Periodic and Purely Periodic Continued Fractions

Definition: An infinite continued fraction $\langle a_0; a_1, a_2, ... \rangle$ is said to be Periodic if there exists an $n \in \mathbb{N}$ such that $a_{n+r} = a_r$ for all $r$ sufficiently large and $n$ is called the Period of the infinite continued fraction.

An example of a periodic infinite continued fraction is:

(1)

\begin{align} \quad \langle 1; 2, 5, 2, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, ... \rangle \end{align}

Note that $n = 2$, and if $r \geq 4$, then we see that $a_{2 + r} = a_r$.

Definition: An infinite continued fraction $\langle a_0; a_1, a_2, ... \rangle$ is said to be Purely Periodic if there exists an $n \in \mathbb{N}$ such that $a_{n+r} = a_r$ for all $r$.

Purely periodic infinite continued fractions are periodic starting at $a_1$. If a purely periodic infinite continued fraction has period $n$ then we can use the following notation to denote it:

(2)

\begin{align} \quad \langle a_0; a_1, a_2, ..., a_n, a_1, a_2, ..., a_n, a_1, a_2, ..., a_n, ... \rangle = \langle a_0; \overline{a_1, a_2, ..., a_n} \rangle \end{align}