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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Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
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209 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
9
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1answer
204 views

Minimum of $|az_x-bz_y|$

I am trying to minimize the following function: \begin{align} &f(z_x,z_y)=|az_x-bz_y| \\ &\text{ s.t. } z_x,z_y \in \mathbb{Z},1 \le z_x \le N_x \text{ and } 1 \le z_y \le N_y \text{ and } ...
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1answer
117 views

Continued fraction expansion of Pi (oeis A001203). [duplicate]

I would like to understand how you get the numbers $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+...}}}}$$ i.e. $\{3,7,15,1,292,...\}$ (A001203). In the comments of A046965 is explained a method ...
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319 views

Where did the negative answer come from in the continued fraction $1+\frac{1}{1+1/(1+\dots)}$?

In this question we increased solution domain by squaring both sides of equation but what about this one ? Here the question is to evaluate ...
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1answer
150 views

Induction proof for continued fractions

Recently while preparing for a maths test, I got this question in a book: Let $a(n) = 3 + \cfrac{1}{3+\cfrac{1}{3+\cfrac{1}{3+\cdots }}}$ till $n$ terms. Prove that $a(n) \cdot a(n-1)=3a(n-1)+1$ ...
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1answer
38 views

Continued fraction inequality: $q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$

In an article it is used the fact that $$q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$$ where $\alpha=[a_0;a_1,\ldots]$ is an irrational number and $q_i$ is the series of the best approximation ...
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1answer
64 views

Approximating a Continued Fraction

From a paper I was reading, If: $$w=\frac {1}{3}\left\{ \frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac ...
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1answer
60 views

L. Jacobsen and H. Waadeland: Glimt fra analytisk teori for kjedebrøker. Del 2.

I am trying to find the aforementioned paper online but have had no luck. I originally spotted it as a reference [26] for the paper Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, ...
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1answer
50 views

Using continued fractions to well-approximate a quadratic form?

Continued fractions are the "best rational approximation" of other numbers. For a real number $\alpha$ the continued fraction algorithm produces a sequence of integers $\alpha = [a_1, a_2, \dots, ...
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74 views

Continued Fraction Expansion

While reading "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the Ladies Diary " [p.602] from $F\left( -\dfrac {1}{2},-\dfrac {1}{2};1;\lambda^{2}\right)=1+\dfrac {\lambda ...
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1answer
121 views

Unsure about infinite continued fraction

How do you/is it possible to express $a=\cfrac{1}{2+\cfrac{3}{4+\cfrac{5}{6+\cdots}}}$ in the form $\frac{p}{q}(k+\sqrt{n})$? I'm still in high school, so I'm not familiar with especially ...
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181 views

Find a number by the decimal part of its square root [duplicate]

I have a math problem consisting of two questions: can we find a number N knowing only the decimal part of its square root up to a precision (only an approximation of the decimal part because the ...
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1answer
381 views

How to prove this determinant is $\pi$?

prove or disprove $$\pi=\begin{vmatrix} 3&1&0&0&0&\cdots\\ -1&6&1&0&0&\cdots\\ 0&-1&\dfrac{6}{3^2}&1&0&\cdots\\ ...
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80 views

Is $\{1,1,2,3,4,5,\cdots,i,\cdots \} $ the simple continued fraction algebraic or transcendental?

Is $$1+\cfrac{1}{1+\cfrac{1}{2+\cdots}} $$ or$\{1,1,2,3,4,5,\cdots,i,\cdots \} , i\in \mathbb{N}$ the simple continued fraction algebraic or transcendental? Any reference is appreciated EDIT and ...
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1answer
30 views

Any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Are there any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?
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2answers
93 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
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1answer
51 views

Great Common Division with Continued Fractions

If I have this GCD equation: $$89=16\cdot5+9\\ 16=9\cdot1+7\\ 9=7\cdot1+2\\ 7=2\cdot3+1\\ 2=1\cdot2+0$$ Then my continued fraction will be: $[5: 1, 1, 3, 2]$ But if I will have this GCD equation: ...
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134 views

Upward continued fractions

Has anybody seen "upward continued fractions", such as $$ \frac{1+\large{\frac{1+\large{\frac{1+...}{a_2}}}{a_1}}}{a_0} \quad? $$ These can be formed, for any real number $x$ with $0<x\le 1$, by ...
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218 views

Help understanding a geometric proof of the ergodicity of the Gauss measure for continued fractions

Any $x\in(0,1)$ can be written as a (regular) continued fraction $$ x = \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} = [a_1,a_2,a_3,\dotsc] $$ An irrational number has a unique expansion, ...
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179 views

Semi-convergent of continued fractions

I have read this from here The simple continued fraction for $x$ generates all of the best rational approximations for $x$ according to three rules: Truncate the continued fraction, and ...
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1answer
80 views

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
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123 views

Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
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76 views

Continued fraction and classification of real numbers.

