infty$ as $n \to \infty$

or:

$\displaystyle \lim_{n \mathop \to \infty} x_n \to -\infty$

### Divergent to Infinity

Consider the case where $\sequence {x_n}$ is both unbounded above and unbounded below.

$\sequence {x_n}$ diverges to $\infty$ if and only if:

$\forall H > 0: \exists N: \forall n > N: \size {x_n} > H$

### Complex Sequence

As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, there is no concept of $-\infty$ in discussions relating to $\C$.

So we can use only the following definition:

Let $\sequence {z_n}$ be a sequence in $\C$.

Then $\sequence {z_n}$ tends to $\infty$ or diverges to $\infty$ if and only if:

$\forall H > 0: \exists N: \forall n > N: \cmod {z_n} > H$

where $\cmod {z_n}$ denotes the modulus of $z_n$.

We write:

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