For questions on continued fractions.

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A conjecture about “equiharmonic numbers” of Flajolet via Doron Zeilberger

While semi-randomly browsing, I came across this conjecture which Philippe Flajolet sent to Doron Zeilberger as a "gift" (the "gift" is here, so you can check to see if I have typeset it correctly): ...
0
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1answer
33 views

modification of Dedekind cuts

Dedekind defining real numbers as equivalence classes of Cauchy sequences of rational numbers. $x=y$ means $x-y=0$ ie $x_n - y_n \to 0$. addition and multiplication are defined for each coordinate. ...
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2answers
84 views

Can we say that $\sqrt{2}=2/(2/(2/(2/\ldots)))$?

Can we say that $\sqrt{2}= \cfrac{2}{\cfrac{2}{\cfrac{2}{\cfrac{2}{\ldots}}}}$? We have ...
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1answer
39 views

Continued Fraction

I am working on the following question "Use the continued fraction $[1;0,1,1,2,1,1,4,1,1,6,1,1]$ to get an estimate for $e$." But I got stuck when I tried to compute $q_i$, since $a_1=0$ , $q_1 =0$. ...
2
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0answers
28 views

Continued fraction for $[1,2,3,4,5,6,\dots]$ [duplicate]

Any continued fraction that does not terminate or repeat can't be rational or a quadratic irrational. It is not hard to write something that does not fit these two categories. Can we still get a ...
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0answers
76 views

nth-root of continued fraction with Raney transducers

There are some algorithms for doing basic arithmetic by using regular continued fraction expansions. These algorithms are mainly due to Gosper (1972) and Raney (1973). These two approaches use ...
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1answer
26 views

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$ Using this formula here (it begins in the middle of the page), I obtained $\frac{\sqrt{7}}3=[0;1,\overline{7,2}]$ but ...
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2answers
148 views

Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$

Find a fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ of $\mathbb Q(\sqrt{141})$ I have different corollaries for different numbers, the most appropriate for $141$ ...
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2answers
232 views

What are the reduced elements of $\mathbb Q(\sqrt{30})$?

What are the reduced elements of $\mathbb Q(\sqrt{30})$ ? From the definition here(on page $32$); An element $\beta\in\mathbb Q(\sqrt{d})$ is said to be reduced, if $\beta>1$ and ...
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0answers
21 views

Continued fraction manipulation

I have the following continued fraction $$ \frac{1}{a_1x+}\;\;\frac{1}{b_1+}\;\;\frac{1}{a_2x+}\;\;\frac{1}{b_2} $$ The paper I am reading then converts this to the following continued z-fraction ...
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2answers
43 views

Continued Fraction Algorithm for 113/50

The numbers $a_k$ can be found for $\frac{113}{50}$ by using a continued fraction algorithm. Note that $\frac{113}{50}$ is rational, and as a result it will have to terminate. Can anyone help me ...
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2answers
55 views

Continued Fraction for Root 5 [duplicate]

How can I find the continued fraction expansion for the square root of 5. Do this without the use of a calculator and show all the steps.
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1answer
49 views

continued fraction expansion for √7 [duplicate]

Can someone help me find the continued fraction expansion for $\sqrt{7}$ just like I did for below. For $\sqrt{3}$ I did this: I was given that $x = \sqrt{3} -1 $ $x = \frac{1}{1+\frac{1}{2+x}} $ ...
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1answer
30 views

Geometric Proof for Slopes (Contined Fractions)

I just started learning about continued fractions, and my lecture had a theorem that estimated the slope $a$ of a given line $L$. This was done in terms of the slope of the point $P$ with coordinates ...
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2answers
59 views

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$?

