Exercise 7.3.1 (Golden ratio)

Question

Evaluate the infinite continued fraction $\langle 1,1,1,1,\ldots \rangle$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Put $	heta = \langle 1,1,1,1,\ldots angle$. Write $	heta = \langle a_0,a_1,a_2,a_3,\ldots angle$, where all $a_i = 1$. By Lemma 7.8,
$$	heta =\langle a_0,a_1,a_2,a_3,\ldots angle = a_0 + \frac{1}{\langle a_1,a_2,a_3,\ldots angle},$$
so
$$	heta =\langle 1,1,1,1,\ldots angle = 1 + \frac{1}{\langle 1,1,1,1,\ldots angle} = 1 + \frac{1}{	heta}.$$
Therefore
$$	heta^2 - 	heta - 1 = 0.$$
Thus $	heta = \frac{1 \pm \sqrt{5}}{2}$, where $	heta >0$, so
$$	heta = \frac{1 + \sqrt{5}}{2}.$$
\end{proof}

Exercise 7.3.1 (Golden ratio)

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Exercise 7.3.1 (Golden ratio)

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