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.

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How to calculate this limit related to continued fractions?

Given that $\lim_{n\to\infty}p_n/q_n=\sqrt{2}$, how to calculate $\lim_{n\to\infty}q_n|p_n-\sqrt{2}q_n|$? Here, $p_n$ and $q_n$ are defined by the continued fraction of $\sqrt{2}$, and $p_n=a_np_{n-1}+...
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Pell-Type Equation: show that the solutions $x,y$ are such that $\frac{x}{y}$ is a convergent

Prove that for any solution of the equation $x^2 − 14y^2 = 2$, in positive integers $x, y$ the value $\frac{x}{y}$ is a convergent of $\sqrt{14}$ I am not too sure where to start, I have the continued ...
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Expressing the continued fraction $[k,\dots,k]$ as a closed form

For positive integers $a_1,\dots,a_n\geq 2$, let $[a_1,\dots,a_n]$ denote the continued fraction $$ [a_1,\dots,a_n]=a_1-\frac{1}{a_2-\frac{1}{\cdots-\frac{1}{a_n}}}.$$ Then we have $[2,\dots,2]=(n+1)/...
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A pattern that leads to regular continued fractions of quadratic irrationals [closed]

The following expression can be obtained by converting the continued fraction of quadratic irrationals to single fraction. $$ \sqrt{N} = \frac{b\sqrt{N}+aN}{a\sqrt{N}+b} $$ The equation holds for any ...
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Find an expansion into an arithmetic continued fraction of $\sqrt{x}$, where $x=((4a^2+1)b+a)^2+4ab+1$, $a,b \in \mathbb{N}$

I am trying to find an expansion into an arithmetic continued fraction of $\sqrt{x}$, where $x=((4a^2+1)b+a)^2+4ab+1$, $a,b \in \mathbb{N}$. So far I have: Clearly $x<((4a^2+1)b+a+1)^2$, so the ...
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why is $ 2 = \frac{5}{1+\frac{8}{4+\frac{11}{7 + \frac{14}{10 + \dots}}} } $

Why is $ 2 = \cfrac{5}{1+\cfrac{8}{4+\cfrac{11}{7 + \cfrac{14}{10 + \ddots}}} } $ where the sequences $5,8,11,14,\dots$ and $1,4,7,10,\dots$ are of the form $5 + 3 n$ and $1 + 3n$. (This converges on ...
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Prove that $p/q$ is a convergent of the continued fraction expansion of $x$ given $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$

Prove that, if $x$ is any irrational number, and if $p/q$ is a rational fraction in lowest terms, with $q\geq 1$, such that $$\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$$ then it can be proved that $p/...
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A pattern of periodic continued fraction

I am interested in the continued fractions which $1$s are consecutive appears. For example, it is the following values. $$ \sqrt{7} = [2;\overline{1,1,1,4}] \\ \sqrt{13} = [3;\overline{1,1,1,1,6}] $$ ...
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How is this cube root continued fraction generated?

I'm looking at the examples on the Wiki Page Generalized Continued Fractions . The introduction to general root-finding states a formula: But then, looking at the second example for cube root of 2, ...
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4 votes
3 answers
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Finding Integer Approximations

The Saros is the time period for the draconic month ($T_d$ = 27.212220815 days), synodic month ($T_s$ = 29.530588861 days) and anomalistic month ($T_a$ = 27.554549886 days) to approximately match. ...
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Relationship between Dirichlet's Approximation Theorem and Convergents

For any real number $r$, the convergents to the continued fraction expansion of $r$ satisfy Dirichlet's approximation inequality of $|r - \frac{p}{q}| < \frac{1}{q^2}$. Does this go the other way? ...
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Continued fraction for $\sqrt 2$

I discovered this continued fraction for $\sqrt 2$ but could not find any sources in which it appeared. It goes as follows: \begin{equation} \sqrt{2} = 1 + \frac{1 + \frac{1 + \frac{1 + \dots}{3 + \...
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Non-periodic continued fraction with explicitly known convergents?

Is an irrational number with non-periodic continued fraction expansion known, for which one can give explicit formulas for the convergents $p_n/q_n$ or at least for the denominators $q_n$ (similar to ...
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Pell's equation fundamental solution vs. an earlier convergent of $\sqrt{d}$

Let $(p,q)$ be the fundamental solution to Pell's equation $x^2-dy^2=1$, which makes $\frac{p}{q}$ a convergent of $\sqrt{d}$, per the theorem underlying the continued fraction algorithm for solving ...
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8 votes
2 answers
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Reduce precision of fraction

Say I have a reduced fraction where the numerator and denominator can only be integers: $$ \frac{1071283}{28187739} $$ and I want to reduce it more, accepting the lose of precision. I could just ...
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Proof of equivalence of two continued fractions

The Problem: Show that $$ \underbrace{a\left(x_1+\cfrac{1}{ax_2+\cfrac{1}{x_3+\cfrac{1}{{ax_4+{}}_{\ddots}}}}\right)}_\text{2n quotients}=\underbrace{ax_1+\cfrac{1}{x_2+\cfrac{1}{ax_3+\cfrac{1}{{x_4+{}...
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2 votes
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Book references about continued fractions

I'm searching for one or more books about continued fractions which covers these aspects: An introductory level book about continued fractions. A divulgative book about continued fractions with some ...
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More general continued fraction formulas

Incidentally, I ran into the "Ramanujan Machine" and was wondering if the many formulas for certain constants can be generalized. For starters, what is the most simple nontrivial CF $1/(a_1+...
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2 votes
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An unproven algorithm for generating continued fractions for square roots.

