Definition:Unbounded Divergent Sequence/Real Sequence/Infinity

Definition

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

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

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

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.

Examples

Example: $\paren {-1} n$

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

$a_n = \paren {-1} n$

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

However, $\sequence {a_n}$ is neither divergent to $+\infty$ nor divergent to $-\infty$.