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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.

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Infinite continued fraction for $\pi$ [closed]

Recently I have found the following infinite continued fraction for $\pi$ : $$\pi=4-\cfrac{2}{1+\cfrac{1}{1- \cfrac{1}{1+\cfrac{2}{ 1-\cfrac{2 }{ 1+\cfrac{3}{ 1-\cfrac{3 }{\cdots}}}}}}}$$ Where can I ...
poliagapitos's user avatar
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The measure of a set of continued fractions. [duplicate]

What would be the measure of the set of continued fractions which only contain 1 or 2 in the denominator, including infinite fractions? If I understand measure correctly, the set of continued ...
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Continued fractions convergents proofs

Given a continued fraction, let $ \frac{p_n}{q_n} $ and $ \frac{p_{n+1}}{q_{n+1}} $ be two consecutive convergents. The proof states: $ |x - \frac{p_n}{q_n}| \leq \frac{2}{2q_n} $ Using the ...
Vibhatsu's user avatar
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Divergence of the generalized continued fraction $1+ \frac{-1\mid}{\mid1}+\frac{-1\mid}{\mid1}+\frac{-1\mid}{\mid1}+\dots$

I learned about continued fraction, and I wondered whether this form of fraction can be a complex number. $$1-\cfrac1{1-\cfrac1{1-\cfrac1{1-\cfrac1\ddots}}}$$ This cannot be convergent, but I can't ...
1112박시후's user avatar
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If $y$ is a quadratic irrational, then there exists $M > 0$ such that $\forall p, q$, $q > 0$, $\left| \frac{p}{q} - y \right| > \frac{M}{q^2}$.

Suppose $y$ is a quadratic irrational. I want to show that there exists $M>0$ such that for all $p, q$, $q>0$, $$\left| \frac{p}{q} - y \right| > \frac{M}{q^2}.$$ I'd like a reminder on how ...
Robin's user avatar
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Reason for property of convex decreasing sequences of reciprocals of positive integers; property defined with greedy algorithm

Let $a_0 < a_1$ be positive integers. Define recursively the sequence $(a_n)$ so that $a_{n+1}$ is the greatest positive integer, if one exists, such that $\left( \frac{1}{a_{n-1}}, \frac{1}{a_n}, \...
Adam Rubinson's user avatar
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Natural extension of Gauss measure (continued fractions)

There's a Gauss measure on $[0,1]$ given by its density $$f(x)=\frac 1{\ln 2} \frac{1}{1+x} $$ It's invariant under Gauss map $x\mapsto\big\{\frac 1 x\big\}$ which acts as a left shift if we describe ...
Big Coconut's user avatar
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Faster continued fraction of $\sqrt{x}$

Some continued fractions converge very fast. Example However, the convergence of the famous continued fraction of $\sqrt{2}$, $$a_1 = 1, a_{n + 1} = 1 + \frac{1}{1 + a_n}$$ is linear. $$|a_{n+1}-\sqrt{...
Hayatsu's user avatar
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Infinite nested radical and fractions [duplicate]

By dealing with infinite radicals I'm pretty sure that we always choose the positive roots, as an example: $\displaystyle\sqrt{1+\displaystyle\sqrt{1+...} }$ $\displaystyle\sqrt{1+x}=x$ $x^2=x+1$ This ...
The king Lucfier's user avatar
5 votes
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Do continued fractions always converge?

Say I have a continued fraction in this form: $$a_0 + \dfrac{b_0}{a_1 + \dfrac{b_1}{a_2 + \dfrac{b_2}{\cdots}}}$$ Does it always converge? If not, how can I know if it does? An example that comes to ...
Elvis's user avatar
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Convergents of continued fraction for quadratic implied by an inequality

Assume that $d$ is a positive squarefree integer and $p$ and $q$ are positive integers such that $$-\sqrt{d}+\tfrac{1}{2} < p^2-dq^2 \leq \sqrt{d}.$$ I want to show that $p/q$ is a convergent in ...
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Closed expression of a specific splitting of a continued fraction

Recently in my research I stumbled upon this splitting of a periodic continued fraction. I wondered whether there is any closed expression or literature on this topic. Visualizing continued fractions ...
Bindajoba's user avatar
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Continued fraction of Laplace transform

I first learned of the below identity from MathWorld and the works of Ramanujan, but it's completely crazy with polygammas and Laplace transforms of hyperbolic trig. It seems weird that the Laplace ...
Michael Duffy's user avatar
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On the distribution of odd/even length symmetric continued fractions.

