Infinity

This article is about the symbol $\infty$

Notation

Always qualify $\infty$ with a $+$ or $-$ except where the meaning of $\infty$ can be unambiguously resolved

Examples

Sequences

Consider a sequence of reals $(a_n)_{n=1}^\infty\subset\mathbb{R}$ then the statement:

• $$\lim_{n\rightarrow\infty}(a_n)$$ or $$n\rightarrow\infty$$

  is not ambiguous as $n$ can only get bigger one way (as it's a natural number) we implicitly mean $+\infty$ here. This is fine.

• $$\lim_{n\rightarrow\infty}(a_n)=-\infty$$

  Clearly means the sequence gets more and more negative, tending towards $-\infty$

• $$\lim_{n\rightarrow\infty}(a_n)=+\infty$$

  Clearly means the sequence gets more hugely positive, tending towards $+\infty$

• $$\lim_{n\rightarrow\infty}(a_n)=\infty$$ to mean $$\lim_{n\rightarrow\infty}(a_n)=+\infty$$

  is wrong as this is a great notation for divergence, for example the sequence $a_n=(-1)^nn$ diverges

So we now have 4 behaviours:

Behaviour	Writing	Reading
Convergence	$$\lim_{n\rightarrow\infty}(a_n)=a$$	The sequence $a_n$ (tends towards|converges) to $a$
	$$\lim_{n\rightarrow\infty}(a_n)=+\infty$$	The sequence $a_n$ (tends toward|converges) to [positive] $\infty$
	$$\lim_{n\rightarrow\infty}(a_n)=-\infty$$	The sequence $a_n$ (tends toward|converges) to negative $\infty$
Divergence	$$\lim_{n\rightarrow\infty}(a_n)=\infty$$	The sequence $a_n$ diverges