Definition:Unbounded Divergent Sequence
Definition
Real Sequence
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:
- $\forall H > 0: \exists N: \forall n > N: \size {x_n} > H$
Complex Sequence
As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, there is no concept of $-\infty$ in discussions relating to $\C$.
So we can use only the following definition:
Let $\sequence {z_n}$ be a sequence in $\C$.
Then $\sequence {z_n}$ tends to $\infty$ or diverges to $\infty$ if and only if:
- $\forall H > 0: \exists N: \forall n > N: \cmod {z_n} > H$
where $\cmod {z_n}$ denotes the modulus of $z_n$.
We write:
- $x_n \to \infty$ as $n \to \infty$.