# Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity

## Definition

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

$\sequence {x_n}$ 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:

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

## Also known as

The statement:

$\sequence {x_n}$ diverges to $+\infty$

can also be stated:

$\sequence {x_n}$ tends to $+\infty$
$\sequence {x_n}$ is unbounded above.

## Examples

### Example: $n^\alpha$

Let $\alpha \in \Q_{>0}$ be a strictly positive rational number.

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$a_n = n^\alpha$

Then $\sequence {a_n}$ is divergent to $+\infty$.

### Example: $2^n$

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$a_n = 2^n$

Then $\sequence {a_n}$ is divergent to $+\infty$.