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$ ? $\displaystyle[1;1,2,1,2,\dots]=1+\frac{1}{[1;2,1,2,\dots]}=1+\frac{1}{1+\frac{1}{2+\frac{1}{[1;2,1,2,\dots]}}}$ If I set $x=[1;2,1,2,\dots]$ then; ...
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3answers
106 views

continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$

I would like to find the continued fraction expansion of the roots of: $$x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$$ Eq 1.6 from [1] What makes this problem so ...
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1answer
53 views

continued fraction of $3 + 17\sqrt{3} $

On a computer, I tried to iterate the Euclidean algorithm on the number $3 + 17\sqrt{3}$ and here is what I got: \begin{array}{ccccrcrcrcr} 3 + 17\sqrt{3} &=& 32 &\cdot\;(& ...
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0answers
28 views

Integral of a Continued Fraction

How might one go about evaluating the following integral $\int_{-\infty}^{\infty}\mathrm{K}_{j=0}^{\infty}(F_{j}e^{-x^2})dx$? Where$\mathrm{K}$ denotes a continued fraction and $F_j$ is the jth ...
2
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1answer
75 views

Why are there no continued fraction representation for $\pi$ obeying mathematical rules?

There are several irrational numbers that can be represented with continued fraction such that a mathematical rule arises in this continued fraction. For example, the Euler number $e$ can be ...
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Continued Fraction summation representation

I have a rational fraction of the form: $$s=\frac{p_0+p_1x+p_2x^2+\cdots+p_Mx^M}{1+q_1x+q_2x^2+\cdots+q_Mx^M} $$ The paper I am reading converts this to the form: $$s = ...
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2answers
170 views

Simple Finite Continued Fraction

I am working on my senior thesis and have encountered, unexpectedly, a finite continued fraction that I would be interested in resolving. I already know the answer (by an informed guess based on where ...
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0answers
38 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
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On the asymptotics of a continued fraction

I know the coefficients of the continued fraction representation of a function that diverges like $O(\sqrt{x})$, where the $a_k$ depend on $x$, $ f(x)=1+a_0/\left(1+\underset{k=1}{\overset{\infty ...
4
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1answer
64 views

Continued Fraction, Closest Neighbours

For setting the proper divisor/multiplier registers in a phase locked loop, I use a continued fraction expansion, which I stop if either the numerator or denominator of the fraction becomes larger ...
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1answer
60 views

Good and best rational approximations

Lately I was reading a bit about continued fractions and came up with a question that I couldn't find an answer for. Here are the definitions I will use in the question: Let $x \in \mathbb{R}$. A ...
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1answer
36 views

Closest rational approximation of $\sqrt x$ with denominator having prime powers $\lt n$

I am representing denominators in rational numbers with powers of their prime factors for easy multiplication and division in lowest terms (by adding and subtracting the prime powers). I would like ...
2
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1answer
37 views

Good rational approximations using continued fractions?

My textbook claims that the best rational approximations (relative the size of the numerator and denominator) of an irrational number by using continued fraction are those whose expansions are ...
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0answers
37 views

Coefficients of the polynomials generated by $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$.

Let $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$ for $i\geq0$, i.e., $f_i=\dfrac{\sqrt{f_{i+1}^2\mp4}+f_{i+1}}2$ I have observed that $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ ...
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Continued fraction approximation

Let $\theta\in\Bbb{R}_{\gt0}$. A) Prove that the convergents for the continued fraction expansion of $\theta$ give us better and better rational approximations to $\theta$. B) Suppose $\theta\notin ...
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How is Lagrange's $2\sqrt{D}$ bound on partial denominators proven for periodic regular continued fractions of quadratic irrationals

For the quadratic surd: $$ \zeta = \dfrac{P + \sqrt D}Q $$ the wikipedia article on periodic continued fractions mentions that Lagrange proves that the largest partial denominator of a regular ...
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1answer
111 views

Continued fraction of $e^{-2\pi n}$

I found this problem on a well-known problem solving website. It is apparently from Ramanujan. With $$\LARGE{a = \frac{1}{1 + \frac{e^{-2\pi}}{1 + \frac{e^{-4\pi}}{1 + \ddots}}}},$$ what is ...
45
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3answers
543 views

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Using a symbolic computation software (Mathematica), I got the following interesting results: $$ \begin{align} \sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{1+\frac{1}{1+n^2}}}} &= ...
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1answer
20 views

Question Mark Function and continued fraction representations

One could defined Minkowki's question mark question by : $$?(x) = a_0 + 2 \sum_{n= 1}^\infty \dfrac{(-1)^{n+1}}{2^{a_0 +\dots +a_k}},$$ with $x = [a_0;a_1,a_2,\dots]$. Is Minkowski's question mark ...
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2answers
189 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...
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1answer
46 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the ...
6
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1answer
115 views

What is the value of this continued fraction?

