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Periodic continued fraction. In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form. where the initial block of k +1 partial denominators is followed by a block of m partial denominators that repeats ad infinitum. For example, can be expanded to the periodic continued fraction .
A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . A continued fraction is a mathematical expression that can be writen as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple ...
Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ...
An infinite periodic continued fraction is a continued fraction of the form = + + + + + + where k ≥ 1, the sequence of partial numerators {a 1, a 2, a 3, ..., a k} contains no values equal to zero, and the partial numerators {a 1, a 2, a 3, ..., a k} and partial denominators {b 1, b 2, b 3, ..., b k} repeat over and over again, ad infinitum.
However, the periodic representation does not derive from an algorithm defined over all real numbers and it is derived only starting from the knowledge of the minimal polynomial of the cubic irrational. [5] Rather than generalising continued fractions, another approach to the problem is to generalise Minkowski's question-mark function.
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 ...
Such periodic orbits have an equivalent periodic continued fraction, per the isomorphism established above. There is an initial "chaotic" orbit, of some finite length, followed by a repeating sequence.
Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of a 0 through a k+m. Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.
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