For questions on continued fractions.

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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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1answer
74 views

How to deal with infinite continued fractions in formal language?

A continued fraction 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, then writing this other number ...
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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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414 views

Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is ...
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3answers
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A question about $[c_0,c_1,\ldots,c_n]$ notation for continued fractions

I try to understand why by definition $[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also $[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ . Those are continued fractions, and ...
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2answers
312 views

How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
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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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2answers
94 views

Nth number of continued fraction

Given a real number $r$ and a non-negative integer $n$, is there a way to accurately find the $n^{th}$ (with the integer being the $0^{th}$ number in the continued fraction. If this can not be done ...
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2answers
216 views

Project Euler Problem 65

I am working on solving Project Euler problem #65 and run upon the following statement: What is most surprising is that the important mathematical constant, e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , ...
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2answers
157 views

The average denominator of the continued fraction expansion of $\pi$.

I was interested in the long term behavior of continued fraction denominators, so I plotted the average of the first $n$ terms in the continued fraction expansion of $\pi$ as a function of $n$ and got ...
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1answer
419 views

Faster arithmetic with finite continued fractions

I was curious about different representations of rational numbers and came across the finite continued fraction (see wp:Finite_continued_fractions ). Note: I will refer to traditional rational ...
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3answers
251 views

Continued fractions for $\sqrt{x} $ and beyond, valid formula?

For $x > 0$, is this trick valid? I use $$ ( \sqrt{x}-1)(\sqrt{x}+1)=x-1 $$ then $$ \sqrt{x}+1 = \frac{x-1}{\sqrt{x}+1-2} $$ so I can use iterations to get the rational approximant $$ \sqrt{x} ...
6
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1answer
128 views

Are all numbers that have a non-repeating, non-terminating continued fraction sequence transcendental? [duplicate]

(By continued fraction sequence, I'm specifically talking about the one kind where the numerator of every fraction is 1.) As a kid in middle school, I learned that all irrational numbers have ...
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1answer
103 views

Continued fractions with $n$ layers

Solve the equation $$x=2+\dfrac1{2+\dfrac1{...2+\dfrac1{2+\dfrac1x}}}$$ Where there are n layers in the fraction
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2answers
110 views

Continued fraction question

I have been given an continued fraction for a number x: $$x = 1+\frac{1}{1+}\frac{1}{1+}\frac{1}{1+}\cdots$$ How can I show that $x = 1 + \frac{1}{x}$? I played around some with the first few ...
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3answers
498 views

How to express an irrational number as generalized continued fraction?

With simple continued fraction, i.e. $$a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 \ldots}}}$$ I can use this formula: $$a_k = \lfloor \alpha_k \rfloor$$ $$\alpha_{k+1} = \dfrac{1}{\alpha_k ...
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Continued Fraction [1,1,1,…]

If the continued fractional representation of an irrational number $\alpha$ is given by [1,1,1,...], I can compute that $\alpha = \frac{1+\sqrt{5}}{2}$ by solving the equation $\alpha = 1+ ...
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1answer
55 views

Continued fraction proof from matrix form

By using the definition $$\pmatrix{p_n&p_{n-1}\\q_n&q_{n-1}} = \pmatrix{a_0&1\\1&0} \pmatrix{a_1&1\\1&0} \cdots \pmatrix{a_n&1\\1&0}$$ I need to show that $p_n/q_n$ is ...
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2answers
107 views

Continued fractions help

I'm trying to learn how to express a square root as continued fraction, but I can't get one thing. The following example of $\sqrt{14}$ is from this page (click the image to see it at full size): ...
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2answers
213 views

Can every transcendental number be expressed as an infinite continued fraction?

Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
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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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2answers
127 views

Continued Fractions periodicity and convolution.

Continued fractions for rationals terminate, for transcendentals like pi, they do not terminate and for irrationals (but non transcendentals) they repeat -- is this correct?
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1answer
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Is it possible to define a zero-set of $X$ to be the zero-set of some $f\in C^{*}(X)$?

It is possible to define a cozero-set of $X$ to be the cozero-set of some $f\in C^{*}(X)$, in fact; Every cozero-set in $X$ is the cozero-set of a function taking values in $[0, 1]$. $proof$: ...
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1answer
94 views

Using Maple for continued fraction expansions

I can find the continued fraction expansion of a value using Maple. Is there a simple way for finding the sequence of convergents (approximants) of the continued fraction expansion in Maple? Currently ...
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1answer
161 views

How do I determine appropriate rational approximations to a sum of square roots in order to bound the error accumulation?

