Definition:Divergent To Infinity
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Definition
Real Sequence
Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.
Then $\left \langle {x_n} \right \rangle$ tends to $+\infty$ or diverges to $+\infty$ iff:
- $\forall H > 0: \exists N: \forall n > N: x_n > H$.
That is, whatever (positive) number you pick, for sufficiently large $n$, $x_n$ will exceed $H$.
Similarly, $\left \langle {x_n} \right \rangle$ tends to $-\infty$ or diverges to $-\infty$ iff:
- $\forall H > 0: \exists N: \forall n > N: x_n < -H$.
We write:
- $x_n \to +\infty$ as $n \to \infty$; or:
- $x_n \to -\infty$ as $n \to \infty$.
If we are not concerned about whether it is $+\infty$ or $-\infty$ that $\left \langle {x_n} \right \rangle$ diverges to, we can say:
$\left \langle {x_n} \right \rangle$ tends to $\infty$ or diverges to $\infty$ iff:
- $\forall H > 0: \exists N: \forall n > N: \left|{x_n}\right| > H$.
Complex Sequence
As the Complex Numbers Can Not be Ordered, there is no concept of $-\infty$ in discussions relating to $\C$.
So we can use only the following definition:
Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.
Then $\left \langle {z_n} \right \rangle$ tends to $\infty$ or diverges to $\infty$ iff:
- $\forall H > 0: \exists N: \forall n > N: \left|{z_n}\right| > H$
where $\left|{z_n}\right|$ is the modulus of $z_n$.
We write:
- $x_n \to \infty$ as $n \to \infty$.
