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 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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122 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
61 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
79 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 ...
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1answer
155 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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0answers
27 views

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
188 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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60 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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0answers
18 views

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 ...
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1answer
76 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
119 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
71 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 ...
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1answer
46 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
44 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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45 views

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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29 views

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
182 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 ...
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691 views

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

Let $$ \text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction} $$ So for example, the terms of the sum $\text{S}_6$ ...
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1answer
37 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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277 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
57 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 ...
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1answer
126 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
79 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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1answer
83 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 ...
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2answers
203 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. ...
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0answers
113 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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118 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
269 views

Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$
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1answer
58 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
46 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 $B_p(x,y)$ ...
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1answer
56 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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171 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
76 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 ...
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1answer
54 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! ...
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1answer
50 views

Is a finite continued fraction a closed-form expression?

We had a discussion regarding this answer on Electrical Engineering. The answer in question discussed a finite continued fraction. We're wondering whether it's a closed-form expression or not. ...
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1answer
253 views

The radius of image of a circle under mobius transformation

A Mobius transformation of the plane takes $z \mapsto \frac{az+b}{cz+d}$. These are known to take circles to circles, but given an explicit circle, how do we compute the radius. Let's parameterize ...
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154 views

How to find convergents/approximate ratios for 3 (or more) numbers - (3 number Euclidean algorithm?)

It is easy to find approximate ratios between 2 numbers by using the Euclidean algorithm to calculate continued fractions. However I can not find a method to do this for 3 numbers. I have tried a ...
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1answer
38 views

Comparation between continued fractions

I'm trying to solve the following problem but I'm having some difficulties.. Let $a_0,a_1,\dots,a_n$ and $b_0,b_1,\dots,b_n,b_{n+1}$ be positive integers. Give conditions that make the following ...
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1answer
100 views

Properties of continuity

Let $f,g :[a,b]\to\mathbb{R}$ be continuous functions such that $$\int\limits_c^df(x)\leq \int\limits_c^dg(x)dx$$ whenever a$\leq$c$<$d$\leq$b. I need to show that $f(x)\leq g(x)$. I have the ...
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1answer
112 views

Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...
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1answer
160 views

Continued fraction of the golden ratio

It is known, that the continued fraction of $\phi = \frac{1+\sqrt{5}}{2}$ is $[\bar{1}]$. This can be shown via the equation $x^2-x-1=0$: $$ x^2-x-1=0 \Rightarrow x = 1+\frac{1}{x} = 1+ ...
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1answer
55 views

Formal solution needed to question that looks too easy to be true about the Gauss map

Using the itineraries of the Gauss map write the continued fraction expansion of the number $0 \leqslant \alpha \leqslant 1$ such that $$\displaystyle \alpha = \dfrac{1}{4+\dfrac{1}{3+\alpha}}$$ I ...
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4answers
168 views

Find the continued fraction of the square root of a given integer [duplicate]

How to find the continued fraction of $\sqrt{n}$, for an integer $n$? I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator?
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1answer
51 views

Inequality with continued fractions: $\theta_r \geq a_{r+2}\theta_{r+1} + \theta_{r+1}$

I want to prove that the following inequality is true (or that is false, I don not know but I think it is true). $$\theta_r \geq a_{r+2}\theta_{r+1} + \theta_{r+1}.$$ Here the notation is as follow: ...
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1answer
59 views

Approximating $\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$

What is a good approximation for $$\omega=\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$$ This will be used to find $$T=\frac{t}{1-\omega}$$ Without using Lambert's continued fraction ...
2
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1answer
190 views

New mathematical constant formed by continued fraction with prime numbers?

Notational convention: $$\bigoplus_{k=0}^{\infty}a_k=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}$$ Let $$ P:=\bigoplus_{k=1}^{\infty}p_k$$ where $p_k$ is the k-th prime ...
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33 views

Name/properties of a difference of continuants

(This is cross-posted at http://mathoverflow.net/questions/181619/name-of-a-difference-of-continuants) Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, ...
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127 views

Sums of nested radicals

Is there a known example of an infinite sum of finitely nested radicals that evaluates to a given value? Or an infinite sum of convergents of an infinite continued fraction? The finitely nested ...
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205 views

Continued fraction and double series.

From Euler's continued fraction formula, we have $$x = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}\,$$ and $$x = 1 + \sum_{i=1}^\infty r_1r_2\cdots r_i = ...
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71 views

Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...