I have two numbers, $A$ and $B$, that are sums of integer multiples of a set of square roots of small primes (and 1) and their products: $A = a_0 + a_1\sqrt 2 + a_2\sqrt 3 + a_3\sqrt 5 + a_4\sqrt 6 + ...
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1answer
168 views

Request for a proof of the following continued-fraction identity

I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close): $$ ...
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4answers
2k views

Continued fraction of a square root

If I want to find the continued fraction of $\sqrt{n}$ how do I know which number to use for $a_0$? Is there a way to do it without using a calculator or anything like that? What's the general ...
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1answer
675 views

A new continued fraction for Apéry's constant, $\zeta(3)$?

As a background, Ramanujan also gave a continued fraction for $\zeta(3)$ as $\zeta(3) = 1+\cfrac{1}{u_1+\cfrac{1^3}{1+\cfrac{1^3}{u_2+\cfrac{2^3}{1+\cfrac{2^3}{u_3 + \ddots}}}}}\tag{1}$ where the ...
6
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1answer
134 views

General Continued Fractions and Irrationality

A while back I came across a result about non-simple continued fractions that allows proving that some numbers are irrational. The result in modern terminology is: If, in the continued fraction ...
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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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1answer
240 views

Continued fractions with rational functions

Express the following rational function in continued-fraction form: $${4x^2+3x-7\over 2x^3+x^2-x+5}$$ The answer is : $${4 \over 2x- \frac{1}{2}} + { \frac{23}{8} \over ...
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2answers
163 views

Continued Fractions Approximation

I have come across continued fractions approximation but I am unsure what the steps are. For example How would you express the following rational function in continued-fraction form: $${x^2+3x+2 ...
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0answers
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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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1answer
578 views

Calculate the continued fraction of square root

I was having difficulty understanding the algorithm to calculate Continued fraction expansion of square root. I know the process is about extracting the integer part in repeat and maintaining the ...
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3answers
141 views

Continued fractions proof?

Let $b_1=1$ and $$b_n=1+\frac{1}{1+b_{n-1}}$$ for $n\ge 2$. Note that $b_n \ge 1$ for all $n$ in $\mathbb N$. ($\mathbb N$ represents the positive integers) Show that $b_{2k-1}^2<2$ for all $k ...
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825 views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
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1answer
146 views

A question about continued fractions and Gauss map

For $\alpha \in (0,1)$, write $\alpha$ as a continued fraction like $\alpha=[a_1, a_2, \ldots]$ (note that the implicit $0$th coefficient $a_0=0$ has been omitted), and let $\frac{p_n}{q_n}$ be the ...
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1answer
146 views

Why is this set of continued fractions perfect?

Would somebody please explain why the set of continued fractions in this answer http://math.stackexchange.com/a/1067/20873 i.e. "the set of all irrationals with continued fractions consisting only ...
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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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191 views

How do I solve a Continued Fraction of solution to quadratic equation?

I know that it is possible to make a CF (continued fraction) for every number that is a solution of a quadratic equation but I don't know how. The number I'd like to write as a CF is: $$\frac{1 - ...
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0answers
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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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3answers
656 views

Why are some mathematical constants irrational by their continued fraction while others aren't?

Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...
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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 ...
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4answers
488 views

How to detect when continued fractions period terminates

I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates? Let me give you an example: $\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
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2answers
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Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?

I have been looking at concise ways to represent irrational numbers using only integers. I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...
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1answer
132 views

optimality of 2 in a continued fraction theorem

I'm giving some lectures on continued fractions to high school and college students, and I discussed the standard theorem that, for a real number $\alpha$ and integers $p$ and $q$ with $q \not= 0$, if ...
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1answer
236 views

Convergent fraction for constant $e$?

I've just learned about e. I am very much the novice and my problem is that while trying to calculate the convergent fractions for e. For instance: ...
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318 views

finding the rational number which the continued fraction $[1;1,2,1,1,2,\ldots]$ represents

I'd really love your help with finding the rational number which the continued fraction $[1;1,2,1,1,2,\ldots]$ represents. With the recursion for continued fraction $( p_0=a_0, q_0=1, p_{-1}=1, ...
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1answer
314 views

Are there simple algebraic operations for continued fractions?

I thought about continued fractions as a cool way to represent numbers, but also about the fact they are difficult to treat because standard algebraic operations like addition and multiplication don't ...
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185 views

Solution to $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$ [duplicate]

Possible Duplicate: Why does this process, when iterated, tend towards a certain number? (the golden ratio?) Please post your favorit solution to the following Compute ...