Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [continued-fractions]

A 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.

0
votes
1answer
24 views

Sequence Converging to the Square Root of an Integer $S \gt 1$

I noticed this answer to the question $\quad$ Continued fraction of a square root and the comment So I felt obliged to take this on using the theory of sequences. Confession: I find it difficult ...
1
vote
0answers
43 views

Distribution function of random continued fractions [on hold]

Let $F(x)$ be the cumulative distribution function of $X$, $$X=\frac{X_1}{1+\frac{X_2}{1+\frac{X_3}{1+…}}}$$ where $X_1,X_2,X_3,\cdots$ are independent and every $X_i$ has a standard uniform ...
4
votes
2answers
85 views

Integral over recurrence relationship

I'm interested in evaluating the following definite integral \begin{equation} I_n = \int_0^{\gamma} F_n(x)\:dx \end{equation} Where $\gamma \gt 0$ and $F_n(x)$ is based on the recurrence ...
4
votes
2answers
105 views

Proof of this formula for $\sqrt{e\pi/2}$ and similar formulas.

\begin{align} \sqrt{\frac{e\pi}{2}}=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}} \end{align} as seen here. ...
0
votes
1answer
22 views

Simple continued fraction for irrational numbers.

I read it here that: "What you must have read is that a number with an infinite simple continued fraction expansion is irrational. A continued fraction is "simple" if all the partial numerators are ...
1
vote
1answer
58 views

I have found a way of computing Euler's number. Is there any possible intuition of how that might be the case?

So a few days ago I just kind of messed around with my calculator, when I had an idea about a new continued fraction. I inputted it, and I found that it converged really quickly, and, quite wondrously,...
0
votes
1answer
93 views

Continued fraction of $\phi^3$

I found that $$\phi^3=4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ How should I prove this? Attempt: Suppose$$x= 4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ To ...
1
vote
1answer
26 views

Identifying the space of geodesics on the hyperbolic plane as a topological space.

On pg. 2 of this PDF, the author defines $\mathcal G=(-\infty, 0)\times(0, 1)$ and mentions that $\mathcal G$ can be thought of as "a space of geodesics on the hyperbolic $2$-space $\mathbb H^2$." ...
7
votes
1answer
149 views

How to prove that $\sum_{k=1}^{\infty}\frac{k^{n+1}}{k!}=eB_{n+1}$

Through some calculation, it can be shown that $$e = 1+\cfrac{1+\cfrac{1+\cfrac{1+\cfrac{\vdots}{4}}{3}}{2}}{1}\tag{1}$$ $$2e = 1+\cfrac{2+\cfrac{3+\cfrac{4+\cfrac{\vdots}{4}}{3}}{2}}{1}\tag{2}$$ $$5e ...
1
vote
0answers
29 views

How do I prove that $\mathcal{P}_d(\mathbb{Z})^+\neq\varnothing$ and that Pell's equation solutions are convergents

I recently had to skip a number theory lecture because I was sick, and they proved that the set $$\mathcal{P}_d(\mathbb{Z})^+=\lbrace(x_0,y_0)\in\mathbb{Z}^2_{\geq1}:x_0^2-dy_0^2=1\rbrace\neq\...
1
vote
1answer
51 views

$N \in \mathbb{N}$ is not a square, show that the continued fraction expansion of $\sqrt N/\lfloor\sqrt N\rfloor$ is $[1,\overline{a_1,a_2,\dots,2}]$

Let $N \in \mathbb{N}$ not a square, show that the continued fraction expansion of $\sqrt{N}/\lfloor\sqrt{N}\rfloor$ is $[1,\overline{a_1,a_2,\dots,2}]$. My notations: the fractional part of $a$ is ...
0
votes
0answers
36 views

Proof a set of Continued fraction is perfect

so in my book there is a problem which asked to proof the set of Continued fraction in $\mathbb{R}$ with euclidean metric ,call it $K$, with only $3$ and $4$ is perfect. I know i have to show that $K=...
4
votes
1answer
98 views

