For questions on continued fractions.

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3
votes
1answer
68 views

Can the number of digits in the denominator of a node in Stern-Brocot-Tree decrease in its children?

The Stern-Brocot-Tree looks like this: (image source files) It is an infinite binary tree that contains every positive rational number as exactly one node. The children of a node $\frac{a}{b}$ ...
10
votes
1answer
394 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
3
votes
1answer
46 views

Why does this pattern fail (sometimes) for the continued fraction convergents of $\sqrt{2}$?

This is connected to my post on the continued fraction convergents of pi. Motivated by Calvin Lin's comment whether a similar pattern exists for other constants, I checked $\sqrt{2}$. Its convergents ...
5
votes
1answer
49 views

Convergence of a sequence to a value and not another.

I was tinkering with the following equation and produced an infinite nested fraction: $$ (x-6)(x-3)=0 $$ $$ x^2-9x+18=0 $$ $$ x=9-\frac{18}{x} $$ $$ x=9-\frac{18}{9-\frac{18}{9-\frac{18}{...}}} $$ ...
1
vote
1answer
112 views

Does the continued fractions $3+\frac{1}{5+\frac{1}{7+\cdots}}$ equal $\pi$?

$3+\frac{1}{5+\frac{1}{7+\cdots}}=\pi$ Is it true? If yes, how to show it? Please help. Thank you.
-1
votes
1answer
256 views

Relation between e and pi [closed]

I found the following relation $\pi=3+\frac{1}{5+\frac{1}{7+\frac{1}{9+\dotsb}}}$ known and $e=3-\frac{1}{5-\frac{1}{7-\frac{1}{9-\dotsb}}}$ Can we relate these directly?
4
votes
1answer
138 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 + ...
6
votes
1answer
159 views

Are there any real (especially irrational) numbers whose decimal expansion and continued fraction are the same?

If a number with more than one digit occurs in the fraction, it should be expanded to as many digits in the expansion. I will be even more impressed, however, if the fraction consists entirely of ...
0
votes
1answer
41 views

What is the rate of decay of $\min\{k\xi-\lfloor k\xi\rfloor|k\in\{1,\dots,n\}\}$, for irrational $\xi$?

I wish to establish bounds on the sequence of infima of $\{n\xi\}_{n\in\Bbb N}$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part function and $\xi$ is irrational. I can prove that ...
4
votes
1answer
47 views

After $n$ iterations of the continued fraction algorithm, what kind of rational numbers will have terminated?

For a positive real number $r_0$, we have the continued fraction recursive algorithm: \begin{align} &r_n\in\mathbb{Z}\implies\text{terminate the algorithm}\\ &\text{else } r_{n+1} = ...
3
votes
1answer
99 views

Inequality related to the continued fraction expansion of sqrt(3)

I am working on a problem related to the continued fraction expansion of $\sqrt3$. If $p_k$ and $q_k$ denote the numerator and denominator, respectively, of the $k$th convergent, I should show that ...
7
votes
1answer
89 views

On the Pell-like $Ax^2-By^2 = 1$

This is connected to the post, Mere coincidence? (prime factors). I was looking at NeuroFuzzy's dataset and noticed the line, {{{1, {4, 2}}, {1, 4, 2, 4, 2}, 23762}} It seems this could be ...
0
votes
0answers
88 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} ...
3
votes
1answer
54 views

Proving that $\sqrt{n^2+1}-n = F(n), n \in \mathbb{N}_{>0}$

Let $F(n)$ denote a infinite continued fraction of form such that: $$F(n) = \cfrac{1}{2n + \cfrac{1}{2n + \cfrac{1}{2n + \cfrac{1}{2n + \cfrac{1}{\dots}}}}}$$ Consider the following equation: ...
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
120 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
433 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
907 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
129 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
100 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
361 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
91 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
281 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
122 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
113 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
489 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
238 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
92 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
310 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
86 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
215 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
722 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
443 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
87 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
121 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
93 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
78 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
559 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
179 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
100 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
315 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
68 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
85 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
79 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
41 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 ...