I found that the following algorithm works correctly for $N=1 \sim 1000$ although I haven't been able to prove it. If this is already known, please point it out. If not, please prove it. Algorithm ...
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Is there a pattern to the roots of continued-fraction equations with depth $n$?

I've found what looks like the start of a pattern, but I don't have the tools or background to know if one really exists here. Does anyone know how to analyze this mathematical behavior and continue ...
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Products of generalized continued fractions of rational numbers

For integers $a_1, \ldots a_n$, define the generalized continued fraction expression \begin{align} [a_n; a_{n-1}, \ldots, a_1] = a_n - \frac{1}{[a_{n-1}; a_{n-2}, \ldots, a_1]} \quad \text{and} \...
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Formula for the period length of the simplest continued fraction of the square root of n

The period length of the simplest continued fraction of the square root of any natural number n is always finite. Is there a formula for the period length of the simplest continued fraction of the ...
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Algorithm to add continued fractions

Lets say I have two continued fractions: $$a=[a_0; \overline{a_1, a_2,a_3...}]$$ $$b=[b_0; \overline{b_1, b_2,b_3...}]$$ How do I add them together?
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Proof of Parabola Theorem in "two families of twin convergence regions for continued fractions"

I am attempting to understand a proof of the Parabola theorem, which states: If all elements of the continued fraction $\mathop{\Large\text{K}}\frac{a_n}{b_n}$ lie in a parabolic region $ P_{\alpha} = ...
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Periodicity of $\frac{1}{x}-\frac{1}{\frac{3}{x}-\frac{1}{\frac{5}{x}-\frac{1}{\cdots}}}$ and others

The function $$f(x)=\frac{\color{blue}{1}}{x}-\frac{1}{\frac{\color{blue}{3}}{x}-\frac{1}{\frac{\color{blue}{5}}{x}-\frac{1}{\cdots}}}$$ is periodic with minimal period $\pi$. But what if we replace ...
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Trying to solve problem related to continuous functions [closed]

A function $f$ is defined by $$ \begin{equation} f(x)= \begin{cases} -2x^3-3 \text{ for } x < -1 \\ ax+b \text{ for } -1 \leq x < 1 \\ 2x^2+1 \text{ for } 1 \leq x\\ \end{cases} \...
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Closed form expression for continued fraction

While I was trying to determine the value of the following infinite series: $$\displaystyle\sum_{n=1}^{\infty}\displaystyle\prod_{k=1}^n\frac{1}{2k+1}$$ I realized that it is equal to the value of the ...
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Is there a general way to compute a continued fraction, knowing the pattern?

Finding the value of some continued fractions is easy. For instance, $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\ddots}}}}}$. To evaluate that, we can let the expression be $x$, so what ...
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A remarkable Continued Fraction for $\pi$

How to analyze the Continued Fraction $3+\cfrac{\color{red}1^2}{5+\cfrac{\color{blue}4^2}{7+\cfrac{\color{red}3^2}{9+\cfrac{\color{blue}6^2}{11+\cfrac{\color{red}5^2}{13+\cfrac{\color{blue}8^2}{15+\...
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7 votes
1 answer
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Closed expression for a continued fraction

Does anyone know a closed expression for the following continued fraction? $$G(x) = \cfrac{1}{x+1+\cfrac{1}{x+3+\cfrac{4}{x+5+\cfrac{9}{x+7+\cfrac{16}{x+9+\cdots}}}}}$$ All I know is that $G(0) = \...
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1 vote
1 answer
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Closed expression for a given continued fraction

Does anyone know a closed-form expression for the continued fraction $$F(x) = \cfrac{x}{x+\cfrac{x}{x+1+\cfrac{2x}{x+2+\cfrac{3x}{x+3+\cfrac{4x}{x+4+\cdots}}}}}?$$ According to page 181 of An ...
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How to approximate a rational number $m/n$ with fractions $a/b$ such that the product $ab$ is given

Consider a certain rational number $\alpha=m/n$ and let $N$ be a positive integer. Is there a way to find the better rational approximation $x/y$ of $\alpha$ between all the fraction such that $xy=N,$ ...
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fractional part of square root 2 to contain a subsequence that converges to 0

the question is Consider the sequence $a_n = \{({{n*\sqrt{2}}})\}, n ≥ 1. $ where {} means the fractional part of $a$. Show that the sequence an contains a subsequence that converges to 0 here's my ...
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How to transform continued fraction expansion into infinite series?

