Definition:Unbounded Divergent Sequence/Real Sequence/Infinity
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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 $-\infty$.