---
tag: 0BK6JBK58W
kind: definition
name: Continued fraction
aliases: [generalized-continued-fraction]
msc: [11A55]
defines: 04P2RJ4WPS
sources:
  - url: https://en.wikipedia.org/wiki/Continued_fraction
    locator: Formulation
---

## Definition

A continued fraction is given by a number $b_0$ and a finite or infinite sequence of pairs $(a_1, b_1), (a_2, b_2), \dots$ of numbers.
It stands for the expression
$$b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}$$
The $a_n$ are the partial numerators and the $b_n$ are the partial denominators.
The $n$-th convergent is the value of the expression cut off after $(a_n, b_n)$. The value of an infinite continued fraction is the limit of its convergents, when this limit exists.

## Remarks

When $a_n = 1$ for all $n$, $b_0$ is an integer and $b_n$ is a positive integer for $n \ge 1$, the continued fraction is a [[simple continued fraction|06XF7BCFFJ]].
For example, $\sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$.
Other continued fractions give formulas for $\pi$ and for functions such as $\tan x$.

## Explanations

### L1 human

A continued fraction is a fraction whose denominator contains another fraction, and so on.
Cut it off at some level, and you get an ordinary fraction. These fractions often approximate a number very well.