Show that $\pi ≈ 355/113$ is the best rational approximation of $\pi$ with a three-digit denominator

Given that the continued fraction expansion of $\pi$ is $[3, 7, 15, 1, 292, ...]$. Prove that $\pi ≈ [3, 7, 15, 1]= 355/113$ is the best approximation of $\pi$ with a 3-digit denominator. From the ...
3
votes
2answers
47 views

Symmetric continued fractions property where $q^2\equiv(-1)^n$ mod $p$

Let $[a_0,a_1,a_2,\ldots,a_n,a_n,\ldots,a_2,a_1,a_0]=:\frac{p}{q}\in\mathbb{Q}$ be a symmetric continued fraction. This sequence of $a_i$'s consists of finitely many elements because $\frac{p}{q}$ is ...
0
votes
0answers
77 views

Evaluating the continued fraction

How to evaluate $\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\frac{1}{\ddots}}}}}$, where $a_{j}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{j}$? That is, How to evaluate ...
0
votes
0answers
17 views

Digit Accuracy of the nth Square Root convergent

A spinoff of the question here: Estimation on the accuracy of the convergents of $\sqrt{a}$ Given a square root convergent, the accuracy to $p$ decimal places can be approximated by: $p$ ≈ digits$_{\...
0
votes
1answer
25 views

Predicting the change in the denominator of a continued fraction when reversing the order of $a_1$ through $a_n$.

When reversing the order of $a_1$ through $a_n$ in a continued/extended fraction, (ie. [$a_1$: $a_2$, ... $a_{n-1}$, $a_n$] becomes [$a_n$: $a_{n-1}$, ... $a_2$, $a_1$]) we see that the denominator ...
13
votes
2answers
335 views

Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$ Prove it converges and, evaluate the series. For the first part of the question, I prove it ...
0
votes
0answers
16 views

Better approximation than convergent of continued fraction

Let $\alpha = [a_0;a_1,a_2,\ldots] \in \mathbb{R}$ and $(p_n/q_n)$ be the convergents to the continued fraction of $\alpha$. Prove that, if $q_n \leq q < q_{n+1}$, gdc($p$,$q$)$=1$ and $p/q \not = ...
0
votes
1answer
46 views

Continued fractions approximation using golden ratio

Hello today my friend helped me with my problem, but he did not give me any additional informations why it works like that. Let's suppose that I need to get ln(n) using continued fractions. He told ...
4
votes
1answer
49 views

Closed form for $K(n)=[0;\overline{1,2,3,…,n}]$

I just started playing around with fairly simple periodic continued fractions, and I have a question. The fractions can be represented "linearly": for $n\in\Bbb N$, $$K(n)=[0;\overline{1,2,3,...,n}]$$ ...
4
votes
0answers
69 views

Is there a proof of an infinite number of prime numbers using the irrationality of $e$?

That the set of prime integers is infinite can be proved using the irrationality of $\pi$; see this wikipedia link. It analyzes the representation $\tag 1 {\displaystyle {\frac {\pi }{4}}={\frac {3}{...
1
vote
1answer
30 views

Representation of Transcendental number via continued fractions

My question is quite simple. As far as I know, rational numbers has finite continued fraction representation. And if for any number we have infinite continued fraction, then it necessarily is ...
16
votes
4answers
2k views

Limit of a sequence of integrals involving continued fractions

The following question was asked in a calculus exam in UNI, a Peruvian university. It is meant to be for freshman calculus students. Find $\lim_{n \to \infty} A_n $ if $$ A_1 = \int\limits_0^...
0
votes
0answers
27 views

limit of a recursive sequence (finite “backwards” continued fraction)

Let $\rho_n$ be a sequence of positive numbers converging to $1$. Fix $\lambda > 0$. Let $$u_3 = - \rho_2 / \lambda$$ and consider the following sequence $$u_{n+1} = \frac{-1}{\frac{\lambda}{\...
0
votes
1answer
29 views

Find continued fractions and corresponding rationals

I want to compute the rationals that the continued fractions $[4;2,1,3,1,2,4]$ and $[0;1,2,3,4,3,2,1]$ represent. Also, I want to find the continued fractions of the rationals $-\frac{19}{51}, \frac{...
0
votes
1answer
22 views

For which the below fraction refer to?

