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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Only one solution proofs [closed]

I have $2$ questions in my homework that I can not figure out how to deal with: $1.$ Let $(a,b,c)\in \mathbb{R}^3$, $a>1$ and $0<b \leq 1$. Prove that $a x=b \sin x+c$ has one solution. For ...
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What is the algorithm for performing continued fraction arithmetic

I am trying to write a python package for doing exact arithmetic with continued fractions, I've been looking for a good while now but can't find any good reference anywhere. I've already read gosper's ...
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1answer
72 views

Explanation of continued fraction for Bessel functions

Doing some computational searches, I found some nice continued fractions that could be used to compute Bessel functions of the first kind (I and J): ...
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Patterns in convergents of continued fraction of $\sqrt{D}$?

First, to give some background: If $D$ is an integer, then the continued fraction of $\sqrt{D}$ is always periodic. For example, the continued fraction of $\sqrt{7}$ is $[2; \overline{1,1,1,4}]$. Also,...
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2answers
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Infinite fraction's derivative [closed]

$f(x)=x+\dfrac{1}{x+\dfrac{1}{x+\dfrac{1}{x+\ldots}}}$ $f'\left(\dfrac{3}{2}\right)=?$ I tried to make equation like $y^2=xy+1$ but I can't made it clearly.
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Possible pattern involving $x$ in the continued fraction expansion of $\frac{1}{\sqrt[3]{x^3+1}-x}$

Consider the expression $$\frac{1}{\sqrt[3]{x^3+1}-x}$$ Plugging in $10$ for $x$ and using $W|A$, we find that the continued fraction expansion is $[300; \mathbf{10}, 450, 8, ...]$. For $x=11$, it's $[...
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1answer
82 views

Mirrored Continued fractions

I know how to evaluate the Continued Fraction $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\ddots}}}$ and its dual version: $1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\cfrac{4}{1+\ddots}}}}$ I also understand $\...
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1answer
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(Exponential) Mixing for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of the mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$ ...
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1answer
358 views

What rational numbers are expressible as “multiple-of-$12$” continued fractions?

I need to use a special subclass of continued fraction expansions for a problem I'm studying. Namely, expansions of the form $$[a_0;a_1;a_2;\dotsc; a_{n}] = a_0+\displaystyle\frac{1}{a_1+\displaystyle\...
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1answer
70 views

Fraction with the smallest entries

This question was asked in INMO 2005 and here is the problem: $Let$ $(a,b)$ $\in$ $\mathbb{N}$ $s.t$ $\cfrac{43}{197}$ $\lt$ $\cfrac{a}{b}$ $\lt$ $\cfrac{17}{77}$. $\text{Find}$ $\min(b)$. My first ...
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Generalized continued fraction and irrationality

We adopt the notation of the continued fraction, for $a_i,b_i\in\mathbb Z_{>0}$, $$\dfrac{a_1}{b_1\pm}\dfrac{a_2}{b_2\pm}\dfrac{a_3}{b_3\pm}\dots:=\dfrac{a_1}{b_1\pm\dfrac{a_2}{b_2\pm\dfrac{a_3}{...
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3answers
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Closed Continued Fraction

I am trying to find a closed form of a continued fraction: $$[1,2,3,5,3,5,3,5...]$$ I understand I have to form a quadratic of some sort and factorise and to give me an irrational number but I am not ...
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240 views

Continued fraction representation of quadratic irrationals

I just learned Continued Fractions and I was asked to evaluate the simple continued fractions $[\bar{1}]$ , $[\bar{2}]$ , and $[1,\bar{2}]$ so far all I know about Quadratic Irrationalities and ...
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1answer
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Convergence of a finite continued fraction for $b_i \in [-1,0]$ and $a_i=1$

based on wiki page here, finite continued fraction is as follows: $$a_0+\cfrac{b_0}{a_1+\cfrac{b_1}{\ddots+\cfrac{\ddots}{a_{n-1}+\cfrac{b_{n-1}}{...}}}}$$ I want to find the limit of finite continued ...
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1answer
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How to simplify the finite continued fraction?

