Periodic continued fraction

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

${\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {1}{a_{k+1}+{\cfrac {\ddots }{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\ddots }}}}}}}}}}}}}}}}}}$

where the initial block ${\displaystyle [a_{0};a_{1},\dots ,a_{k}]}$ of k+1 partial denominators is followed by a block ${\displaystyle [a_{k+1},a_{k+2},\dots ,a_{k+m}]}$ of m partial denominators that repeats ad infinitum. For example, ${\displaystyle {\sqrt {2}}}$ can be expanded to the periodic continued fraction ${\displaystyle [1;2,2,2,...]}$.

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 a_i (i ≥ 1) are positive integers. The general case, where the partial denominators a_i are arbitrary real or complex numbers, is treated in the article convergence problem.