📂Lemmas

Continued Fraction

Definition

A fraction of the form as shown below is called a continued fraction.

$$a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{a_{3} + \dfrac{1}{\ddots + \dfrac{1}{a_{n}}}}}} \tag{1}$$

Explanation^{1 2}

$(1)$ is denoted as $[a_{1}, a_{2}, \dots, a_{n}]$.

Naturally, one can also consider taking the limit of it. For example, let’s consider a sequence with the recurrence relation given by $a_{n+1} = 1 + \dfrac{1}{1 + a_{n}}$ and $a_{1} = 1$. Then, the following holds: $$a_{n} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{\ddots 2 + \dfrac{1}{1 + a_{1}}}}}$$

Since the limit of $a_{n}$ is $\sqrt{2}$, $\sqrt{2}$ can be written as $[1, 2, 2, 2, \dots]$.

$$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \ddots}}} \tag{2}$$

The same form as the right-hand side of $(2)$ is called the continued fraction expansion of $\sqrt{2}$.