I want to simplify the following expression for $q_i$ and $p_i$, where $i \in \{1,2,...,n \}$ and $q_i \in [0,1]$ and $p_i \in [0,1] $ , is there any standard method for it? For example, based on ...
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1answer
487 views

Continued fraction for Apéry's constant conjectured by The Ramanujan Machine

Recently the following identity was conjectured by The Ramanujan Machine: $$ \frac{8}{7\zeta(3)}=1-\frac{u_1}{v_1-\frac{u_2}{v_2-\frac{u_3}{v_3-\ddots}}}, $$ where $u_n=n^6$ and $v_n=(2n+1)(3n^2+3n+1)...
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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 ...
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Length of continued fractions

Consider, for any pair of relatively prime positive integers $0 <j <n$, the expansion as a continued fraction of the quotient $\frac{n}{j}$ $$ \frac{n}{j}=b_1-\frac{1}{b_2-\dfrac{1}{\cdots-\...
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1answer
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Finding continued fractions of a root of polynomial (Solution verification)

I was trying to solve this problem from my number theory class. I think the solution had a typo, so I wanted to verify my understanding. (Just to make sure I'm not missing something and mistaking a ...
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3answers
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Proof for a theorem about relations beween convergents in continued fractions

Upon reading about some properties of numerators and denominators in a textbook called Continued Fractions (here, chapter 2.3), I was unable to understand the following transmutation of the expression ...
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2answers
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Prove that any rational number is positive $\frac{m}{n}$ Can be displayed in the form of a “continued fraction”

"Continued fraction" is an expression of the form $[a_{0},a_{1},a_{2}...,a_{n}]= > a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+_{\frac{1}{...+}}}}}$ $a_{0},a_{1},a_{2}...,a_{n}$ ...
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1answer
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How can Zaremba's Conjecture be true if it fails for $d=1$?

In the following photo is the definition of some sets and the conjecture of Zaremba. Am I missing something obvious or does the conjecture not certainly fail as $d=1$ will never be in the set $\...
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Bounded, aperiodic irrationals with bounded, aperiodic sum

For an irrational $q$ with continued fraction expansion $[q_0;q_1,q_2 \dots]$, say $q_i$ is the $i$-th patial quotient of $q$. I have the following question: Are there any nontrivial examples of ...
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Denominators in a continued fraction

I'm doing some "research" for a high school competition, and I stumbled upon a problem I'll describe below. I'll be thankful for any references where can I find any results on this topic – I ...
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Existence of continued fraction $\sqrt{n}$ with any period $k$

In this paper it is conjectured that for any positive integer $k$ there are infinitely many primes $p$ with the continued fraction expansion of $\sqrt{p}$ having length $k$ (Conjecture 5.1, https://...
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Why/How was Hardy impressed with Ramanujan’s work prior to seeing any proofs of their validity?

Hardy’s famous correspondence with Ramanujan highlights an interesting aspect of the mathematics discipline; the appreciation of a mathematical expression prior to knowing its validity. Why/how was ...
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107 views

nth term of $2,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}},2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}},2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}}…$

What is the nth term of the sequence: $$2,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}},2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}},2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}}...$$ in terms of $s_{n-1}$. I ...
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Can I use Euler's formula to prove a series converges to a rational value?

Given the following infinite series with rational $x < 1$ && integer $n \geqslant 2$: $$1-\frac{x}{n}+\frac{(1+n)x^2}{2n^2}-\frac{(1+n)(1+2n)x^3}{6n^3}+\frac{(1+n)(1+2n)(1+3n)x^4}{24n^4}-\...
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Solving infinite continued fraction with natural logarithm.

How would one go about solving the following continued fraction? $$ 1+\cfrac{z}{2+\cfrac{z}{3+\cfrac{2^{2} z}{4+\cfrac{2^{2} z}{5+\cfrac{3^{2} z}{6+\cfrac{3^{2} z}{\ldots}}}}}}=\frac{z}{\ln (1+z)} $$ ...
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Present as a continued fraction

I want represent the fraction $$\frac{a+b+a b c+a b d}{1+a c+b c+a d+b d+a b c d}\qquad\qquad\qquad (1)$$ as a continued fraction. Here $a,b,c,d$ free variables. I could only get $$\frac{a+b+a b c}{1+...
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How do I solve $3=\frac{k}{5+\frac{k}{5+\frac{k}{5+\cdots}}}$ for $k$?

How do I solve this for $k$? $$3=\frac{k}{5+\frac{k}{5+\frac{k}{5+\cdots}}}$$ I know $k$ can be estimated by graphing, but is there a way to solve using algebra? If so, how?
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What is the J-type continued fraction of the generating function of the Fibonacci numbers?

Let \begin{equation} \begin{array}{ll} \displaystyle F(z) &\displaystyle = \ 1 \ + \ z \ + \ 2z^2 \ + \ 3z^3 \ + \ 5z^4 \ + \ \cdots \\ &\displaystyle = \ {1 \over {1 -z - z^2}} \end{array} ...
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1answer
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The Infinitely Nested Radicals Problem and Ramanujan's wondrous formula

In mathematics, a nested-radical is any expression where a radical (or root sign) is nested inside another radical, eg. $\sqrt{2 + \sqrt{3}}$. By extension, an Infinitely nested radical (aka, a ...
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1answer
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continued fraction formula . pls help

I'm self studying this book "Methods of Solving Number theory Problems by Elina" since many days but currently stuck on this formula of continued fractions. For example $a=87/ 55 = [1,1,1,...
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4answers
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Computing continued fraction expansions

My question concerns the numerical accuracy of a continued fraction expansion. A typical algorithm for computing a continued fraction can be written in Python as : ...
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Roth's theorem: contradiction?

