Definition:Periodic Continued Fraction

Definition
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.