Definition:Value of Continued Fraction

Definition
The value of a finite continued fraction is the number which results from calculating out the fractions.

Note that formally the continued fraction and its value are considered to be distinct; a continued fraction is an arithmetic representation of its value.

However, this is a nicety of interpretation and may usually be ignored &mdash; $x = \left[{a_1, a_2, a_3, \ldots, a_n}\right]$ is often used to mean that $x$ is the value of the given continued fraction.

Infinite continued fraction
The value of an infinite continued fraction is the limit of its sequence of convergents, if it exists.

Also see

 * Value of Simple Infinite Continued Fraction, where it is shown that the sequence of convergents of a simple infinite continued fraction does indeed converge to a limit.