Exercise 7.3.3 (Some infinite continued fractions)

Question

Evaluate the infinite continued fractions:

\begin{tabular}{ll}
(a) $\langle 2,2,2,2,\ldots angle$; &(b)  $\langle 1,2,1,2,1,2,\ldots angle$;\
(c)  $\langle 2,1,2,1,2,1,\ldots angle$; &(d)  $\langle 1,3,1,2,1,2,1,2,\ldots angle$.
\end{tabular}

Richard Ganaye · Accepted Answer

\begin{proof} 
Only with Sagemath this time!
\begin{verbatim}
age: cf = continued_fraction([(),(2,)]); cf
[(2)*]
sage: cf.value()
sqrt2 + 1
sage: cf = continued_fraction([(),(1,2)]); cf
[(1, 2)*]
sage: cf.value()
1/2*sqrt3 + 1/2
sage: cf = continued_fraction([(),(2,1)]); cf
[(2, 1)*]
sage: cf.value()
sqrt3 + 1
sage: cf = continued_fraction([(1,3),(1,2)]); cf
[1; 3, (1, 2)*]
sage: cf.value()
-sqrt3 + 3
\end{verbatim}

This gives
\begin{align*}
\langle 2,2,2,2,\ldots angle &= 1 + \sqrt{2},\
\langle 1,2,1,2,1,2,\ldots angle &= \frac{1+ \sqrt{3}}{2},\
\langle 2,1,2,1,2,1,\ldots angle &= 1 + \sqrt{3}\
\langle 1,3,1,2,1,2,1,2,\ldots angle &= 3 - \sqrt{3}.
\end{align*}
\end{proof}

(a) $⟨ 2, 2, 2, 2, \dots ⟩$ ;	(b) $⟨ 1, 2, 1, 2, 1, 2, \dots ⟩$ ;
(c) $⟨ 2, 1, 2, 1, 2, 1, \dots ⟩$ ;	(d) $⟨ 1, 3, 1, 2, 1, 2, 1, 2, \dots ⟩$ .

Exercise 7.3.3 (Some infinite continued fractions)

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Exercise 7.3.3 (Some infinite continued fractions)

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