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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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Plotting complicated function on sagemath

I am not sure if this is the right place to ask about coding a program sagemath. But it is the only math online community I know, so I hope I can get some suggestions here. The problem is I want to ...
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1answer
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Understanding Proof about Continued Fraction convergent sequences

I copied a proof from lecture and don't understand the end of it. It is intro number theory on continued fractions. Hopefully someone can explain it to me Background: The sequences {$h_n$} and {$k_n$...
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Prove that this sequence of continued fractions $\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}},\dots$ tends to $1$.

The Problem: I'll write up a couple more terms: $$\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}}, \frac{20}{19+\frac{18}{17+\frac{16}{15+\frac{14}{13}}}}, \frac{30}{29+\...
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Showing that the simple continued fraction of $\sqrt{d}$ has period length 1 iff $d=a^2+1$

Given that I know if $d$ is an integer that $\sqrt{d}=[\alpha_0,\bar{\alpha_1},...\bar{\alpha_n},\bar{2\alpha_0}]$. I want to show that $\sqrt{d}$ has period length 1 if and only if $d=a^2+1$, for ...
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51 views

Clarification on this continued fractions question.

I would rather this not be solved for me , as its a homework question , I just want some clarification on whether I'm understanding it correctly. Say we have the continued fraction $\alpha=[3,\bar{2}...
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49 views

Finding the quadratic polynomial of a continued fraction

Given $\alpha=[3,\bar{2},\bar{4},\bar{5}]=3+\tfrac{1}{2+\tfrac{1}{4+\tfrac{1}{5+...}}}$ I want to (i) find the quadratic integer polynomial P for which $\alpha$ is a root and (ii) hence write $\alpha$...
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Summation Formula for Tangent/Secant Numbers

I came across the following expressions: $$\begin{align} \widehat{S}_{2n} &:= \sum_{1 \leq k_1<\cdots<k_n \leq 2n} \prod_{\ell=1}^n (k_\ell-2\ell)^2, \\ \widehat{T}_{2n+1}&:=\sum_{1 \...
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Limit of continued fraction

Let $(c_n)$ be a sequence of positive numbers converging to $1$ as $n\to\infty$. That is, $c_n > 0$ for every $n$ and $\lim\limits_{n\to\infty}c_n=1$. Let $g_1=-c_1$ and define, for $n \geq 1$,$$g_{...
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What is the significance of the same leading term in a continued fraction expansion? [closed]

The continued fraction expansion for the golden mean $\varphi = \frac{1+\sqrt{5}}{2}$ gives: ContinuedFraction[(1 + Sqrt[5])/2, 50]: $$ 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
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Algorithm to find floors of multiples of the golden ratio

What is an algorithm to calculate $\lfloor n\phi \rfloor$ given some integer $n$, where $\phi$ is the golden ratio? I am thinking the easiest way will involve calculating multiples of its continued ...
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70 views

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, ...
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Confusion while evaluating the complex continued fraction $\frac{1}{i+\frac{1}\ddots}$

I faced some problems when I tried to evaluate the following continued fraction $$\cfrac{1}{i+\cfrac{1}{i+\cfrac{1}{i+\cfrac{1}{i+\cfrac{1}{i+\cfrac{1}{i+\cdots}}}}}}\ $$ The common trick as usual is ...
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Continuous function and limit $ f(x,y)=\left\{\begin{matrix} (x-y)\sin\frac{1}{x}\sin\frac{1}{y} & , xy\neq 0 \\ 0 &, x=y=0 \end{matrix}\right. $

I have this function:$$ f(x,y)=\left\{\begin{matrix} (x-y)\sin\frac{1}{x}\sin\frac{1}{y} & , xy\neq 0 \\ 0 &, x=y=0 \end{matrix}\right. $$ a) Show that $ \lim_{x\rightarrow 0}[\lim_{y\...
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Continued fraction with the given sequence of signs

If $\epsilon_i$ denote a sequence of signs ($\pm1$). Then there is a continued fraction expansion of every real number with $\epsilon_i$ as the $i$-th partial numerator of the continued fraction ...
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Doubt in a solution provided to IMO Shortlist 2013

The following question is present in the IMO shortlist 2013: "Let n be a positive integer and let $a_1, . . . , a_{n-1}$ be arbitrary real numbers. Define the sequences $$u_0, . . . , u_n$$ and $$v_0,...
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arithmetic with continued fraction abbreviated notation?

say we have the string $[4;3,2,2]$ and [3;1,1]. they represents $73/17$ and $7/2$. the product of these numbers is $511/34$, which is $[15;34]$ in continued fraction abbreviated notation form. is ...
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Gauss measure and continued fraction

