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

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.

Since all the partial numerators in a regular continued fraction are equal to unity we can adopt a shorthand notation in which the continued fraction shown above is written as

where, in the second line, a vinculum marks the repeating block. Some textbooks use the notation

where the repeating block is indicated by dots over its first and last terms.

If the initial non-repeating block is not present – that is, if k = -1, a₀ = a_m and

the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction ${\displaystyle [1;1,1,1,\dots ]}$ of the golden ratio φ is purely periodic, while the regular continued fraction ${\displaystyle [1;2,2,2,\dots ]}$ of ${\displaystyle {\sqrt {2}}}$ is periodic, but not purely periodic. However, the regular continued fraction ${\displaystyle [2;2,2,2,\dots ]}$ of the silver ratio ⁠${\displaystyle \sigma ={\sqrt {2}}+1}$ is purely periodic.