Definition:Periodic Continued Fraction

Periodic Continued Fraction
Let $$\left[{a_1, a_2, a_3, \ldots}\right]$$ be a simple infinite continued fraction.

Let the partial quotients be of the form:
 * $$\left[{r_1, r_2, \ldots, r_m, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, \ldots}\right]$$

that is, ending in a block of partial quotients which repeats itself indefinitely.

Such a SICF is known as a periodic continued fraction.

The notation used for this is $$\left[{r_1, r_2, \ldots, r_m, \left \langle{s_1, s_2, \ldots, s_n}\right \rangle}\right]$$, where the repeating block is placed in angle brackets.

Purely Periodic Continued Fraction
A purely periodic continued fraction is a SICF whose partial quotients are of the form:
 * $$\left[{\left \langle{s_1, s_2, \ldots, s_n}\right \rangle}\right]$$.

That is, all of its partial quotients form a block which repeats itself indefinitely.

Cycle
The repeating block in a periodic (or purely periodic) continued fraction is called the cycle of the SICF.