For questions on continued fractions.

learn more… | top users | synonyms

2
votes
0answers
88 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 ...
3
votes
1answer
125 views

Limit of a continued fraction

Given the continued fraction: $$f(x,N)=\left[2,3,4,...N,x\right]$$ $$f(x,N)=\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{...+\cfrac{1}{x}}}}}$$ is it possible to find an expression for the integral: ...
10
votes
4answers
736 views

Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$

I'm currently a high school junior enrolling in AP Calculus, I found this website that's full of "math geeks" and I hope you can give me some clues on how to solve this problem. I'm pretty desperate ...
6
votes
2answers
571 views

Continued fraction expansion related to exponential generating function

A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series: $$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
2
votes
2answers
202 views

Continued fraction for $\sqrt{14}$

I'm referencing this page: An Introduction to the Continued Fraction, where they explain the algebraic method of solving the square root of $14$. $$\sqrt{14} = 3 + \frac1x$$ So, $x_0 = 3$, Solving ...
5
votes
1answer
109 views

For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?

Inspired by this question I ask this. For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$? The original question concerned $a=e$, the usual ...
30
votes
1answer
587 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
12
votes
1answer
157 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 ...
6
votes
1answer
156 views

Finding near-integers in a range

I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
1
vote
1answer
80 views

Defining piecewise summation of continued fractions and rationality of sums

Let $a=[a_1,a_2\dots]$ and $b=[b_1,b_2\dots]$ be two real numbers and their continued fraction representations. They may be infinite or finite. Let us define a thing $+^*$ so that ...
2
votes
1answer
82 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 ...
18
votes
2answers
330 views

Evaluation of a 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 ...
0
votes
0answers
87 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 ...
2
votes
3answers
54 views

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 ...
2
votes
2answers
224 views

Calculate an infinite continued fraction

Is there a way to algebraically determine the closed form of any infinite continued fraction with a particular pattern? For example, how would you determine the value of ...
3
votes
0answers
151 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 ...
1
vote
2answers
117 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 ...
2
votes
2answers
262 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 , ... , ...
3
votes
2answers
192 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 ...
6
votes
1answer
233 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 ...
8
votes
3answers
273 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} ...
1
vote
1answer
105 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
5
votes
5answers
322 views

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+ ...
4
votes
1answer
57 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 ...
6
votes
2answers
116 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): ...
1
vote
3answers
297 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?
7
votes
0answers
85 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 ...
4
votes
2answers
155 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?
1
vote
1answer
48 views

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 $[0, 1]$. $proof$: ...
2
votes
1answer
42 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 ...
13
votes
2answers
429 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 ...
0
votes
1answer
134 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 ...
1
vote
1answer
198 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 + ...
12
votes
1answer
198 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): $$ ...
5
votes
4answers
4k 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 ...
6
votes
1answer
506 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 ...
6
votes
1answer
159 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 ...
1
vote
0answers
101 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 ...
1
vote
1answer
293 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 ...
1
vote
2answers
206 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 ...
1
vote
0answers
61 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. ...
3
votes
1answer
777 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 ...
0
votes
3answers
158 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 ...
6
votes
1answer
1k 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 ...
6
votes
1answer
188 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 ...
1
vote
0answers
94 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 ...
1
vote
3answers
222 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 - ...
1
vote
0answers
156 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 ...
8
votes
1answer
156 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 ...
5
votes
2answers
523 views

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 ...