In this question it is shown that if a continued fraction of some rational $\frac{p}{q}=[a_0; a_1, \dots, a_n]$ is symmetrical (i.e. $a_k = a_{n-k}$) then $q^2 \equiv(-1)^n \pmod{p} $. A converse is ...
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Is the sum of the continued fraction sequence / number of steps in Euclid algorithm a metric?

It seems to me that the sum of the continued fraction sequence minus one can serve as a metric on positive rational numbers. Given a positive rational number $q=[a_0, a_1, ...]=a_0+\frac{1}{a_1+\dots}$...
Alexandre's user avatar
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Continuous analogue of the discrete simple continued fraction

Background The classical Riemann integral of a function $f : [a,b] \to \mathbb{R}$ can be defined by setting $$\int_{a}^{b} f(x) \ dx := \lim_{\Delta x \to 0} \sum f(x_{i}) \ \Delta x. $$ Here, the ...
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What is meant by "q-generalisation"?

I was reading Prof Gaurav Bhatnagnar, "How to prove Ramanujan q-continued fractions", on the first page he mentions: $$\text{the q-generalisation of } \\ 1+1+1 \cdots + 1 = n \\ \text{is } ...
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Method to difference between two continuous fractions and keep continued fraction form

I warmly welcome an approach on differencing between two continuous fractions Without applying the appropriate algebra to reduce the continuous fraction. I cannot find a way without completing the ...
Nishi's user avatar
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1 vote
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The Equivalence Transformation (??) for Generalized Continued Fractions

The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/...
Lucien Jaccon's user avatar
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Continued Fraction Representation of sin(x)

To provide context, the continued fraction in the form $\frac{a_0}{1-\frac{a_1}{1+a_1-\frac{a_2}{1+a_2-...}}}$ evaluated to the $n$th denominator equals $\sum_{k=0}^{n}\prod_{j=0}^{k}a_j$. If one ...
Lucien Jaccon's user avatar
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If $\sqrt{5}<k(\alpha)<29 / 10$, show that there exists $N \in \mathbb{N}$ such that $a_n=2$ for all $n > N$, and conclude that $k(\alpha)=2 \sqrt{2}$

I need help solving this number theory exercise related with best aproximation by continued fractions. Im using BROCHERO, F., MOREIRA, C.G., SALDANHA, N., TENGAN, E. – Teoria dos números – um passeio ...
Arkantos9's user avatar
1 vote
1 answer
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Representing the continued fraction of phi as inversion in a circle

Does the continued fraction of the golden ratio (phi) have visual representations in inversive or projective geometry? I was looking at the circles of Apollonius represented as harmonic conjugates in ...
Svenn's user avatar
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Another weird limit involving gamma and digamma function via continued fraction

Context : I want to find a closed form to : $$\lim_{x\to 0}\left(\frac{f(x)}{f(0)}\right)^{\frac{1}{x}}=L,f(x)=\left(\frac{1}{1+x}\right)!×\left(\frac{1}{1+\frac{1}{1+x}}\right)!\cdots$$ Some ...
Miss and Mister cassoulet char's user avatar
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Continued fraction sequence converges [duplicate]

I'm stuck in the following problem from Lima's "Análise Real": Given $a > 0$, recursively define the sequence $(x_n)$ by letting $x_1 = 1 / a$ and $x_{n + 1} = 1/(a + x_n)$. Consider the ...
Henrique Fonseca's user avatar
6 votes
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Does the continued fraction $\frac{2}{3 - \frac{2}{3 - \ddots}}$ have two values?

I was trying to prove that $$1 = \frac{2}{3 - \frac{2}{3 - \dots}}$$ It's clear that $$\begin{aligned} & & x = \frac{2}{3 - \frac{2}{3 - \dots}} \\ & \Rightarrow & x = \frac{2}{3 - x} \...
SchellerSchatten's user avatar
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1 answer
502 views

Closed form of integral $\int_{0}^{\infty}\frac{e^{-x}}{1+x/\cdots}dx$ with continued fraction

Problem: Let $$f(x)=\frac{1}{1+\frac{x}{1+\frac{x}{\cdots}}}$$ Then does $$\int_{0}^{\infty}e^{-x}f(x)dx=:I$$ have a closed form ? The first step in the Continued Fraction involves not surprisingly ...
Miss and Mister cassoulet char's user avatar
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Understanding the proof of a transcendental criterion regarding palindromic continued fraction.

