Search results
Results from the WOW.Com Content Network
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).
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 ...
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
By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.
As in the binary search technique for generating the Stern–Brocot tree, the Farey sequences can be constructed using mediants: the Farey sequence of order n + 1 is formed from the Farey sequence of order n by computing the mediant of each two consecutive values in the Farey sequence of order n, keeping the subset of mediants that have ...
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the ...
The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction. [3] [4] [5] For the first definition, one has to consider also the semiconvergents. [1]