# Definition:Continued Fraction/Infinite

< Definition:Continued Fraction(Redirected from Definition:Infinite Continued Fraction)

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## 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**.