I am trying to understand the proof of theorem 2.1 from the following paper: https://aif.centre-mersenne.org/item/10.5802/aif.2306.pdf. Basically the authors prove that given $\alpha=[0;a_1,a_2,\dots,...
WiggedFern936's user avatar
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1 answer
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Series involving continued fractions

For an irrational $\alpha>1$, define the sequence $\alpha_n$ such that $\alpha_0=\alpha$ and for $n\geq 1$, $\alpha_n=\frac{1}{\alpha_{n-1}-\lfloor\alpha_{n-1}\rfloor}$. We can also define the ...
Alan Abraham's user avatar
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A ratio of Bessel functions with a similar continued fraction expansion to $e^{1/m}\,$?

This 2021 post discusses the continued fraction, $$F_4=1+\cfrac{1/1}{1+\cfrac{1/3}{1+\cfrac{1/5}{1+\cfrac{1/7}{1+\ddots}}}}= \frac{2}{ 1+\displaystyle\frac{I_1(1/4)}{I_0(1/4)}}=1.7793\dots$$ where $...
Tito Piezas III's user avatar
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Given that x has the continued fraction expansion x = [1,$\bar{s}$], show that $ x = 1+ \frac{2}{s+\sqrt{4+s^2}}$

Let x be a real number and have the continued fraction expansion $x = [1,\bar{s}]$, where s is a natural number and $\bar{s}$ denotes the infinite string $s, s, \ldots $ Let $y = [s, \bar{s}]$ and ...
Colonel Armfeldt's user avatar
3 votes
2 answers
122 views

Other "natural" simple continued fractions with closed-forms?

Given the simple continued fraction, $$x =b_0 + \cfrac{1}{b_1 + \cfrac{1}{b_2 + \cfrac{1}{b_3 + \ddots}}} $$ for some well-defined sequence of positive integers $b_k$ for $k>0$. More compactly, for ...
Tito Piezas III's user avatar
4 votes
2 answers
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Natural continued fraction

My question concerns the following sequence : $$u_0=0+\frac{1}{2} \quad u_1 =0+\frac{1}{2+\frac{3}{4}} = \frac{4}{11} \quad u_2=0+\frac{1}{2+\frac{3}{4+\frac{5}{6}}}=\frac{29}{76} \quad u_3=0+\frac{1}{...
alati ahmad's user avatar
3 votes
1 answer
388 views

Functional equation: $f(x) = x + 1 + \frac{f(x+1)}{f(x+2)}$ , $f(t) = 0$?

Consider $$f(x) = x+1+\dfrac{x+2+\dfrac{x+3+\dfrac{x+4+\cdots}{x+5+\cdots}}{x+4+\dfrac{x+5+\cdots}{x+6+\cdots}}} {x+3+\dfrac{x+4+\dfrac{x+5+\cdots}{x+6+\cdots}}{x+5+\dfrac{x+6+\cdots}{x+7+\cdots}}} $$ ...
mick's user avatar
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3 votes
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Proof $x^2-dy^2=-1$ is unsolvable in integers if $\sqrt{d}$ has continued fraction expansion of even period

I want to prove that the negative Pell's equation $$x^2-dy^2=-1$$ is unsolvable in integers if $\sqrt{d} = [a_0, \overline{a_1, \dots, a_n}]$ with $n$ even and minimal (minimal since all $\sqrt{d}$ ...
Robin's user avatar
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-1 votes
1 answer
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A new Dottie number via Continued Fraction?

It's a bit fancy equation . Let $x>0$ : Then define the CF in even times: $$f\left(x\right)=\frac{1}{x+\frac{\cos\left(x\right)}{x+\frac{2\cos\left(\cos\left(x\right)\right)}{x+\frac{3\cos\left(\...
Miss and Mister cassoulet char's user avatar
1 vote
0 answers
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Continued Fraction of the Square Root of the Ratio of Two Primes $\equiv 3 \mod 4$

I have been playing around with continued fractions, and I have noticed that if $p,q$ are two primes $\equiv 3 \mod 4$, then the continued fraction representation of $\sqrt{\frac{p}{q}}$ has an even ...
poeplva's user avatar
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Trouble understanding Khinchin's proof of the recursion formulae $p_k=a_kp_{k-1}+p_{k-2};q_k=a_kq_{k-1}+q_{k-2}$ for continued fractions

I am trying to understand a particular conundrum that has been bothering me in Khincin's proof of his Theorem 1 in his book Continued Fractions. My precise question is at the end of this post marked ...
Cartesian Bear's user avatar
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Stuck at showing recursion property $q_n=c_nq_{n-1}+q_{n-2}$ of the continuants $q_n$ of $x$'s continued fraction expression $x=\frac{p_n}{q_n}$.

