Definition:Unbounded Divergent Sequence/Real Sequence
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Definition
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:
- $\ds \lim_{n \mathop \to \infty} x_n \to +\infty$
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:
- $\ds \lim_{n \mathop \to \infty} x_n \to -\infty$
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:
- $\forall H > 0: \exists N: \forall n > N: \size {x_n} > H$
Also see
- Definition:Unbounded Divergent Complex Sequence
- Definition:Divergent Sequence
- Definition:Infinite Limit at Infinity
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.26$: Divergent Sequences