List questions from continued-fractions

Published on 25 Mar 2022 - 19:45

368

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How is this cube root continued fraction generated?

Published on 09 Apr 2022 - 19:14

267

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A pattern of periodic continued fraction

Published on 26 Mar 2026 - 7:43

243

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Prove that $p/q$ is a convergent of the continued fraction expansion of $x$ given $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$

Published on 18 Apr 2022 - 11:06

680

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why is $ 2 = \frac{5}{1+\frac{8}{4+\frac{11}{7 + \frac{14}{10 + \dots}}} } $

Published on 25 Apr 2022 - 11:24

81

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Write $\sqrt{x}$ as a continued fraction, where $x=((4a^2+1)b+a)^2+4ab+1$

Published on 28 Apr 2022 - 22:01

119

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A pattern that leads to regular continued fractions of quadratic irrationals

Published on 05 May 2022 - 16:21

160

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Expressing the continued fraction $[k,\dots,k]$ as a closed form

Published on 18 May 2022 - 7:06

62

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How to calculate this limit related to continued fractions?

Published on 26 May 2022 - 10:02

138

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Why does this simple continued fraction evaluate to a modified Bessel function on WolframAlpha? (without proof)

Published on 28 Mar 2026 - 21:06

171

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How do I show convergence of continued fraction?

Published on 22 Jun 2022 - 15:22

53

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Rational approximation / gear ratios / watchmaking

Published on 28 Jun 2022 - 6:01

83

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How to explain this result about $\pi$ and $\arctan$ function?

Published on 28 Jun 2022 - 16:35

74

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What would the cubic formula be if roots never existed?

Published on 23 Feb 2026 - 8:38

30

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What's the relationship between the numerators and denominators of convergents of a continued fraction?

Published on 06 Jul 2022 - 5:25

List Question

Finding Integer Approximations

How is this cube root continued fraction generated?

A pattern of periodic continued fraction

Prove that $p/q$ is a convergent of the continued fraction expansion of $x$ given $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$

why is $ 2 = \frac{5}{1+\frac{8}{4+\frac{11}{7 + \frac{14}{10 + \dots}}} } $

Write $\sqrt{x}$ as a continued fraction, where $x=((4a^2+1)b+a)^2+4ab+1$

A pattern that leads to regular continued fractions of quadratic irrationals

Expressing the continued fraction $[k,\dots,k]$ as a closed form

How to calculate this limit related to continued fractions?

Why does this simple continued fraction evaluate to a modified Bessel function on WolframAlpha? (without proof)

How do I show convergence of continued fraction?

Rational approximation / gear ratios / watchmaking

How to explain this result about $\pi$ and $\arctan$ function?

What would the cubic formula be if roots never existed?

What's the relationship between the numerators and denominators of convergents of a continued fraction?

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