Definition:Periodic Continued Fraction

Definition

Let $\sqbrk {a_1, a_2, a_3, \ldots}$ be a simple infinite continued fraction.

Let the partial denominators be of the form:

$\sqbrk {r_1, r_2, \ldots, r_m, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, \ldots}$

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

Such a SICF is known as a periodic continued fraction.

The notation used for this is $\sqbrk {r_1, r_2, \ldots, r_m, \sequence {s_1, s_2, \ldots, s_n} }$, where the repeating block is placed in angle brackets.

Purely Periodic Continued Fraction

A periodic continued fraction is a purely periodic continued fraction if its partial denominators are of the form:

$\sqbrk {\sequence {s_1, s_2, \ldots, s_n} }$

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

Cycle

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