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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Why does this infinite repeating fraction seem to be equal to 2 different values?
Let's start how to find the value of the infinite repeating fraction $$1 + \cfrac{2}{1 + \cfrac{2}{1 + \cdots}}$$
We set the entire expression to $x$, and we notice that the entire expression under ...
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Sum of two continued fractions
Prove the equality
$$\cfrac{{1 + \cfrac{{1 + \cfrac{{1 + \cfrac{{1 + \cfrac{{...}}{{18}}}}{{14}}}}{{10}}}}{6}}}{2} + \cfrac{1}{{1 + \cfrac{2}{{1 + \cfrac{4}{{1 + \cfrac{6}{{1 + \cfrac{8}{{...}}}}}}}}}}...
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Simplifying a complicated continued fraction expression.
This probably relates to continued fractions or numerical methods but I do not see how.
There is probably some kind of induction or telescoping.
$$\sqrt 3 = 1+\dfrac{2+\dfrac{3+\dfrac{4+\cdots}{5+\...
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Continued Fraction with Arithmetic Sequence Convergence [closed]
I'm trying to figure out what the following series of continued fractions $c_m$ will converge to as I take the limit of $m\rightarrow \infty$.
$$c_m = a_1-\frac{1}{a_2-\frac{1}{a_3-\cdots\frac{1}{a_m}}...
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How to explain it ? Difference between two continued fraction very small
Well I find it as a coincidence but how to explain :
$$\frac{1}{1+\frac{a}{1+\frac{a^{2}}{1+\frac{a^{3}}{1+\frac{\cdot\cdot\cdot}{a^{2n}}}}}}-\frac{1}{1+\frac{b}{1+\frac{b^{2}}{1+\frac{b^{3}}{1+\frac{\...
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Continued fraction inequality with well know constant as $\pi$ and golden ratio.
I found this inequality beautiful let me share it :
Let :
$$d=\frac{1}{1+\frac{a}{1+\frac{2a^{2}}{1+\frac{3a^{3}}{1+\cdot\cdot\cdot}}}}+\frac{1}{1+\frac{b}{1+\frac{2b^{2}}{1+\frac{3b^{3}}{1+\cdot\...
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Showing that there exists infinitely many rational numbers $\frac{p}{q}$ satisfying $|x - p/q|\leq \frac{1}{q^{2+t}}$ for irrational $x$ and $t>0$
Let $t>0$ be a fixed parameter and $x\in\mathbb{R}$ be an irrational number. I am trying to show that there exists infinitely many rational numbers $\frac{p}{q}$ such that $\left|x - \frac{p}{q}\...
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Negative continued fraction with maximal numerator
For integers $a_1,\dots,a_n \geq 2$, let $[a_1,\dots,a_n]$ denote the 'negative' continued fraction $$[a_1,\dots,a_n]=a_1-\dfrac{1}{a_2-\dfrac{1}{\cdots -\dfrac{1}{a_n}}}.$$
We would have $[a_1,\dots,...
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For Which $a$ In $[0,1]^n$ Is $\text{frac}(ta)$ Maximally Aperiodic
I know that there is no 'nice' solution to this question, but I'm just curious if there exists a solution at all...
Let $c$ be the $n$-cube $[0,1]^n$.
Given a vector $a\in c$, we can look at the value ...
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Taylor Series and Continued Fractions
I was looking into applications of continued fractions in Physics, I found that it is possible to find a rational estimate of an irrational function; which I guess can be used for creating bode plots ...
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Define $f\left(x\right) = \frac{x}{x + \frac{x}{x + \frac{x}{x + \frac{x}{x + \vdots}}}}$. What is $f'\left(x\right)$, the derivative?
Since $f\left(x\right)$ is indeed an infinitely deep continued fraction, I have seen that $f\left(x\right) = \frac{x}{x + f\left(x\right)}$, but taking the derivative of both sides from there has not ...
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How do you properly notate $\frac{1}{x+\frac{1}{x+\frac{1}{x+...}}}$?
I want to write $\frac{1}{x+n}$ where $n=\frac{1}{x+n}$. My current solution is $\frac{1}{x+\frac{1}{x+\frac{1}{x+...}}}$. Is this a proper notation?
I noticed that this is also equal to $\frac{x(...)+...
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Find the Lagrange number of a given infinite simple continued fraction.
This question arises in Ch. 2 of Martin Aigner's beautiful book Markov's Theorem and 100 years of the uniqueness conjecture (page 36, Remark 2.4). These definitions are from Aigner's book summarized ...
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Numbers Whose Multiples Don't Get Near To Each Other Modulo 1
Let $d(x,y):=\min(x-y-\lfloor x-y\rfloor,y-x-\lfloor y-x\rfloor)$ (think about the geodesic distance of points on the circle), for each real number $a$ there is the monotone sequence $(\min_{0\le i<...
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Constructions for finding two squares which sums to a prime
This question is related to Efficiently finding two squares which sum to a prime.
