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Questions tagged [continued-fractions]

A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ai are called the coefficients or terms of the continued fraction.

Links:

Continued Fraction at Wolfram MathWorld

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Adding 1 to each entry of continued fractions

Here we denote $[a_0,...,a_n]$ as the continued fraction of some rational number. If I take $p/q=[a_0,a_1,...,a_n]$ to $p'/q'=[a_0+1,a_1+1,...,a_n+1]$, are there any nice properties I can say about $p'/q'$?
continuedFractionQs
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A theorem about Higher order Continued fractions

I discovered a nice Continued Fraction for Higher order roots. (Obviously, there are other and better ways to approximate a higher order root.) The general theorem is: $\cfrac{\sqrt[n]{A}+1}{\sqrt[n]{A}-1}=1\cdot C+\cfrac{1-(1\cdot n)^2}{3\cdot C…
Paul vdVeen
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A remarkable Continued Fraction for $\pi$

How to analyze the Continued Fraction…
Paul vdVeen
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Evaluation of the 'Odd Harmonic' Continued Fraction

How to prove that this continued fraction $1+\cfrac{1/1}{1+\cfrac{1/3}{1+\cfrac{1/5}{1+\cfrac{1/7}{1+\ddots}}}}$ evaluates to $\displaystyle\frac{2}{\displaystyle 1+\frac{I_1(\frac14)}{I_0(\frac14)}}$? The recurrence relation of these…
Paul vdVeen
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Continued fraction [1; 2, 2, 3, 3, 3, 4, 4, 4, 4....]

Is the continued fraction $\left[1;2,2,3,3,3,4,4,4,4\dots\right]$, where every positive integer $n$ is repeated $n$ times in order starting at $1$, a known value? What properties does it have? I've been unable to find any reference to this number…
hakr14
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How to determine the values of $x$ such that $\frac{1}{x}=[1,1,1,\dots,x]$ with $n$ $1'$s?

I'm not sure I understood the question correctly. In the hope of finding some pattern, I made a code in Mathematica that expands $[1,1,1,\dots,x]$ with $n$ $1$'s and solve that equation, for $n=2,3,4,\dots,20$ I…
Red Banana
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What is a 2-fraction?

I am working with a 4-term recurrence relation coming from the application of the Frobenius method on a ODE, in particular I am studying the convergence of the resulting series. Trying to find out a method to find conditions that impose convergence…
mattiav27
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Infinite Continued Fraction

$$A = a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \cdots}}},$$ where $a_n = f(n), f:\mathbb{Z^+} \rightarrow \mathbb{Z^+}$. Is there an easy way of calculating the value of $A$ using algebra and/or simple calculus for a given $f(n)$? And…
Tahir Imanov
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Continued fraction with odd partial numerators and even partial denominators

Given the continued fraction $$\epsilon=\cfrac{1}{2+\cfrac{3}{4+\cfrac{5}{6+\cfrac{7}{8+\ddots}}}},$$ in https://groups.google.com/forum/#!topic/sci.math/mlZ0VCTUJi8, Robert Israel ensures that…
ksoriano
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Continued fraction of $\frac{1+\sqrt{5}}{2}$

I am trying to get a better understanding of continued fractions (CF) and was watching a view tutorial clips e.g. this here and looking through some stackexchange articles. Than found this article, where someone was able to get the CF of…
PatrickSteiner
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derivative formula related to Rogers-Ramanujan continued fraction

What is the derivative of Rogers-Ramanujan Continued Fraction? In the answer of upper link, Nikos Bagis claimed that $$ R'(q)=5^{-1}q^{-5/6}f(-q)^4R(q)\sqrt[6]{R(q)^{-5}-11-R(q)^5}\textrm{, }:(d1)$$ Where can I get a proof for this formula?
Youngseok Choe
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Solving Quadratics Using Continued Fractions/Nonsimple to Simple Continued Fractions

Let's say we want to find the continued fraction that solves the equation $x^2 - 2x - 1 = 0$. Solution: $$ x = 2 + \frac1x = 2 + \dfrac1{2 + \dfrac1x} = 2 + \dfrac1{2 + \dfrac1{2 + \dfrac1x}} = [2;\overline2] $$ However, what happens if the…
Misha Shklyar
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Formula for the simple reapeted infinitely continued fractions

I was thinking about infinite fractions of the form $1+\frac{a}{1+\frac{a}{1+...}}$ but realised it would be much more useful and satisfying and only a little bit harder to solve fractions of the form $a+\frac{b}{a+\frac{b}{a+...}}$. One thing that…
Aaron Quitta
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Easiest way to find the polynomial satisfied by a continued fraction representation

Given a repeating simple continued fraction $x = [a_{0};a_{1}, \ldots]$, what is the easiest way to find the (quadratic) polynomial satisfied by it? Certainly, you can just do it by hand. For example, the number $x = [\overline{2}]$ satisfies the…
coolpapa
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Continued Fraction pattern

I have been give then continued fraction $\dfrac{1}{\dfrac{1}{\dfrac{1}{x-1}-1}-1}$ If I let x=5, I get the following pattern... $\frac{1}{4}, \, -\frac{4}{3}, \,-\frac{3}{7}, \,-\frac{7}{10}, \,-\frac{10}{17}, \,...$ It appears that (excluding the…
Seth Killian
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