I have got this fraction representation :$$a=\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots}}}} $$ but i can't know for which it's refer to , I mixed that with Golden ratio however the ...
0
votes
0answers
19 views

Let $C_k = \frac{q_k}{p_k}$ denote the $k^{th}$ convergent of $[a_0;a_1,a_2,\dots]$. Prove that $p_kq_{k-1}-p_{k-1}q_k = (-1)^{k-1}$.

Let $C_k = \frac{q_k}{p_k}$ denote the $k^{th}$ convergent of $[a_0;a_1,a_2,\dots]$. Prove that $$p_kq_{k-1}-p_{k-1}q_k = (-1)^{k-1}.$$ I am not sure how to prove this and I am not sure what it even ...
1
vote
1answer
46 views

Convergence of the continued fraction [1,1,1,…]

I'm trying to show that the continued fraction $[1,1,\dots]$ converges. After that it is easy to determine the limit, so I'm interested in a proof of convergence specifically. I don't think it's ...
0
votes
0answers
15 views

Prove $aB_m-bA_m=[q_n,\cdots,q_m]$, where $\frac{a}{b}$ is a rational and $\frac{A_m}{B_m}$ is a convergent.

Let the natural numbers $q_0,q_1,\cdots ,q_n$ be the $n$ terms in the continued fraction expansion of the rational number $\frac{a}{b}$, that is $$\frac{a}{b}=q_0 +\frac1{q_1+}\cdots\frac1{q_n}=\frac{[...
2
votes
4answers
72 views

Convergence to $\sqrt{2}$

It is a very good way to approximate $\sqrt{2}$ using the following; Let $D_{k}$ and $N_{k}$ be the denominator and the numerator of the $k$th term, respectively. Let $D_1=2$ and $N_1=3$, and for $k\...
1
vote
1answer
47 views

Book on the Lambert W function.

The Lambert W function is the inverse of function $x\mapsto xe^x$. It is traditionally denoted by $W(x)$. The function $W(x)$ is bivalued in interval $(-\frac{1}{e},0)$. See Wikpedia and Wolfram for ...
1
vote
1answer
92 views

Finding the limit of the sequence $(1, \frac{1}{2+1}, \frac{1}{2+\frac{1}{1+1}},\cdots)$

I am struggling to find the limit of the sequence: $$1, \cfrac{1}{2+1}, \cfrac{1}{2+\cfrac{1}{1+1}}, \cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+1}}}, \cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+1}}}},\...
1
vote
1answer
29 views

continued fraction development of the numbers of the following form, $\sqrt{n^2-1}$

Determine the continued fraction development of the numbers of the following form, $\sqrt{n^2-1}$, with $n>1$ an integer. I wasn't sure how to tackle this, so I just tried to write it out and ...
5
votes
1answer
207 views

Is it true that there are infinitely many square numbers of the form $\lfloor n\sqrt{2}\rfloor$?

Is it true that there are infinitely many square numbers in the sequence $\left \lfloor n \sqrt{2} \right \rfloor $ with $n$ is a whole number? If not, how can we prove it?
15
votes
1answer
227 views

Coefficients of binomial continued fractions

For a natural number $n$, let $$ \begin{equation} \beta_n(z)=\frac{(1+z)^n+(1-z)^n}{(1+z)^n-(1-z)^n}. \end{equation} $$ Then the coefficients of the numerator and denominator of $\beta_n$ are binomial....
2
votes
1answer
128 views

Determinants of products of binary matrices and binomial coefficients

Consider two binary semi-infinite matrices with obvious patterns: $$ C= \begin{bmatrix} 1 &0 &0 &0 &0 &0 &0 &\cdots\\ 1 &0 &0 &0 &0 &0 &0 &\...
0
votes
0answers
26 views

forming the continued fraction of the euler number

Good afternoon The euler number is a irrational number. And you can have a infinite continued fraction of euler number. But how can you form the coninued fraction of euler number?
-1
votes
1answer
64 views

description in mathematics [closed]