The following 3 methods are found in Chapter 5.3 of The Higher Arithmetic by H. Davenport. However, I failed to ...
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Program computing the Hirzebruch-Jung continued fraction
For relatively prime positive integers $a>b>1$, it is known that there are uniquely determined integers $a_1,\dots,a_n\geq2$ such that $$\frac{a}{b}=a_1-\frac{1}{a_2-\frac{1}{\cdots-\frac{1}{a_n}...
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Hurwitz Test: continued fraction expansion not working
I'm trying to make a Hurwitz fraction expansion of $\;p(s) = s⁴ + s³ + 3s² + 3s +2 \;$ , to check the stability, but can't see what I'm missing. I followed the instructions given in this note, so I ...
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Cube roots and Continued fractions
I once read that it was not known that for any third degree real number if for its continued fraction (c0; c1, c2, c3, ...) all the iterates are bounded or not. Is this still an open question? I ...
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Proving that there are only "finitely" many irreducible fractions that are sufficiently close to a given rational number.
Here is the question in my elementary number theory class:
Given a rational number $q$ , prove that there are only finitely many pairs of coprimes $(a,b)$ such that $|\,q-\frac{a}{b}\,|<\frac{1}{b^...
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Show that the following function is infinitely continuously differentiable
$$
f(x) = \begin{cases}
e^\frac{1}{x} &\text{für } x<0 \\
0 &\text{für } x \geq 0
\end{cases}
$$
Problem: I know that to first I have to show that the function is ...
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Is there a simple way to interpolate smoothly between levels of a complex-valued continued fraction?
I have two complex numbers, $a = x_1 + y_1 i$ and $b = x_2 + y_2 i$. These serve as inputs to an infinite continued fraction of the form $f_n = a + \frac{b}{f_{n - 1}}$, with $f_1 = a$. Thus the first ...
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How can I determine the reason for the mismatch between the value of this infinite continued fraction and the expected result?
How can I determine the reason for the mismatch between the value of this infinite continued fraction and the expected result?
$$6.9431070487= \lim\limits_{n\rightarrow\infty}7- \frac{7}{129- \dfrac{...
user1135362
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A proof about convergence of $\prod_{n=1}^{\infty} (1-a_n)$, given $\sum_{n=1}^{\infty} a_n$ convergence. [duplicate]
Hypothesis
$a_n > 0$, for all $n\ge 1$.
$\sum_{n=1}^{\infty} a_n$ converges.
Why I am asking
I'm reading Continued Fractions by A. Ya. Khinchin. This statement helps to prove the convergence of ...
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Is the maximum entry in the simple continued fraction $6$?
I came across the sum $$S:=\large \sum_{j=1}^{\infty} \frac{1}{2^{2^j}}$$ which I think is a Liouville-number and therefore transcendental (is this correct ?). To clarify : the denominator is $2^{2^j}$...
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Can some continued fraction with all terms less than 4 represent some fraction that has any given denominator (except 6)?
Consider fractions with denominator q such that $\frac{p}{q}$ < 1 and p & q are coprime, e.g. q = 10: $\frac{1}{10}$, $\frac{3}{10}$, $\frac{7}{10}$, $\frac{9}{10}$. Now take the continued ...
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Splitting a rectangle into MxN pieces, with target aspect ratio
Note: I have an "okay" solution to this problem (see below), but I'm wondering if there's a better algorithm, since mine is fairly conservative.
The problem:
We have a rectangle with a some ...
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How do I know that a continued fraction of an irrational number converges?
I am familiar with the algorithm to calculate the continued fraction of an irrational number, but I was wondering how we can be sure that the fraction actually converges to the irrational we are ...
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Does anyone know where I can get the continued fraction representation of the Lambert W function?
Wikipedia's image for the representation
I was just wondering if anyone has an extended version of it. (Like a longer fraction so it is more accurate)
I need to get further than the 94423 term in
x/(1+...
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A computation with continued fractions
I am puzzled with the following property on continued fractions. Let's say I denote $x=(n_0, n_1, ...)$ the negative continued fraction (with minuses), where the $n_i > 1$ for $i \geq 1$. If I let $...
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Resolving problem in proof of Theorem 1 in V. Sós "On the distribution mod 1 of the sequence $n\alpha$"
I am reading V. Sós' "On the distribution mod 1 of the sequence $n\alpha$" which can be found on page 127 of its journal issue. This is one of the early proofs of the "three gap ...
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Reversing the coefficients of the continued fraction expansion of a rational number
On this Wikipedia page about the Markov constant, there's a surprising theorem. Basically if $\alpha$ is a real number with continued fraction expansion $[a_0;a_1,a_2,...]$, the claim is that the ...
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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'...
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Is there a constant for the proportion of $1$s in the SCF of a "general" real number, analogous to the Khinchin's constant?
SCF stands for Simple Continued Fraction.
In other words, how much among all numbers in "almost all" real numbers' SCF expansions will be "1".
