# Definition:Continued Fraction/Infinite

## Definition

Let $F$ be a field, such as the field of real numbers $\R$.

Informally, an infinite continued fraction in $F$ is an expression of the form:

$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n + \cfrac 1 {\ddots}}} }}}$

where $a_0, a_1, a_2, \ldots, a_n, \ldots \in F$.

Formally, an infinite continued fraction in $F$ is a sequence, called sequence of partial quotients, whose domain is $\N_{\geq 0}$.

An infinite continued fraction should not be confused with its value, when it exists.

## Also known as

An infinite continued fraction is often abbreviated ICF.