Definition:Unbounded Divergent Sequence/Real Sequence

Definition
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:
 * $\forall H > 0: \exists N: \forall n > N: \left|{x_n}\right| > H$.

Also see

 * Definition:Divergent to Infinity/Complex Sequence


 * Definition:Divergent Sequence
 * Definition:Infinite Limit at Infinity