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 partial denominators that repeats over and over again, ad infinitum. For example can be expanded to a periodic continued fraction, namely as .
The partial denominators {ai} can in general be any real or complex numbers. That general case is treated in the article convergence problem. The remainder of this article is devoted to the subject of regular continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers.
Read more about Periodic Continued Fraction: Purely Periodic and Periodic Fractions, Relation To Quadratic Irrationals, Reduced Surds, Length of The Repeating Block, See Also
Other articles related to "continued, periodic continued fraction, continued fraction":
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... An infinite periodic continued fraction is a continued fraction of the form where k ≥ 1, the sequence of partial numerators {a1, a2, a3.. ... Ak, and Bk are the numerators and denominators of the k-1st and kth convergents of the infinite periodic continued fraction x, it can be shown that x converges to one of the ... If r1 and r2 are finite then the infinite periodic continued fraction x converges if and only if the two roots are equal or the k-1st convergent is ...
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