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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Are these continued fractions of integrals known?

Motivated by this paper on polynomial continued fractions (Bowman, 2000), I thought about various extensions to current definitions of these fractions. What if we defined a continued fraction such that the numerator of each fraction is the integral…
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Limit of a sequence of integrals involving continued fractions

The following question was asked in a calculus exam in UNI, a Peruvian university. It is meant to be for freshman calculus students. Find $\lim_{n \to \infty} A_n $ if $$ A_1 = \int\limits_0^1 \frac{dx}{1 + \sqrt{x} }, \; \; \; A_2 = …
James
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The most complex formula for the golden ratio $\varphi$ that I have ever seen. How was it achieved?

I am fascinated by the following formula for the golden ratio $\varphi$: $$\Large\varphi = \frac{\sqrt{5}}{1 + \left(5^{3/4}\left(\frac{\sqrt{5} - 1}{2}\right)^{5/2} - 1\right)^{1/5}} - \frac{1}{e^{2\pi\,/\sqrt{5}}}\,\mathop{\LARGE…
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What's the formula for this series for $\pi$?

These continued fractions for $\pi$ were given here, $$\small \pi = \cfrac{4} {1+\cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} {2+\ddots}}}} = \sum_{n=0}^\infty \frac{4(-1)^n}{2n+1} = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} +…
Tito Piezas III
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Are there simple algebraic operations for continued fractions?

I thought about continued fractions as a cool way to represent numbers, but also about the fact they are difficult to treat because standard algebraic operations like addition and multiplication don't work on them in a simple way. My question is: do…
Blex
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a conjectured continued-fraction for $\displaystyle\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\cot\left(\frac{z\pi}{4z+2n}\right)$ is…
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Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$
E.H.E
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Coefficients of binomial continued fractions

For a natural number $n$, let $$ \begin{equation} \beta_n(z)=\frac{(1+z)^n+(1-z)^n}{(1+z)^n-(1-z)^n}. \end{equation} $$ Then the coefficients of the numerator and denominator of $\beta_n$ are binomial. For example:…
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Continued Fraction and Random Variable

Let $X_1,X_2,\dots$ are independent r.v. such that $P(X_i=1)=p=1-P(X_i=\epsilon_i)$, $0<\epsilon_i<1$ $$Y=X_1+\frac{X_1}{X_2+\frac{X_2}{X_3+\frac{X_3}{X_4+\dots}}}$$ 1.What is the distribution of $Y$? 2.What is the characteristic function of…
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Is there an irrational number containing only $0$'s and $1$'s with continued fraction entries less than $10$?

The number $$0.10111001001000000000001$$ has continued fraction $$[0, 9, 1, 8, 9, 5, 1, 1, 5, 3, 1, 3, 1, 1, 4, 6, 1, 1, 8, 2, 5, 8, 1, 9, 9, 5, 2 , 8, 1, 1, 6, 4, 1, 1, 3, 1, 3, 5, 1, 1, 5, 9, 8, 1, 9]$$ So, the maximum values is $9$. But we are…
Peter
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why is $ 2 = \frac{5}{1+\frac{8}{4+\frac{11}{7 + \frac{14}{10 + \dots}}} } $

Why is $ 2 = \cfrac{5}{1+\cfrac{8}{4+\cfrac{11}{7 + \cfrac{14}{10 + \ddots}}} } $ where the sequences $5,8,11,14,\dots$ and $1,4,7,10,\dots$ are of the form $5 + 3 n$ and $1 + 3n$. (This converges on both even and uneven iterates) I was surprised…
mick
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Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
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Exact value for the continued fraction of $\tiny 1+\cfrac{1}{3+\cfrac{3}{5+\cfrac{5}{7+\cfrac{7}{9...}}}}$?

Does anyone know the exact value for the continued fraction of $$1+\cfrac{1}{3+\cfrac{3}{5+\cfrac{5}{7+\cfrac{7}{9+\ddots}}}}?$$ I already know that $$1+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{9\ddots}}}}=\frac{e^2+1}{e^2-1},$$ but I only…
HarryXiro
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Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\cfrac { 1 }{ 1+\cfrac { 1 }{ 2+\cfrac { 1 }{ 3+\cfrac { 1 }{ 4+\ddots } } } } $$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I couldn't make out whether it converges or diverges. Can…
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Continued fraction estimation of error in Leibniz series for $\pi$.

The following arctan formula for $\pi$ is quite well known $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\tag{1}$$ and bears the name of Madhava-Gregory-Leibniz series after its discoverers. The formula has an easy proof via…
Paramanand Singh
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