Roth's theorem says that for irrational algebraic number $\alpha$ and $\epsilon>0$, there are finitely many solutions to this: $$\displaystyle \left|\alpha-\frac pq\right|<\frac 1{q^{2+\epsilon}}...
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What is the value of $1+\frac{1}{2+\frac{1}{3+\dots}}$?

This is just a curiousity question. HISTORY:- $Motivation:$ I first started wondering about this question about 3 months ago when I first got interested in continued fractions but I really didn't ...
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Convergence criteria for continued fraction

I have $$ \overline{\Bbb C^2}\ni(x,y)\mapsto\cfrac{\epsilon_{2n+1}}{(x-a_{2n+1})+\cfrac{\epsilon_{2n}}{(y-a_{2n})+\cfrac{\epsilon_{2n-1}}{\dots}}} $$ where $\overline{\Bbb C^2}$ is the one-point ...
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Analytic Continuation of Fractions

Excuse my lack of expertise, I study natural sciences (physics) and not mathematics so I will be off with my terminology and mathematical vocabulary. Please feel free to poke and build at this idea ...
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1answer
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Is there a general way to convert a continued fraction into a recursive formula?

I was looking into this continued fraction: $$e = 3 - \frac{1}{4 - \frac{2}{5 - \frac{3}{6 -...}}}$$ I was able to duplicate the approximations of the continued fraction with this recursive formula: ...
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1answer
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355/113 and small odd cubes

An important approximation to $\pi$ is given by the convergent $\frac{355}{113}$. The numerator and the denominator of this fraction are at the same distance of small consecutive odd cubes. $$\frac{...
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continuous fraction for 30/pi^2

I read that $ 30/\pi^2 $ can be represented as a continued fraction of the form $$ 3 + \frac{1}{25 + \frac{16}{69 + \frac{...}{...}}} $$ over here: http://www.ramanujanmachine.com/wp-content/uploads/...
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2answers
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Irrationality of ${\pi}$ and $e^{x/y}$

I attempt to prove that ${\pi}$ is an irrational number. For this, I use the beautiful continued fraction given by Brouncker who rewrote Wallis' formula as a continued fraction, which Wallis and later ...
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1answer
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Negative continued fraction of Euler Mascheroni constant

$\gamma$, the Euler-Mascheroni constant, has the following simple regular continued fraction: $$\gamma=[0; 1, 1, 2, 1, 2, 1,\dots]=0+ \cfrac{1}{ 1+\cfrac{1}{ 1+\cfrac{1}{ 2+\cfrac{1}{ ...
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1answer
51 views

Rationality of negative continued fractions

I was wondering which is the effect of changing additions by substractions on a simple continued fraction, and its mathematical interpretation (if any). For instance, $$\sqrt{2}=[1;2,2,2,2,\dots]=1+ \...
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1answer
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Solving equation involving finite continued fraction

I'm interested in solving the equation $$ \color{red}{x}=1+\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\ddots \cfrac{a_n}{b_n+\color{red}{x}}}}, $$ where $a_i,b_i$ are positive real numbers. Is there a formula to ...
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2answers
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Evaluate 360 (1/15+1/105+1/315+⋯).

Evaluate $360$ ($\frac{1}{15}$+$\frac{1}{105}$+$\frac{1}{315}$+$\cdots$). My Work :- Well we can clearly see that $15 = 1*3*5$ ; $105 = 3*5*7$ ; $315 = 5*7*9$. So I basically know the pattern, but I ...
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1answer
38 views

Integral part of an infinite simple continued fraction

Let $[a_0,a_1,a_2,\ldots]$ be an infinite simple continued fraction. How can one show that $a_0$ is the integral part of $[a_0,a_1,a_2,\ldots]$? There is a similar theorem for finite continued ...
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Approximation: Understanding a property of continued fractions and convergents

In a paper by Simons & Weger in 2005, they talk about approximating the value of $\delta = \dfrac{\log3}{\log2}$ through continued fraction theory using integers $p$ and $q$. Specifically, they ...
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2answers
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the second order continued fraction expansion of an irrational number and the distance to its closest integer

Let $u$ be a posive irrational number and let $$u=a_0+ \frac{1}{a_1+\frac{1}{a_2 \cdots} }$$ be its continued fraction expansion. Consider the second-order finite continued fraction expansion of $u$: ...

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