Periodic continued fraction
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In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form
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where the initial block [a0, a1,...ak] of k+1 partial denominators is followed by a block [ak+1, ak+2,...ak+m] of m partial denominators that repeats ad infinitum. For example, can be expanded to the periodic continued fraction [1,2,2,2,...].
This article considers only the case of periodic regular continued fractions. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers. The general case, where the partial denominators ai are arbitrary real or complex numbers, is treated in the article convergence problem.