I have a general question. I would like to describe the different ways of representing euler's number. My question is how to describe something in mathematics. The Euler number can be represented ...
0
votes
0answers
21 views

Continued fractions formula

How would I find the continued fraction of and number $a/b$? For example, $5/8$ = $1/(1+3/5)$ I tried using its decimal expansion but couldn’t find anything, and I want to be able to describe the ...
13
votes
4answers
304 views

What is this function related with continued fractions?

Playing with continued fractions, I came with the idea of iterating the limit of the simplest one: $$1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}}}\ = \Phi$$ And then I ...
18
votes
1answer
259 views

Continued fraction involving Fibonacci sequence

What is the limit of the continued fraction: $$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}\ $$ that involves the Fibonacci sequence terms as denominators? I'...
2
votes
0answers
47 views

$d \in \mathbb{N}$, that is not a square, show that the continued fractions for following numbers are purely periodic

Let $d \in \mathbb{N}$ such that $d$ not a square number. Now show that the continued fractions for $\sqrt{d} + \lfloor\sqrt{d}\rfloor$ and $\frac{1}{(\sqrt{d} - \lfloor\sqrt{d}\rfloor)}$ are ...
1
vote
0answers
80 views

Continued Fraction of $ \frac{7 + 2 \sqrt{2}}{9 + \;\,\sqrt{2}}$ with coefficients in $\mathbb{Z}[\sqrt{2}]$

We can read from various sources that $\mathbb{Q}(\sqrt{2})$ has class number one, and that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain. However, it also has a group of units: $\mathbb{Z}[\sqrt{2}]^\...
6
votes
0answers
112 views

Proving that the sequence $1, \frac12, \frac{1/2}{3/4}, \cdots$ converges to $\frac{\sqrt 2}2$ [duplicate]

I saw this on a Facebook page: \begin{align*} 1, \qquad \frac12,\qquad\frac{\frac12}{\frac34}, \qquad\frac{\frac{\frac12}{\frac34}}{\frac{\frac56}{\frac78}},\qquad \dots\to \frac{\sqrt{2}}2. \end{...
1
vote
3answers
52 views

expressing $\frac{a_0 + a_1 + \cdots}{b_0 + b_1 +\cdots}$ in terms of $\frac{a_0}{b_0}, \frac{a_1}{b_1},\ldots$

Is it possible to rewrite fractions of the form $\frac{a_0 + a_1 + \dots + a_N}{b_0 + b_1 + \cdots + b_N}$ in terms of $\frac{a_0}{b_0}, \frac{a_1}{b_1},\ldots, \frac{a_N}{b_N}$? under which ...
0
votes
1answer
54 views

Why is this value a reduced quadratic irrational

The paper: http://web.math.princeton.edu/mathlab/jr02fall/Periodicity/alexajp.pdf Note: The paper defines a reduced quadratic irrational $\alpha$ as: $\alpha > 1$ Its the root of a quadratic with ...
2
votes
0answers
46 views

Is there a distinguishable characteristic between the summation / continuous fraction method in algebraic and transcendental numbers?

I try to illustrate the formation and derivation of algebraic numbers and transcendental numbers. I found that both categories of numbers can be made by continuous summation or division/fraction ...
3
votes
0answers
81 views

Binary eigenvalues matrices and continued fractions

I'm working on a rational approximation of the square root function by continued fractions in the complex plane. The following kind binomial coefficients Hurwitz matrices (for $n=4$ and $6$) play ...
0
votes
0answers
61 views

Use continued fractions to find the fraction k/r that satsfies $\left|\frac{427}{512} - \frac{k}{r}\right| \leq \frac{1}{1024}$

Math wizards: I believe there is only one solution to this for which r>k and that is 5/6. As to the how you get there, I have no idea. $ \vert {\frac{427}{512} - \frac{k}{r}} \vert \leq \frac{1}{...