I would be grateful if anyone can tell if there are any methods to classify real numbers using continued fraction. eg: Suppose $[a_0;a_1,a_2,\ldots,a_n]$ is the representation of some real number ...
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177 views

Generalizations of $\sum_{m=3n+2}^{\infty}\phi^m=\phi^{3n}$ and $\sum_{m=13n+1}^{\infty}(\sqrt2-1)^m=\dfrac{(\sqrt2-1)^{13n}}{\sqrt2}$

I noticed that the following identies hold with the help of wolfram alpha and oeis. I'm sure they're well-known, but I'd like to know how they generalize. ...
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1answer
566 views

An infinite series plus a continued fraction by Ramanujan

Here is a famous problem posed by Ramanujan Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + ...
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1answer
23 views

A weak inequality than Hurwitz

How can i prove that among any two consecuent convergents to x, al least one of the satisfy $|x-h_{n}/k_{n}|$ $< 1/2k_{n}^2$ I know, by the Thoerem of Hurwitz, that among any three consecutive ...
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30 views

About an rational aproximation to an irrational

How to show that if $x$ is an irrational number, then $x= a_{0} + \sum_{n=0}^{\infty} \frac{(-1)^n}{k_{n}k_{n+1}}$ where the $k_{n}$ are the denominators of the $n$th convergents to $x$? Maybe a ...
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1answer
450 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
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1answer
33 views

How to find the terms of the continued fraction representation for $e^\pi$

The question is - Find the first ten terms of the continued fraction representation for $e^ \pi $
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1answer
96 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
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1answer
158 views

Proof of continued fraction convergence theorem

How does one prove that $$|\alpha - \frac{p_n}{q_n}| < |\alpha - \frac{p_m}{q_m}|$$ for all $n>m$? I know that the left side is less than $\frac{1}{2q_n^2}$ and the right side is less than ...
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1answer
110 views

Continued fraction proof

I am really confused about the proof of this theorem: For any continued fraction, $$q_n\alpha - p_n = \frac{(-1)^n}{\alpha_{n+1}q_n + q_{n-1}}$$ I got the base induction case for $n=0$ but I can't ...
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1answer
146 views

Rational aproximations of golden ratio

I read a blogpost that mentions that for golden ratio, the sets of best rational approximations of the first kind and the second kind are the same. Is this true? If so, why? Are there other numbers ...
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46 views

PDF describing nth term in continued fraction

For a real number r chosen uniformly at random in the range (0,1), what's the marginal Probability Density Function that describes the nth term in the continued fraction representation of r? What ...
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1answer
42 views

How to write continued fraction as polynomial?

I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write $r(x)$ such that nominator and ...
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2answers
76 views

Evaluating $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } }$

What is the value of the fraction $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $? I let $x=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $ and hence $x=\frac{2}{3-x} $ which gives $x=1$ or $x=2$. ...
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38 views

functions on a continued fraction expansion

Let $x$ be an irrational number with continued fraction expansion $[a_0;a_1,a_2,\ldots]$. Is there an $x$ and a non-identity function $f$ such that $f(x)=[f(a_0);f(a_1),f(a_2),\ldots]$. Given that I ...
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Cube root of two continued fraction

I know there is a nice way of getting the continued fraction expansion of quadratic irrationals mainly because they recur after a point, and if they recur after a point they are quadratic irrationals. ...
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174 views

Question in fraction (not simple )

I have a question and its answer but I don't know how can i solve $$\frac {37}{13} = 2+ \frac {1}{x+\frac{1}{5+\frac{1}{y}}} $$ the answer $ x =1, y=2$ Could any one explain how to solve this ?? ...
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Updated:Sum of entries in continued fraction of $\sqrt d$ and $\sqrt{d}-\lfloor \sqrt{d}\rfloor$ equals (divides) $d$.

(1)I noted as a joke in class, for $\sqrt{13}$ which has continued fraction expansion $[3;\overline{1,1,1,1,6}]$ that $3+1+1+1+1+6=13$. Another eg. $\sqrt{22}=[4;\overline{{1,2,4,2,1,8}}]$, as ...
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89 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
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1answer
61 views

Proving something about $|f(x)|$ when the lim of $f(x)/x^2$ is known

I've been trying to crack this issue for 2 days and I got pretty much nothing Given that $f$ is a continuous function and the following limits exists and are finite: $$ (1) ...
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454 views

What's the value of $n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}$ for $n\in\mathbb{C}$?

Write $$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\vdots}}$$ so that $\phi_n=n+\frac{n}{\phi_n},$ which gives $\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$ We know $\phi_1=\phi$, the Golden Ratio, so ...
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715 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
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49 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
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3answers
577 views

Convergents of square root of 2

On wikipedia I read about the continued fraction of the square root of 2: $$1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{...}}}}$$ The first convergents are ...
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4k views

How to find continued fraction of pi

I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are: $\pi=[3:7,15,1,292,.....]$ and many others given here. Similarly some nice continued ...
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1answer
126 views

Is there a bijection between real numbers and continued fractions?

It is known that there is a bijection between rational numbers and finite continued fractions, so every rational number is uniquelly identified by a finite continued fractions and vice versa. It is ...
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46 views

Champernowne constant - summation and behavior of terms in continued fraction expansion

Is there an infinite summation that gives the Champernowne constant? Wikipedia has one, and so does Wolfram MathWorld. Are they valid? If so, could someone explain why, i.e how do they work? Also, ...