(Setting:) Take it as granted that every real number can be expressed as a continued fraction $x = [c_0;c_1, c_2,\dots]$ for $c_0;c_1,c_2,\dots \in \mathbb{Z}$ with $x = c_0 + \frac{1}{c_1 + \frac{1}{...
Cartesian Bear's user avatar
3 votes
1 answer
154 views

An infinite continued fraction of $x^n$

Below this answer @WeiZhong commented that it is wrong, I also think it is wrong. Reducing each numerator to $1$: $$f(x)=\cfrac{x}{x+\cfrac{x^2}{x^2+\cfrac{x^3}{x^3+\lower2ex\ddots}}}\\ =0+\cfrac{1}{1+...
hbghlyj's user avatar
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0 votes
1 answer
90 views

Deriving the continued fraction for Pi [closed]

So I was searching online for methods to approximate Pi and found this continued fraction that supposedly approximates to Pi when continued infinitely. I've tried searching all over the internet for ...
Isshin's user avatar
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7 votes
1 answer
408 views

Matrix and Taylor Expansion of a Finite Continued Function

I noticed a pattern between matrix and Taylor series of a finite continued fraction function. However, I don't know how to prove it or why they are related. Let $$ f_{1}(z)=\frac{1}{-z-1} $$ $$ f_{2}(...
Apple's user avatar
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0 answers
180 views

Exercise for student about $y\simeq P$

Inspired by On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$? I made an exercise for the student ...
Miss and Mister cassoulet char's user avatar
1 vote
0 answers
51 views

Does the rest of this family of continued fractions have closed forms?

The pattern for the continued fractions below is quite straightforward. $F_1$ has numerators with all the integers but, $F_2\; \text{is missing}\; 2m+1 = 3,5,7,\dots\\ F_3\; \text{is missing}\; 3m+1 = ...
Tito Piezas III's user avatar
2 votes
1 answer
108 views

largest term in simple continued fraction of $\sqrt{n}$

Let $n$ be a positive integer that is not a perfect square. It is well known that the simple continued fraction of $\sqrt{n}$ is of the form $$\sqrt{n} = [a_0;\overline{a_1,a_2,\ldots,a_{k-1},2a_0}],$...
Jesse Elliott's user avatar
9 votes
2 answers
329 views

On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$?

We have Ramanujan's well-known, $$\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots\color{blue}+\,\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\...
Tito Piezas III's user avatar
6 votes
1 answer
262 views

Continued fraction of $\frac{\prod_{k=1}^{p-1}(2^k-1)}{2^p-1}$ for prime p.

Investigating the q-factiorals. And found experimentally (at least frist 1000 primes) that the continued fraction of $$\frac{\prod_{k=1}^{p-1}(2^k-1)}{2^p-1}$$ for prime p has level 3 and the last ...
Gevorg Hmayakyan's user avatar
4 votes
0 answers
167 views

Expressing Ramanujan's $\sqrt{\frac{\pi\,e}{2}}$ as $two$ continued fractions

Due to a recent comment by Akiva about this post, I decided to revisit Ramanujan's beautiful continued fraction (plus series) relating $\pi$ and $e$, $$\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\...
Tito Piezas III's user avatar
1 vote
1 answer
56 views

Convert Rational Function to Finite Continuted Functions.

If we have a finite continued rational functions, it is easy to clear the demoninator and write as a rational function (quotient of polynomials). For example,$$ \frac{1}{x+\frac{1}{x+\frac{1}{x+1}}}=\...
Apple's user avatar
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Reverse Calculate Continued Fraction of Log, to get Antilog.

Log2(5) is 2^x = 5. The integer part would be 2 because 2^2 = 4. And what's left is 2^(2+x) = 5. So 2^x = 5/4 We know that x actually has to be between 2 and 3 without being 3 itself because 2^3 is 8. ...
HayOne's user avatar
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0 answers
69 views

Number of elements in a continued fraction

When we work with rational numbers, our continued fraction will have a finite number of elements. Are there ways to estimate the number of elements of a continued fraction when expanding a rational ...
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