I am curious about the value of the continued fraction $$1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\dots}}}}}.$$ Can we evaluate it ? Is it a nice value ? Clearly it should ...
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2answers
72 views

From the continued fraction

What would be the irrational number $\dfrac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers given by this expression: $$ \left( ...
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51 views

Multiply all terms in continued fraction by a constant

I noticed that continued the fraction for $\sqrt{12}$ is $3;2,6,2,6,2,\ldots$ and the continued fraction for $\sqrt{7\times12}$ is $9;6,18,6,18,6,\ldots$ all the terms in the continued fraction are ...
3
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2answers
169 views

Continued fraction to irrational number

Let $[1;\overline{2,1}$] be a continued fraction. I want to find the corresponding number. I know how to transform fractions of the form $[a;\overline{b}]$ but I am having a hard time here. Thank you. ...
3
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62 views

Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
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89 views

How find this Continued fraction $[1,3,5,7,9,11,\cdots]$ value.

show this: $$\alpha=[1,3,5,7,9,11,\cdots]=1+\dfrac{1}{3+\dfrac{1}{5+\dfrac{1}{7+\dfrac{1}{\cdots}}}}=\dfrac{e^2+1}{e^2-1}$$ I found wiki Continued fraction also not have this problem,maybe this ...
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1answer
245 views

Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$
1
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1answer
51 views

What is a counterexample for this one?

Let $x$ be an irrational number. Let $\{a_0\}$ be the sequence of positive integers except for $a_0$ such that $x=a_0+K(1/a_n)$. Let $a,b$ be integers such that $b>0$ and $gcd(a,b)=1$ and ...
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0answers
31 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
2
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1answer
29 views

Convergents of continued fraction proof

Let $\frac{P_n} {Q_n} and \frac{P_{n+1}} {Q_{n+1}}$ be two consecutive continued fraction convergents for $b$. Then prove that: $$\left|{\frac{P_n} {Q_n}-b}\right|< \frac{1}{2Q_n^2}$$ or ...
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If there exists infinitely many fractions $P/Q$ Continued fractions then prove:

Prove that if b is a real number, then b is irrational iff there exist infinitely many fractions $P/Q$ such that : $|b-\frac{P}{Q}|<\frac{1}{Q^2}$ Thanks in advance!
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0answers
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Can inequality $-1<(x-\tfrac{1}{2})^2 - 3 (y-\tfrac{1}{2})^2 < 1$ be solved with continued fractions?

It's known at Pell's equation $x^2 - 3 y^2 = 1$ can be solved using the periodic continued fraction expansion of $\sqrt{3}= [1;\overline{1,2}]$. Eventually we get convergents $\tfrac{p}{q} \approx ...
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96 views

Divergent continued fractions?

The solutions to $$ x^2-6x+10=0 \tag 1 $$ are $$ 3\pm i\tag2. $$ Rearranging $(1)$ just a bit, we get $$ x = 6 -\frac{10}x \tag3 $$ and then substituting the right side of $(3)$ for $x$ within the ...
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1answer
55 views

Approximate irrational numbers with the same denominator

Let $\alpha$ be a irrational number, then using the continued fraction expansion we can find two sequences $\{p_n\}$ and $\{q_n\}$ with $q_n\rightarrow\infty$ as $n\rightarrow\infty$ such that ...
0
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1answer
39 views

Continued Fraction Form of sqrt(6) [duplicate]

I have to find the continued fraction form of sqrt(6). I have tried it, and have the answers but I can't get to the correct answer. If someone could help me that would be much appreciated. Thank you! ...