# Convergent (continued fraction)

﻿
Convergent (continued fraction)

A convergent is one of a sequence of values obtained by evaluating successive truncations of a continued fraction. The nth convergent is also known as the nth approximant of a continued fraction.[1]

## Representation of real numbers

Every real number can be expressed as a regular continued fraction in canonical form. Each convergent of that continued fraction is in a sense the best possible rational approximation to that real number, for a given number of digits. Such a convergent is usually about as accurate as a finite decimal expansion having as many digits as the total number of digits in the nth numerator and nth denominator. For example, the third convergent 333/106 for π (Pi) is roughly 3.1415094, which is not quite as accurate as the 6-digit 3.14159; the fourth convergent 355/113 = 3.14159292 is more accurate than the 6-digit decimal.

By the determinant formula it appears that the successive convergents Ak/Bk of a regular continued fraction are connected by the formula

$A_{k-1}B_k - A_kB_{k-1} = (-1)^k \,$

This implies, in particular, that the greatest common divisor (AkBk) = 1; in other words, each convergent of a regular continued fraction, as given by the fundamental recurrence formulas, is automatically expressed in lowest terms.

More detailed properties of best rational approximations and convergents of π are discussed in the continued fraction article.

## Convergents and convergence

In mathematical analysis a continued fraction is usually written as

$x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4} {b_4 + \ddots\,}}}}$

where the ai and the bi are integers. The ai are the partial numerators of the continued fraction x. The bi are the partial denominators, and the ratios ai / bi are the partial quotients. The convergents of this fraction can be computed by using the fundamental recurrence formulas.

An infinite continued fraction converges if the sequence of convergents approaches a limit. If the sequence of convergents does not approach a limit, the continued fraction is divergent.

Because of the way the partial denominators and partial numerators interact with each other as the successive convergents are calculated, the convergence problem for continued fractions is inherently more difficult than it is for infinite series. The Śleszyński–Pringsheim theorem provides one sufficient condition for convergence.

## Notes

1. ^ Nechaev, V.I. (2001), "Continued fraction", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Continued fraction — Finite continued fraction, where a0 is an integer, any other ai are positive integers, and n is a non negative integer. In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the… …   Wikipedia

• Continued fraction of Gauss — In complex analysis, the continued fraction of Gauss is a particular continued fraction derived from the hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several… …   Wikipedia

• Generalized continued fraction — In analysis, a generalized continued fraction is a generalization of regular continued fractions in canonical form in which the partial numerators and the partial denominators can assume arbitrary real or complex values.A generalized continued… …   Wikipedia

• Euler's continued fraction formula — In the analytic theory of continued fractions, Euler s continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. First published in 1748, it was at first regarded as a simple… …   Wikipedia

• Gauss's continued fraction — In complex analysis, Gauss s continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several …   Wikipedia

• convergent — 1. adjective a) That converges or focuses b) A sequence in a metric space with metric d is convergent to a point , denoted as , if for every there is a natural number N such that for every : . 2. noun the rational number obtained when a …   Wiktionary

• Solving quadratic equations with continued fractions — In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is:ax^2+bx+c=0,,!where a ne; 0.Students and teachers all over the world are familiar with the quadratic formula that can be derived by completing …   Wikipedia

• List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

• Convergence problem — In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ai and partial denominators bi that are sufficient to guarantee the convergence of the continued fraction This… …   Wikipedia

• Methods of computing square roots — There are several methods for calculating the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below. Contents 1 Rough estimation 2 Babylonian method 2.1 Example …   Wikipedia