For questions on continued fractions.

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Continued fraction explanation

This is about simple infinite continued fraction. I don't understand the line '...then $C_0 < x < C_1$'. $C_k$ here refers to $C_k=[a_0;a_1,a_2,...,a_k]$ where $1 \leq k \leq n$. $C_o=a_0$. ...
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2answers
108 views

Infinite continued fraction expansion

How can we find the first six partial quotients of the infinite continued fraction expansion of $\sqrt[3]2$? I know how to do this by expanding when we have a square root function... but I"m not sure ...
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1answer
123 views

Evaluation of a slow continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
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1answer
107 views

Evaluation of Rogers-Ramanujan continued fraction $R(e^{-2\pi/5})$

Let $A = \{(\sqrt{5} + 1)/2\}^{5}$ and let $\alpha,\beta$ be positive reals such that $\alpha\beta = \pi^{2}/5$. Then it is known that $$\left\{A + R^{5}(e^{-2\alpha})\right\}\left\{A + ...
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1answer
38 views

Applications of hypergeometric continued fractions

http://en.wikipedia.org/wiki/Gauss%27s_continued_fraction Using a technique due to Gauss a lot of special functions can be expressed as continued fractions. What applications of this are there ...
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1answer
40 views

Math and Taxes - Items versus Sum Total differs

First, The question is a bit confusing as I am not really sure how to word this problem as a question. The math problem I encountered which is a bit of an anomaly is this : Suppose you are ...
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1answer
21 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
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1answer
62 views

Continued fraction : How to find the first 3 terms

I can't calculate the exact first tree terms $F_0$, $F_1$ and $F_2$ of this continued fraction : $$F_n=\cfrac{1}{-\text{i$\omega $}\,+A\,\cfrac{(n+1)^2}{{4 (n+1)^2-1}}F_{n+1}}$$ $A$ and ...
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0answers
137 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
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0answers
74 views

Properties of a continued fraction convolution operation

Usually the partial numerators of a continued fraction are all 1s. Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
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0answers
431 views

What can Euler's identity teach us about (generalised) continued fractions?

We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ ...
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202 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
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0answers
94 views

Different ways of operating an infinite continued fraction

Given the continued fraction below, $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}+\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}} $$ I wanted to know to which number it converged, so I ...
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0answers
106 views

Finding a closed expression for a calculated value.

Sometimes, when getting some numerical results when investigating number theory sequences with a computer, I find myself suspecting that a decimal value ($a$) I have found might be a quadratic ...
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0answers
259 views

Evaluating matrix-continued fractions?

I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity $A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...
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146 views

Algorithm For Continued Fraction of $\pi$ without error.

Is there an algorithm that will output the numbers in the continued fraction of $\pi$ without error? If one uses the usual method of calculating continued fractions, an approximation of $\pi$ is ...
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0answers
75 views

Is that observation really a property of the log of coefficients of continued fractions (example: cf(log(3)/log(2))

I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small ...
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0answers
138 views

How to prove that this series $f(z)=1+\sum_{k=1}^{\infty}2^{-k z}$ converges using the theory of continued fractions?

Consider the following series \begin{equation} f(z)=1+\sum_{k=1}^{\infty}\frac{1}{2^{k z}} =1+ \sum_{n=1}^{\infty}\left( \prod_{k=1}^{n}\frac{1}{2^{z}} \right) \end{equation} Using Euler's continued ...
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0answers
250 views

continued fractions and convex hulls

If I remember correctly, there is a nice correspondence between continued fractions and convex hulls of lattice points in the plane. If $\theta>0$ is the slope of a line in $\mathbb{R}^2$ passing ...
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183 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
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73 views

Interesting Recursive Continued Fraction Limit

I was messing around with recursive functions the other day and came up with something that could be interesting: Definition of $\bar{\Xi}(n)$:\ Let $\Xi ...
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47 views

How long should you descent in Stern-Brocot-Tree to get a fixed approximation guarantee?

I've read in Wikipedia: By stopping once the desired precision is reached, floating-point numbers can be approximated to arbitrary precision. If a real number x is approximated by any rational ...
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0answers
87 views

Lower bound for the length of continued fraction

Define $\mathscr L: \mathbb Q \mapsto \mathbb N$ as the minimal number of terms in the continued fraction of a rational number. Example: the continued fraction of $\frac{5}{8}$ is ...
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0answers
71 views

Cantor set as a set of continued fractions?

