Continued fraction

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. For example, $\sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$ . Other continued fractions give formulas for $\pi$ and for functions such as $\tan x$ .

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Tag	`0BK6JBK58W`
URI	`m2://0BK6JBK58W`
Aliases	`generalized-continued-fraction`
Kind	definition
Writing	W1 stated
MSC	11A55
Defines	Continued fraction

Continued fraction

Definition

Remarks

Where this leads

Further reading

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