 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]}
Contents
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 6digit 3.14159; the fourth convergent 355/113 = 3.14159292 is more accurate than the 6digit decimal.
By the determinant formula it appears that the successive convergents A_{k}/B_{k} of a regular continued fraction are connected by the formula
This implies, in particular, that the greatest common divisor (A_{k}, B_{k}) = 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
where the a_{i} and the b_{i} are integers. The a_{i} are the partial numerators of the continued fraction x. The b_{i} are the partial denominators, and the ratios a_{i} / b_{i} 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.
See also
Notes
 ^ Nechaev, V.I. (2001), "Continued fraction", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/C/c025540.htm
Categories: Continued fractions
 Mathematical analysis
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