Does the classical cantor set have a nice description as a set of continued fractions? I made a (superficial) search and didn’t find anything, but I’m very tired right now, so please forgive me that ...
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0answers
185 views

On the continued fraction of $e$

The following question comes up during analysis of Padé approximants to $e^x$ (see my related question in MathOverflow for more background). Recall that the continued fraction expansion of $e$ is $$ e ...
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Champernowne constant - summation and behavior of terms in continued fraction expansion

Is there an infinite summation that gives the Champernowne constant? Wikipedia has one, and so does Wolfram MathWorld. Are they valid? If so, could someone explain why, i.e how do they work? Also, ...
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0answers
57 views

Is this a bounded sequence ? (about continued fraction)

Represent $\sqrt{2}$ in the form $$\sqrt{2}=1+\frac{8}{A_1+\displaystyle\frac{8}{A_2+\displaystyle\frac{8}{A_3+\ddots}}},$$ where $A_n$ is a positive integer and $A_n \geq 8$ for all $n$. So we have ...
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0answers
52 views

Lyapunov-exponent and rate of convergence of continued fractions

I'm writing an essay on the Lyapunov-exponents of the Gauß map \begin{align} T:[0,1]&\to[0,1],\\ x&\mapsto \begin{cases}\frac 1 x \mathrm{mod}\,1 & \text{for $x>0$,}\\ 0 & \text{for ...
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27 views

Explanations of the Euler's continued fractions to compute exponential

After looking for explanations of the Euler's continued fractions to compute exponential on internet and after reading Euler's explanations about, I still don't understand how Euler find this ...
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80 views

A lower bound for continued fraction approximation.

It is known that, for a continued fraction expansion of an irrational $\alpha$ we have that: $$ \left| \alpha - \frac{p_n}{q_n} \right| = (\alpha_{n+1}q_n^2 + q_nq_{n-1})^{-1} $$ Show that the ...
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60 views

Proof of a Continued Fraction Identity using basic CF definition.

Two definitions (the first is informal) of continued fraction: This is the basic Continued Fraction algorithm for real numbers. Let $\alpha \in \mathbb{R}$. If $[\alpha]=\alpha$, then we are done. ...
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86 views

continued fraction multivariate normal distribution?

After searching for a while, I wonder if a continued fraction representation exists for the multivariate Mills ratio $P(Z \geq x)/\phi_Z(x)$ where $Z \tilde\, N(\mu,\Sigma)$. The representation ...
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125 views

Uniqueness of continued fraction representation of rational numbers

I have problems proving the uniqueness of simple continued fraction representation of rational numbers as claimed in http://en.wikipedia.org/wiki/Continued_fraction#Finite_continued_fractions. Let ...
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0answers
68 views

Proof of a striking identity of Tito Piezas III

In the q series blog of Tito Piezas here . He gives a very striking relation I am wondering on how to prove that ?
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21 views

uniform spanning tree of $2 \times n$ graph

In Probability on Trees and Networks Chapter 1 study the uniform spanning tree on the ladder graph: _ |_| |_| |_| ... |_| |_| The probability the bottom rung ...
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83 views

Calculate an infinite continued fraction as special function

It is possible to convert this infinite continued fraction $\cfrac{1}{-a+\cfrac{b\;f(0)}{a+\cfrac{b\; f(1)}{-a+\cfrac{b\; f(2)}\ddots}}}$ to a special function ? Please, how do it? where : $(a,b) ...
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0answers
79 views

Why is this finite continued fraction expression equal to $1$?

How find this value I use computer relsut is $1$, maybe this problem have the general relsut But How can prove this by hand? maybe have nice methods? Thank you
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0answers
65 views

Siegel's Proof and the Continued Fraction Constant

I am working through a result of C.L. Siegel that uses the continued fraction: $$ \frac{v}{v'} = \lambda + 1 + \cfrac{z}{\lambda + 2 + \cfrac{z}{\lambda + 3 + \cfrac{z}{\lambda + 4 + ...
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40 views

Intermediate convergents of a continued fraction.

I have been studying continued fractions and convergent's properties, and i have a questions about "intermediate convergents" I have read that the expression of the intermediate convergents (those ...
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82 views

Continued fraction of $\gamma+1$ using recursion

Number $\gamma,$ the Euler-Mascheroni constant, is defined as the value of $$\gamma = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} - \ln(n).$$ We know that $$\lim_{n\to\infty} ...
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0answers
77 views

continued fraction

$[a_0,a_1,a_2,\cdots,a_n]:=1/(a_0 + 1/(a_1 + 1/(a_2 + \cdots + 1/(a_n)\cdots )))$ I am so curious that what is the shape of $ f_n(x) $ such that $$ \sum_{n \geq 0} f_n(x) y^n = [-y,1] +[-y,y,1]x + ...
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82 views

By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?

this is a question from a book I'm struggling with, please could you provide a clear proof Consider the sequence of rational numbers $a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1 ...
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145 views

continued fraction to rational polynomial in maple?

In maple is there a way to convert a continued fraction into a rational polynomial? I'm using the minimax function and for a particular function I want to approximate it returns a continued fraction ...