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The sequence {} is then called an exceptional sequence for the continued fraction. See Chapter 2 of Lorentzen & Waadeland (1992) for a rigorous definition. There also exists a notion of absolute convergence for continued fractions, which is based on the notion of absolute convergence of a series: a continued fraction is said to be absolutely ...
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 identity connecting a finite sum with a finite continued fraction in such a way that the extension ...
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence {} of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated ) continued fraction like
The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy. Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic. [4]
and so forth. Notice how the fractions derived as successive approximants to √ 2 appear in this geometric progression. Since 0 < ω < 1, the sequence {ω n} clearly tends toward zero, by well-known properties of the positive real numbers.
Farey sequences are very useful to find rational approximations of irrational numbers. [15] For example, the construction by Eliahou [16] of a lower bound on the length of non-trivial cycles in the 3x+1 process uses Farey sequences to calculate a continued fraction expansion of the number log 2 (3).
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