For $x \in [0,1)$ then the continued fraction representation of $$x=0 + \cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\cfrac{1}{a_3(x)+\cfrac{1}{\dots}}}}$$ can be written as $[0; a_1(x), a_2(x), a_3(x), \dotsc]$ ...
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Let $x=1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{…}}}}$; then the value of $(2x-1)^2$ equals…

Let $$x=1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{...}}}};$$ then the value of $(2x-1)^2$ equals... I don't how to start this question. Please help.
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20 views

How to Compute Infinite Continued Factions

I'm supposed to find the value of the infinite continued fracton $[2;1,3,1,3,1,3,1,3...]$. How would I go about doing this?
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The continued Fraction Algorithm Proofs

I am trying to prove that for any positive integer $\sqrt{n^{2}+1}= [n; \overline{2n}]$, where $\overline{2n}$ is infinitely repeating. I think the best way to do this is to use the continued fraction ...
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Find the Farey triple given the base vertex of any isosceles triangle in Farey diagram.

There is a visualization of the circular Farey diagram where all triangles are isosceles. Observe that any rational $\frac{a}{b}$ distinct from $\frac{0}{1}$ and $\frac{1}{0}$ is always the vertex ...
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56 views

Algorithm for regular continued fraction of a square root

Say I have a number $n$, and want to find the expression of $\sqrt{n}$ as a regular continued fraction. How would I do such a thing systematically? A naive computer algorithm wouldn't work, due to ...
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how to convert continued fractions into normal fractions?

i couldnt find anything on google so i just tried opening it normally and recording each step. so i got: [d,c,b,a] = ((((a)*b+1)*c+A)*d+B)/C. [e;d,c,b,a]=(((((a)*b+1)*c+A)*d+B)*e+C)/D. etc.. (X ...
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Non convergent simple continued fractions?

Let $<a_0;a_1,a_2,\dots>$ be an infinite sequence of integers such that $$0<n\implies a_n>0.$$ For any natural $n$ we know there exists a convergent: one rational number $r_n$ such that ...
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94 views

Sequence Converging to the Square Root of an Integer $S \gt 1$

I noticed this answer to the question $\quad$ Continued fraction of a square root and the comment So I felt obliged to take this on using the theory of sequences. Confession: I find it difficult ...
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Integral over recurrence relationship

I'm interested in evaluating the following definite integral \begin{equation} I_n = \int_0^{\gamma} F_n(x)\:dx \end{equation} Where $\gamma \gt 0$ and $F_n(x)$ is based on the recurrence ...
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Proof of this formula for $\sqrt{e\pi/2}$ and similar formulas.

\begin{align} \sqrt{\frac{e\pi}{2}}=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}} \end{align} as seen here. ...
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29 views

Simple continued fraction for irrational numbers.

I read it here that: "What you must have read is that a number with an infinite simple continued fraction expansion is irrational. A continued fraction is "simple" if all the partial numerators are ...
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203 views

I have found a way of computing Euler's number. Is there any possible intuition of how that might be the case?

So a few days ago I just kind of messed around with my calculator, when I had an idea about a new continued fraction. I inputted it, and I found that it converged really quickly, and, quite wondrously,...
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110 views

Continued fraction of $\phi^3$

I found that $$\phi^3=4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ How should I prove this? Attempt: Suppose$$x= 4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ To ...
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Identifying the space of geodesics on the hyperbolic plane as a topological space.

On pg. 2 of this PDF, the author defines $\mathcal G=(-\infty, 0)\times(0, 1)$ and mentions that $\mathcal G$ can be thought of as "a space of geodesics on the hyperbolic $2$-space $\mathbb H^2$." ...
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How to prove that $\sum_{k=1}^{\infty}\frac{k^{n+1}}{k!}=eB_{n+1}=1+\cfrac{2^n+\cfrac{3^n+\cfrac{4^n+\cfrac{\vdots}{4}}{3}}{2}}{1}$

Through some calculation, it can be shown that $$e = 1+\cfrac{1+\cfrac{1+\cfrac{1+\cfrac{\vdots}{4}}{3}}{2}}{1}\tag{1}$$ $$2e = 1+\cfrac{2+\cfrac{3+\cfrac{4+\cfrac{\vdots}{4}}{3}}{2}}{1}\tag{2}$$ $$5e ...
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How do I prove that $\mathcal{P}_d(\mathbb{Z})^+\neq\varnothing$ and that Pell's equation solutions are convergents

I recently had to skip a number theory lecture because I was sick, and they proved that the set $$\mathcal{P}_d(\mathbb{Z})^+=\lbrace(x_0,y_0)\in\mathbb{Z}^2_{\geq1}:x_0^2-dy_0^2=1\rbrace\neq\...
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61 views

