Definition:Unbounded Divergent Sequence/Real Sequence/Positive 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:
- $\forall H \in \R_{>0}: \exists N: \forall n > N: x_n > H$
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 $+\infty$.
Also see
- Definition:Divergent Real Sequence to Negative Infinity
- Definition:Unbounded Divergent Real Sequence
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.26$: Divergent Sequences