Infinity

This article is about the symbol [ilmath]\infty[/ilmath]

Notation

Always qualify [ilmath]\infty[/ilmath] with a [ilmath]+[/ilmath] or [ilmath]-[/ilmath] except where the meaning of [ilmath]\infty[/ilmath] can be unambiguously resolved

Examples

Sequences

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

• [math]\lim_{n\rightarrow\infty}(a_n)[/math] or [math]n\rightarrow\infty[/math]

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

• [math]\lim_{n\rightarrow\infty}(a_n)=-\infty[/math]

  Clearly means the sequence gets more and more negative, tending towards [ilmath]-\infty[/ilmath]

• [math]\lim_{n\rightarrow\infty}(a_n)=+\infty[/math]

  Clearly means the sequence gets more hugely positive, tending towards [ilmath]+\infty[/ilmath]

• [math]\lim_{n\rightarrow\infty}(a_n)=\infty[/math] to mean [math]\lim_{n\rightarrow\infty}(a_n)=+\infty[/math]

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

So we now have 4 behaviours:

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