Periodic Continued Fraction

In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form

$x = a_0 + cfrac{1}{a_1 + cfrac{1}{a_2 + cfrac{ddots}{quadddotsquad a_k + cfrac{1}{a_{k+1} + cfrac{ddots}{quadddotsquad a_{k+m-1} + cfrac{1}{a_{k+m} + cfrac{1}{a_{k+1} + cfrac{1}{a_{k+2} + cfrac{1}{ddots}}}}}}}}},$

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.

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