For questions on continued fractions.

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2
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2answers
107 views

All Even-Numbered Convergents of a Finite Continued Fraction Are Less Than the Value

Let $x=[a_0;a_1,a_2]$ be shorthand notation for the continued fraction $$x=a_0+\frac{1}{a_1+\frac{1}{a_2}}.$$ Then every $x\in\mathbb{Q}$ can be represented as a finite continued fraction ...
4
votes
1answer
173 views

How can I find power series of $f(x)$?

$$f(x)=\dfrac{1}{1+\dfrac{x}{1+\dfrac{x^2}{1+\dfrac{x^3}{1+\dfrac{x^4}{\ddots}}}}}$$ How can a power series be found given the continued fraction $f(x)$? I'm trying to find $f(x) ...
0
votes
2answers
97 views

Narrowing a Stern-Brocot tree

Say I only wanted to enumerate the rational numbers between 0 and $a$. Is there a way to "narrow" a Stern-Brocot tree to provide this? I tried keeping my left bound at "$\frac{0}{1}$" and setting my ...
3
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0answers
75 views

Is that observation really a property of the log of coefficients of continued fractions (example: cf(log(3)/log(2))

I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small ...
5
votes
2answers
173 views

Calculating the continued fraction of $\sqrt{47}$ using a different result

I have calculated the continued fraction of $\alpha=\frac{6+\sqrt{47}}{11}$ which equals $\overline{[1,5,1,12]}$. Now I am asked to calculated the cont. fraction of $\sqrt{47}$ using this result. I am ...
1
vote
1answer
89 views

Calculate the quadratic irrational number given by a certain periodic cont. fraction

Calculate the quadratic irrational number $\alpha$ given by the periodic continued fraction $\alpha = \overline{ [1,2,1] } $. To be honest I am not sure how to tackle this one. I know the algorithm ...
3
votes
2answers
77 views

Unique continued fraction

If $x$ is a uniformly random number in $[0,1]$, what distribution should the $n$-th term in its continued fraction expansion follow? What is the expected vale of $a_n$ in $[a_0;a_1,a_2,\dots]$? Here ...
8
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1answer
92 views

Do best lower approximations of a quadratic irrational always form a linear recurrence sequence?

Let $\theta$ be an irrational number and let $$ {\cal L}= \bigg\lbrace (a,b) \in {\mathbb Z} \times {\mathbb N}^{*} \bigg| \frac{a}{b} \leq \theta \bigg\rbrace $$ and $$ {\cal B}= \bigg\lbrace ...
2
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0answers
185 views

On the continued fraction of $e$

The following question comes up during analysis of Padé approximants to $e^x$ (see my related question in MathOverflow for more background). Recall that the continued fraction expansion of $e$ is $$ e ...
3
votes
3answers
170 views

What do High-Water Marks in Continued Fractions mean?

While reading through several articles concerned with mathematical constants, I kept on finding things like this: The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, ...
3
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0answers
138 views

How to prove that this series $f(z)=1+\sum_{k=1}^{\infty}2^{-k z}$ converges using the theory of continued fractions?

Consider the following series \begin{equation} f(z)=1+\sum_{k=1}^{\infty}\frac{1}{2^{k z}} =1+ \sum_{n=1}^{\infty}\left( \prod_{k=1}^{n}\frac{1}{2^{z}} \right) \end{equation} Using Euler's continued ...
7
votes
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; ...
1
vote
1answer
2k views

Converting a square root into a simple continued fraction

One can convert a square root $\sqrt{n}$ into a continued fraction $[a_0; \overline{a_1, a_2, \dots , a_k}]$ following the algebraic algorithm explained here: ...
0
votes
1answer
68 views

Continued Fraction: Why do we get with $\gamma \in \mathbb{R}\setminus \mathbb{Q}$ the CF $\frac{1}{\gamma}=\langle0;a_0,a_1,\dotsc\rangle$

I have a question concerning continued fractions: If we have $\gamma \in \mathbb{R} \setminus \mathbb{Q}$ and $\gamma=\langle a_0;a_1,a_2,\dotsc\rangle$. Why do we get $$\frac1\gamma = \langle ...
3
votes
0answers
250 views

continued fractions and convex hulls

If I remember correctly, there is a nice correspondence between continued fractions and convex hulls of lattice points in the plane. If $\theta>0$ is the slope of a line in $\mathbb{R}^2$ passing ...
3
votes
3answers
456 views

Continued fraction: Show $\sqrt{n^2+2n}=[n; \overline{1,2n}]$

I have to show the following identity ($n \in \mathbb{N}$): $$\sqrt{n^2+2n}=[n; \overline{1,2n}]$$ I had a look about the procedure for $\sqrt{n}$ on Wiki, but I don't know how to transform it to ...
5
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0answers
202 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
2
votes
1answer
128 views

Reciprocal of a continued fraction

I have to prove the following: Let $\alpha=[a_0;a_1,a_2,...,a_n]$ and $\alpha>0$, then $\dfrac1{\alpha}=[0;a_0,a_1,...,a_n]$ I started with ...
2
votes
1answer
83 views

Show $p(n)=n(p_{n-1}+p_{n-2})+(n-1)p_{n-3}+(n-2)p_{n-4}+…+3p_1+2p_0+2$

I have to show the following: Let $N_k=\frac{p_k}{q_k}$ with $\alpha=\langle 1;2,3,4,...,n,n+1\rangle$ and $n \in \mathbb{N}$. Then $\forall n \in \mathbb{N}$ with $n\geq 3$, ...
5
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2answers
171 views

