Editing Limit of a sequence (section)

===Infinite limits===

A sequence <math>(x_n)</math> is said to '''tend to infinity''', written
:<math>x_n \to \infty</math>, or 
:<math>\lim_{n\to\infty}x_n = \infty</math>,
if the following holds:
:For every real number <math>K</math>, there is a natural number <math>N</math> such that for every natural number <math>n \geq N</math>, we have <math>x_n > K</math>; that is, the sequence terms are eventually larger than any fixed <math>K</math>.

Symbolically, this is:
:<math>\forall K \in \mathbb{R} \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies x_n > K \right)\right)\right)</math>.

Similarly, we say a sequence '''tends to minus infinity''', written
:<math>x_n \to -\infty</math>, or
:<math>\lim_{n\to\infty}x_n = -\infty</math>,
if the following holds:
:For every real number <math>K</math>, there is a natural number <math>N</math> such that for every natural number <math>n \geq N</math>, we have <math>x_n < K</math>; that is, the sequence terms are eventually smaller than any fixed <math>K</math>.

Symbolically, this is:
:<math>\forall K \in \mathbb{R} \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies x_n < K \right)\right)\right)</math>.

If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence <math>x_n=(-1)^n</math> provides one such example.