If it is, I think it's a(nother) ...
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Measuring how good the best possible rational approximation of a real number is
The Markov constant of a real number $r$ is
$$
\limsup_{d \to \infty} \frac{1}{|r-n/d|d^2}
$$
where we choose the best possible $n$ for each corresponding $d$, e.g. $n = \text{round}(r\cdot d)$.
This ...
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Diophantine approximation theory and "logarithmic" rather than "linear" error
There is an enormous amount of literature on Diophantine approximations, including the general theory of continued fractions, the Stern-Brocot tree, the notion of "badly approximable number" ...
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How well approximated by rationals are almost all real numbers?
Let $\psi:\Bbb{N}\to\Bbb{R}_{\geq 0}$ be such that $q\mapsto q\psi(q)$ is a (weakly) decreasing function. We say a real number $x$ is $\psi$-approximable if $|qx-p|<\psi(q)$ has solutions for ...
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this recursive expression is less or equal to zero at some point?
Let $N \in \mathbb{N}$ where $N \geq 3$, $r \in \left(\frac{2N}{N+2},2\right]$ and $p-1 \in \left(1,\frac{N+2}{N-2}\right)$. Define for each $j \in \mathbb{N}$ recursively the expressions:
$$\frac{1}{...
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How does this theorem on continued fractions relate to the Riemann-Stieltjes integral?
I'm going to include two precursor sections here to introduce all of the content referenced in my actual question (if you're familiar with both continued fractions and the Riemann-Stieltjes integral, ...
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Proving Rate of Convergence
I am investigating the following coupled sequence:
\begin{align*}
y_0 &= 1\\
x_{n+1} &= \sqrt{1 + \frac{1}{y_n}}\\
y_{n+1} &= \sqrt{1 - \frac{1}{x_{n+1}}}\\
\end{align*}
I am trying to ...
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p-smooth numbers that are adjacent on the Stern-Brocot tree
An interesting question from looking at the Stern-Brocot tree:
Two rational numbers, call them $a/b$ and $c/d$, are "adjacent" on the Stern-Brocot tree if and only if we have $ad-bc = \pm 1$,...
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Does this number have degree $>2$ over $\mathbb{Q}(i)$?
I’m thinking about the number $x=r \alpha e^{2i\pi \theta}$ where
$$
r = \sqrt{\frac{\sqrt{2}}{3-\sqrt{3}}},\quad \alpha = 1-\frac{\sqrt{3}}{2} + \frac{i}{2},\quad \theta = \frac{1}{48}.
$$
Each one ...
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Relationship between Gauss–Legendre quadrature and continued fractions
Wikipedia says the following:
Carl Friedrich Gauss was the first to derive the Gauss–Legendre quadrature rule, doing so by a calculation with continued fractions in 1814.
https://en.wikipedia.org/...
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How do we prove that the simple continued fraction for $e^{2/n}=[1;\frac{n-1}{2},6n,\frac{5n-1}{2},1,1,...]$?
Motivation: remarkably, the simple continued fraction - which is unique - for $e^{1/n},e^{2/n}$ is known for every $n\in\Bbb N$. The expansions for $e^{3/n}$, or even $e^{p/q}$, are not known in ...
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When should continued fraction expansions which start from an original value, converge to that same value? $\tan x=x/1-x^2/3-x^2/5-\dots$
$\newcommand{\K}{\operatorname{\large\mathcal{K}}}$I am asking about a very common practice in proofs, that I see online, concerning continued fractions. There is an implicit assumption which I’d like ...
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How does one come up with the continued fraction for the arctangent? $\arctan x=\frac{x}{1+}\frac{x^2}{3+}\frac{(2x)^2}{5+}\cdots$
$\newcommand{\K}{\operatorname{\large{K}}}$The question has already been asked here. However, I find the accepted answer unsatisfactory since it details all the parts which are, in my opinion, trivial,...
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How did Euler obtain this ratio of infinite series from a continued fraction, the terms of the series not being equal to the convergents?
If one feels disinclined to read the contextual preamble, I have made some partial progress in clarifying the question. Skip to the bottom! The motivation behind this is to understand the derivation ...
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Rewriting continued fraction for spheroidal eigenvalue function $\lambda_{n,m}(z)$
I know little about spheroidal functions and browsing Wolfram functions, the Spheroidal Eigenvalue function $\lambda_{n,m}(z)$ was intriguing. According to the DLMF section 30.3(iii):
$$b_p-\lambda-\...
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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]{...
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A property of the convergents of the continued fraction expansion of a rational number
I'm looking for a proof of the following result (theorem 6.14 of the book Cryptography. Theory and practice by Paterson and Stinson):
Theorem 6.14 Suppose that $\text{gcd}(a,b) = \text{gcd}(c,d) = 1$ ...
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Find a perfect set without rationals.
Give an example of a perfect set in $\mathbb R$ that does not contain any of the rationals.
I found out a proof here using continued fractions. I have trying to understand the proof.
In continued ...
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