I want to know if it is possible to transform a continued fraction expansion into infinite series. If this is not possible in general case, I want to know if we can do it when the function is always ...
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Higher order roots as Continued Fractions

All square roots can be expressed as a regular Continued Fraction. For example: $$\newcommand{\contfrac}{\raise{-0.5ex}\mathop{\Large\mathrm{K}}}\sqrt{2}+1=2+\contfrac_{n = 0}^\infty \frac{1}{2}$$ It ...
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How many simple continued fractions have $p_k=n$?

Let $\alpha$ be a number with continued fraction $[a_0; a_1, a_2, ...]$, and let $\frac{p_k}{q_k}=[a_0;a_1,a_2,...,a_k]$ be the $k$-th convergent. I'm interested in how many different fractions ...
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Infinite continued fractions

I am fascinated by the fact that "important" irrational numbers like the golden ratio, base of the natural exponent, pi, square roots have a "regular" representation as an infinite ...
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Request for proof: Regularity of pi's continued fractions

Notice In this post, a continued fraction is represented as follows $$ a + \cfrac{1^2}{b+\cfrac{3^2}{b+\cfrac{5^2}{\ddots}}} = a +K^\infty_{k=1}\frac{(2k-1)^2}{b} $$ When I was checking the ...
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2 answers
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Is there a function for $[1;x,x,...]$ (as an infinite continued fraction)?

So I was experimenting with continued fractions and came up with an idea for a function $$f(x) =1+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\dots}}}}}},$$ as an infinite ...
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How are Ostrowski Numeration Systems built from a periodic continued fraction?

Ostrowski number systems represent integers and real numbers using the denominators of the convergents of continued fractions. One better known special case of this is Zeckendorf's Theorem and related ...
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Question on continued fraction? [closed]

I have seen in wikipedia that irrational numbers have infinite continued fraction but I also found $$1=\frac{2}{3-\frac{2}{3-\ddots}}$$ so my question is that does that mean $1$ is irrational because ...
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An uncommon continued fraction of $\frac{\pi}{2}$

I'm currently stuck with the following infinite continued fraction: $$\frac{\pi}{2}=1+\dfrac{1}{1+\dfrac{1\cdot2}{1+\dfrac{2\cdot3}{1+\dfrac{3\cdot 4}{1+\cdots}}}}$$ There is an obscure clue on this: ...
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1 answer
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Evaluation of the 'Odd Harmonic' Continued Fraction

How to prove that this continued fraction $1+\cfrac{1/1}{1+\cfrac{1/3}{1+\cfrac{1/5}{1+\cfrac{1/7}{1+\ddots}}}}$ evaluates to $\displaystyle\frac{2}{\displaystyle 1+\frac{I_1(\frac14)}{I_0(\frac14)}}$?...
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Mirrored "Harmonic" Continued fraction

The beautiful Continued Fraction $1+\cfrac{\cfrac{1}{1}}{1+\cfrac{\cfrac{1}{3}}{1+\cfrac{\cfrac{1}{5}}{1+\cfrac{\cfrac{1}{7}}{1+\ddots}}}}$ equals $ \frac{\displaystyle 2}{\displaystyle 1+\frac{\...
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Does this continued fraction have a name? [duplicate]

$1/(2+1/(3+1/(5+...) = [0; 2,3,5,...p_{\infty}] \approx 0.432332$ Does this constant have a name? What is it called? It does appear to converge from my initial calculations, and I'm surprised that I ...
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Sum of an infinite series with continued fractions

I'm trying to determine if it is possible to sum an infinite series of the form: \begin{equation} \frac{1}{x}+\frac{a}{x^2(x-\frac{a}{x})}+\frac{a^2}{x^2(x-\frac{a}{x})^2(x-\frac{a}{x-\frac{a}{x}})}+\...
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Borel-Berstein Theorem 's Proof

There is a theorem stated in the book. It involves a simple continued fraction of the form $\xi=\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\ddots}}}:=[a_1,a_2,a_3,\dots]$, that Theorem 197. If $\...
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-1 votes
1 answer
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Approximations of real numbers on $(0,1)$ with powers of the form $(3/2)^n \pmod 1$

Let $0<r<1$ a real number which is not a fraction of the form $p/2^n$ for any integers $p,n$. Now, for every integer $n\ge 1$ we can find the closest fraction of the form $p/2^n$ to $r$, which ...
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2 votes
1 answer
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A formula about best rational within an interval

In this Wikipedia page, it claims that for real numbers $x,y>0$, if $$x=[a_0;a_1,a_2,...,a_{k-1},a_k,...],$$$$y=[a_0;a_1,a_2,...,a_{k-1},b_k,...],$$ then $$z(x,y):=[a_0;a_1,a_2,...,a_{k-1},\min(a_k,...
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8 votes
1 answer
179 views

New way to obtain partial continued fractions of square roots?

Some background: I was working on a problem that asked for the sum of the cubes of the roots of a cubic, $x^3 - 5x^2 + 5x - 1$. I found that this factored into $(x-1)(x^2 - 4x + 1)$, meaning that its ...
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