$N \in \mathbb{N}$ is not a square, show that the continued fraction expansion of $\sqrt N/\lfloor\sqrt N\rfloor$ is $[1,\overline{a_1,a_2,\dots,2}]$

Let $N \in \mathbb{N}$ not a square, show that the continued fraction expansion of $\sqrt{N}/\lfloor\sqrt{N}\rfloor$ is $[1,\overline{a_1,a_2,\dots,2}]$. My notations: the fractional part of $a$ is ...
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47 views

Proof a set of Continued fraction is perfect

so in my book there is a problem which asked to proof the set of Continued fraction in $\mathbb{R}$ with euclidean metric ,call it $K$, with only $3$ and $4$ is perfect. I know i have to show that $K=...
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135 views

Show that $\pi ≈ 355/113$ is the best rational approximation of $\pi$ with a three-digit denominator

Given that the continued fraction expansion of $\pi$ is $[3, 7, 15, 1, 292, ...]$. Prove that $\pi ≈ [3, 7, 15, 1]= 355/113$ is the best approximation of $\pi$ with a 3-digit denominator. From the ...
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Symmetric continued fractions property where $q^2\equiv(-1)^n$ mod $p$

Let $[a_0,a_1,a_2,\ldots,a_n,a_n,\ldots,a_2,a_1,a_0]=:\frac{p}{q}\in\mathbb{Q}$ be a symmetric continued fraction. This sequence of $a_i$'s consists of finitely many elements because $\frac{p}{q}$ is ...
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Evaluating the continued fraction

How to evaluate $\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\frac{1}{\ddots}}}}}$, where $a_{j}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{j}$? That is, How to evaluate ...
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Digit Accuracy of the nth Square Root convergent

A spinoff of the question here: Estimation on the accuracy of the convergents of $\sqrt{a}$ Given a square root convergent, the accuracy to $p$ decimal places can be approximated by: $p$ ≈ digits$_{\...
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Predicting the change in the denominator of a continued fraction when reversing the order of $a_1$ through $a_n$.

When reversing the order of $a_1$ through $a_n$ in a continued/extended fraction, (ie. [$a_1$: $a_2$, ... $a_{n-1}$, $a_n$] becomes [$a_n$: $a_{n-1}$, ... $a_2$, $a_1$]) we see that the denominator ...
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Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$ Prove it converges and, evaluate the series. For the first part of the question, I prove it ...
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Better approximation than convergent of continued fraction

Let $\alpha = [a_0;a_1,a_2,\ldots] \in \mathbb{R}$ and $(p_n/q_n)$ be the convergents to the continued fraction of $\alpha$. Prove that, if $q_n \leq q < q_{n+1}$, gdc($p$,$q$)$=1$ and $p/q \not = ...
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Continued fractions approximation using golden ratio

Hello today my friend helped me with my problem, but he did not give me any additional informations why it works like that. Let's suppose that I need to get ln(n) using continued fractions. He told ...
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59 views

Closed form for $K(n)=[0;\overline{1,2,3,…,n}]$

I just started playing around with fairly simple periodic continued fractions, and I have a question. The fractions can be represented "linearly": for $n\in\Bbb N$, $$K(n)=[0;\overline{1,2,3,...,n}]$$ ...
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Is there a proof of an infinite number of prime numbers using the irrationality of $e$?

That the set of prime integers is infinite can be proved using the irrationality of $\pi$; see this wikipedia link. It analyzes the representation $\tag 1 {\displaystyle {\frac {\pi }{4}}={\frac {3}{...
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54 views

Representation of Transcendental number via continued fractions

My question is quite simple. As far as I know, rational numbers has finite continued fraction representation. And if for any number we have infinite continued fraction, then it necessarily is ...
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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^...
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Find continued fractions and corresponding rationals

I want to compute the rationals that the continued fractions $[4;2,1,3,1,2,4]$ and $[0;1,2,3,4,3,2,1]$ represent. Also, I want to find the continued fractions of the rationals $-\frac{19}{51}, \frac{...
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23 views

For which the below fraction refer to?

I have got this fraction representation :$$a=\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots}}}} $$ but i can't know for which it's refer to , I mixed that with Golden ratio however the ...
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Let $C_k = \frac{q_k}{p_k}$ denote the $k^{th}$ convergent of $[a_0;a_1,a_2,\dots]$. Prove that $p_kq_{k-1}-p_{k-1}q_k = (-1)^{k-1}$.

Let $C_k = \frac{q_k}{p_k}$ denote the $k^{th}$ convergent of $[a_0;a_1,a_2,\dots]$. Prove that $$p_kq_{k-1}-p_{k-1}q_k = (-1)^{k-1}.$$ I am not sure how to prove this and I am not sure what it even ...