For questions on continued fractions.

learn more… | top users | synonyms

2
votes
0answers
91 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 ...
1
vote
1answer
118 views

find the value of 1/(2+1/(4+1/(4+1/(…))))

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$. I have totally no clue; thanks for the help! How am I suppose to solve ...
3
votes
1answer
101 views

History Question on Continued Fractions

I worked out the periodicity of some infinite continued fractions last night by hand. (Don't ask me why)For example, $\sqrt{13}= [3,1,1,1,1,6,1,1,1,1,6,\ldots]$. Last night I worked out the first ...
4
votes
1answer
425 views

Finding the value of a continued fraction?

I know how to calculate the exact value for continued fractions such as $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}=\frac{1+\sqrt{5}}{2}$$ However, is it possible to find the value of continued ...
16
votes
3answers
901 views

Motivation behind this eccentric Ramanujan Identity

I just visited the MathJaX page due to the Math.SE website showing some problems while loading the page. I saw some demo math equations samples at this page, when this identity actually caught my ...
6
votes
1answer
124 views

Mistake in Khinchin's “Continued Fractions”

I am reading Khinchin's Continued Fractions page 10. $\lbrack a_1;a_2,a_3\ldots\rbrack$ is a continued fraction and $q_k$ is given by $q_k=a_kq_{k-1}+q_{k-2}$. Suppose $\sum_{n=1}^{\infty}a_n$ ...
2
votes
1answer
96 views

Continued Fractional representation of irrational numbers [duplicate]

I know that any rational number can be expressed as a continued fraction, but what about irrational numbers? For example, what is the continued fractional representation of Pi, or e for that matter? ...
7
votes
5answers
351 views

How are continued fractions useful?

On Wolfram Alpha, I see continued fractions being listed in the results. Although I understand continued fractions, and how they can be used for approximations, what is a better approximation than a ...
3
votes
1answer
89 views

Pell-like equations and continued fractions

Why does the continued fraction method work? Could be applied in order to solve, for example, $x^{17}-19y^{17}=1$ ?
14
votes
2answers
276 views

Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
4
votes
2answers
121 views

A numerical coincidence with continued fractions

My brother built a garage that measures 45 feet by 30 feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches, $$ ...
4
votes
1answer
110 views

Finding an upper bound on a fraction

$0<\varepsilon <1$. If $n_k$ and $a_k$ are positive integers for which $$n_{k+1}=a_{k+1}n_k+n_{k-1}$$ Let $L\in\mathbb{N}.$ If $L>a_k \ge 3$, what's the smallest upper bound I can place on ...
6
votes
2answers
471 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 ...
0
votes
2answers
82 views

Bounded partial quotients set is nowhere dense

I've stumbled upon a claim that the set: $$ B_N = \{[a_0;a_1,a_2,...] | \exists n_0 >0\forall n\geq n_0 a_n<N\} $$ for some $N$, is nowhere dense (and closed). Unfortunately, I have found that ...
5
votes
1answer
237 views

How to find the number of continued fraction from a periodic representation?

Problem Find the number that represented by $[2,2,2 \ldots]$ I know it wasn't difficult, but I was absent the last two classes. So I just want to make sure that I got it right. My attempt was, ...
2
votes
1answer
91 views

Why are continued fractions for irrational numbers always convergent?

Like in the title: Why are continued fractions for irrational numbers (i.e. infinite fractions) always convergent?
7
votes
1answer
309 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $A$ be the diagonal matrix w/alternating in sign diagonal entries: $$ A = \begin{pmatrix} (-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\ 0 & ...
0
votes
1answer
83 views

Non-Recursive Fundamental Recurrence Formulas

Is there a non-recursive version of the fundamental recurrence formulas for continued fractions? I am trying to compute $A_{1000}$, and it is taking me an extremely long time. By the way, I am ...
2
votes
2answers
214 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 ...
10
votes
4answers
719 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 ...
3
votes
3answers
433 views

Equivalence of Two Different Irrational Numbers

If $\alpha$ and $\beta$ are two real numbers, we say that $\beta$ is equivalent to $\alpha$ if there are integers $a$, $b$, $c$, and $d$ such that $ad-bc=\pm1$ and $\beta=\frac{a\alpha+b}{c\alpha+d}$. ...
0
votes
0answers
86 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 + ...
3
votes
1answer
119 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: ...
2
votes
1answer
92 views

A trigonometric identity for special angles

Prove that for a natural number $n$, $$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$
2
votes
0answers
75 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 ...
29
votes
1answer
548 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 ...
2
votes
2answers
174 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
98 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 ...
6
votes
1answer
153 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 ...
18
votes
2answers
313 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 ...
1
vote
1answer
67 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 ...
0
votes
0answers
84 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
1answer
78 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 ...
2
votes
1answer
40 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 ...
11
votes
3answers
424 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 ...
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 ...
13
votes
2answers
332 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 ...
3
votes
0answers
149 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
103 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
233 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
169 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
450 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 ...
8
votes
3answers
260 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
votes
1answer
171 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 ...
1
vote
1answer
104 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
4
votes
2answers
113 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 ...
1
vote
3answers
533 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 ...
5
votes
5answers
270 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
56 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
108 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): ...