Exercise 7.3.2 (Evaluate $\langle 2,1,1,1,1,\ldots \rangle$ and $\langle 2,3,1,1,1,1,\cdots \rangle$)

Question

Evaluate the infinite continued fractions $\langle 2,1,1,1,1,\ldots \rangle$ and $\langle 2,3,1,1,1,1,\cdots \rangle$.

Richard Ganaye · Accepted Answer

\begin{proof} 
By Lemma 7.8 and Problem 1,
\begin{align*}
\langle 2,1,1,1,1,\ldots angle &= 2 + \frac{1}{\langle 1,1,1,1,\ldots angle}\
&= 2 + \frac{1}{\left( \frac{\sqrt{5} + 1}{2} ight)}\
&= \frac{2+\sqrt{5} + 4}{\sqrt{5} + 1}\
&= \frac{3 + \sqrt{5}}{2}.
\end{align*}
With Sagemath:
\begin{verbatim}
sage: cg = continued_fraction([(2,),(1,)]);cg
[2; (1)*]
sage: cg.value()
1/2*sqrt5 + 3/2
\end{verbatim}

Similarly,
\begin{align*}
\langle 2,3,1,1,1,1,\cdots angle&= 2 + \frac{1}{3+ \frac{1}{\langle 1,1,1,1,\cdots angle}}\
&= 2 + \frac{1}{3+ \frac{1}{\left( \frac{\sqrt{5} + 1}{2} ight)}}\
&= \frac{7\sqrt{5} + 11}{3 \sqrt{5} + 5}\
&= \frac{5}{2} - \frac{1}{10} \sqrt{5}.
\end{align*}

Verification with Sagemath:
\begin{verbatim}
sage: K.<sqrt5> = QuadraticField(5)
sage: x = 5/2 - (1/10) * sqrt5
sage: x.continued_fraction()
[2; 3, (1)*]
\end{verbatim}
\end{proof}

Exercise 7.3.2 (Evaluate $\langle 2,1,1,1,1,\ldots \rangle$ and $\langle 2,3,1,1,1,1,\cdots \rangle$)

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Exercise 7.3.2 (Evaluate $\langle 2,1,1,1,1,\ldots \rangle$ and $\langle 2,3,1,1,1,1,\cdots \rangle$)

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