A continued fraction involving composite numbers

What is the limit of the continued fraction whose partial denominators are the composites?
2
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2answers
199 views

Taking the negative of a continued fraction

If I have a continued fraction for an irrational number $z= \langle a_0;a_1,a_2,a_3,\ldots\rangle$ it seems that $(-1)*z = \langle-a_0;-a_1,-a_2,-a_3,\ldots\rangle$. Is this true? In general, if you ...
2
votes
1answer
397 views

Convergent of continued fractions the best rational approximation of a number? [duplicate]

Possible Duplicate: A nicer proof of Lagrange's 'best approximations' law? I was reading through the wikipedia article on continued fractions, and they state, essentially, that ...
13
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1answer
618 views

Arithmetic of continued fractions, does it exist?

I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider $$ ...
11
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3answers
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 ...
20
votes
1answer
611 views

A continued fraction involving prime numbers

What is the limit of the continued fraction $$\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{11+\cfrac{1}{13+\cdots}}}}}}\ ?$$ Is the limit algebraic, or expressible in terms of e or ...
5
votes
1answer
228 views

Is it right that the fundamental recurrence of an arbitrary continued fraction cannot be proved without induction?

Let $\dfrac{A_{n}}{B_{n}}$ be the $n^{th}$ convergent (approximant) $$ \frac{A_{n}}{B_{n}}=b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{\begin{array}{c} b_{3}+ \\ \\ \end{array} ...
1
vote
3answers
191 views

Continued fraction form for rational numbers less than $1$

How could we convert a rational number (less than $1$) to the continued fraction form? This is probably an extension of this question. After reading Bill Dubuque's answer here and here, I got ...
1
vote
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 ...
3
votes
1answer
331 views

How to express an irrational as a continued fraction in computer with high precision?

Background I'm writing a C++ library for continued fraction using MPIR (Multiple Precision Integers and Rationals) library http://www.mpir.org/ due to the limitation of built-in ...
4
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0answers
259 views

Evaluating matrix-continued fractions?

I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity $A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...
3
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0answers
184 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
4
votes
3answers
270 views

Is it practical to use infinite continued fraction to generate random numbers?

I observed the pattern of this irrational number: $$\sqrt{1 + \sqrt{2}}$$ and realized that each element $a_i$ occurred very randomly. For the first 100 elements, this is the result: ...
4
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4answers
261 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, e and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
2
votes
1answer
280 views

Is the Iterated Continued fraction from Convergent​s for Pi/2 exactly 3/2?

Iterated continued fraction from convergents are described at https://oeis.org/wiki/Convergents_constant and https://oeis.org/wiki/Table_of_convergents_constants. Do you think there is any error in ...
0
votes
3answers
1k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
7
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1answer
231 views

What causes the convergence of Iterated continued fractions from convergents?

Here is a small discovery I stumbled across a few weeks ago. I hope at least one person will find it interesting enough to help me. The iterated continued fractions from convergents (or convergents ...
16
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3answers
879 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 ...
5
votes
1answer
235 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, ...
3
votes
1answer
136 views

How to find continued fraction of the form $a\sqrt{b}$?

For the form $\sqrt{b}$, I could just apply the recursive quadratic formula: $$P_{k+1} = a_kQ_k - P_k$$ $$Q_{k+1} = \dfrac{d - P^2_{k+1}}{Q_k}$$ $$\alpha_k = \dfrac{P_k + \sqrt{d}}{Q_k}$$ ...
12
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1answer
440 views

Deriving a trivial continued fraction for the exponential

Lately, I learned about the following continued fraction for the exponential function: $$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$ I thought it was ...
6
votes
2answers
202 views

A question on continued fraction

Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive convergents of its simple continued fraction, where $k\ge 1$. Is it possible that both ...
10
votes
3answers
684 views

Proving the continued fraction representation of $\sqrt{2}$

There's a question in Spivak's Calculus (I don't happen to have the question number in front of me in the 2nd Edition, it's Chapter 21, Problem 7) that develops the concept of continued fraction, ...
2
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1answer
263 views

Solving equations using continued fractions?

We solve the pell equation using the continued fraction for square root of 2. What equations can we solve using the continued fraction of cube roots (and other numbers too)?
7
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0answers
431 views

What can Euler's identity teach us about (generalised) continued fractions?

We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ ...
6
votes
1answer
315 views

What is the length of a continued fraction expansion of a rational number?

I was reviewing quantum factorization and am slightly unclear on a classical detail of order-finding. Given a (suitably nice) periodic function $f$ with unknown period $r$ and a power of two $N > ...
10
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1answer
489 views

Continued Fraction expansion of tan(1)

Prove that continued fraction of tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a ...
4
votes
2answers
302 views

Adding integers to an infinite continued fraction expansion doesn't change the value?

I'm learning about continued fractions, and I've enjoyed them so far, but I'm unsure if I've done the following correctly. I have no real experience with analysis, so I'm not sure if my reasoning is ...
5
votes
1answer
413 views

Continued Fraction of an Infinite Sum

What is the continued fraction for $\displaystyle\sum_{i=1}^n\frac{1}{2^{2^i}}$ It seems to be "almost" periodic, but I can't figure out the exact way to express it.
7
votes
3answers
658 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 ...
10
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2answers
868 views

How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?

Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...