For questions on continued fractions.

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2
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2answers
47 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 ...
0
votes
0answers
23 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: ...
4
votes
1answer
54 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 ...
0
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0answers
14 views

Comparing Generalized Continued Fractions

Gosper lays out a method (under Approximations) for comparing regular (a.k.a simple) continued fractions which have all partial numerators set to 1. Continue comparing terms until they differ, then ...
3
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0answers
28 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, ...
0
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0answers
24 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 ...
2
votes
1answer
42 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 ...
0
votes
1answer
59 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 ...
3
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2answers
39 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 ...
1
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0answers
36 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. ...
15
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1answer
158 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) + ...
0
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0answers
21 views

a periodic continued fraction $c_{n+2} = 1 - \frac{c_{n+1}}{1 - c_{n}}$

Howvever, start with two numbers $c_0, c_1$ I read in a paper the following sequence has period 5: $$ c_{n+1} = 1 - \frac{c_n}{1 - c_{n-1}} $$ Example if we have $c_0=1, c_1=2$ the sequence ...
0
votes
1answer
9 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 ...
0
votes
0answers
25 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 ...
6
votes
1answer
173 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 ...
2
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1answer
27 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 $
0
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1answer
73 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 ...
1
vote
1answer
9 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 ...
0
votes
1answer
66 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 ...
3
votes
1answer
59 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 ...
1
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0answers
7 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 ...
0
votes
1answer
27 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 ...
3
votes
2answers
69 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 $x=2$. ...
1
vote
2answers
25 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 ...
3
votes
0answers
32 views

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. ...
2
votes
2answers
161 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 ?? ...
3
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0answers
59 views

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 ...
3
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0answers
44 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 ...
2
votes
1answer
59 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) ...
10
votes
5answers
396 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 ...
18
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2answers
588 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 ...
1
vote
1answer
44 views

Math and Taxes - Items versus Sum Total differs

First, The question is a bit confusing as I am not really sure how to word this problem as a question. The math problem I encountered which is a bit of an anomaly is this : Suppose you are ...
1
vote
1answer
21 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{ ...
1
vote
3answers
95 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 ...
5
votes
3answers
764 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 ...
1
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1answer
48 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 ...
1
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0answers
23 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, ...
6
votes
1answer
64 views

Infinite Continued Fraction Notation

I can't find anywhere via googling; is there some sort of $\sum$ like notation for infinite continued fractions? In other words, for a sum we do this: $$ 1+x+x^2+x^3+... = \sum_{n=0}^\infty x^n $$ ...
1
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2answers
103 views

Reconstruct a quadratic irrational from its periodic continued fraction

How can one find a quadratic irrational when knowing its periodic continued fraction? For example(using Wikipedia notion), how can one find the quadratic irrational that its continued fraction is ...
0
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0answers
80 views

Proof of a striking identity of Tito Piezas III

In the q series blog of Tito Piezas here . He gives a very striking relation I am wondering on how to prove that ?
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1answer
64 views

Solve the equation $285x \equiv 177 \pmod{924}$ using continued fraction

Solve the equation $285x \equiv 177 \pmod{924}$ using continued fraction My attempt(using Wikipedia notion): Continued fraction form for $\frac{924}{285}$ is $[3;4,6,1,9]=[q_1;q_2,q_3,q_4,q_5]$ ...
2
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1answer
32 views

uniform spanning tree of $2 \times n$ graph

In Probability on Trees and Networks Chapter 1 study the uniform spanning tree on the ladder graph: _ |_| |_| |_| ... |_| |_| The probability the bottom rung ...
1
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1answer
65 views

Nested square root of continued fraction references

$$\sqrt {a_0 + \cfrac{b_1}{\sqrt{a_1 + \cfrac{b_2}{\sqrt{a_2 + \cfrac{b_3}{ \ddots }}}}}}$$ Are there any texts that explain how to deal with expressions like this?
1
vote
2answers
67 views

Find value of unending continued fraction

I am trying to find the limit if it exists for the following unending continued fraction: $$1+{1\over{2+{1\over2+{1\over{2+...}}}}}$$ I have discovered this is the continued fraction for $\sqrt2$, ...
1
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0answers
63 views

Is this a bounded sequence ? (about continued fraction)

Represent $\sqrt{2}$ in the form $$\sqrt{2}=1+\frac{8}{A_1+\displaystyle\frac{8}{A_2+\displaystyle\frac{8}{A_3+\ddots}}},$$ where $A_n$ is a positive integer and $A_n \geq 8$ for all $n$. So we have ...
4
votes
0answers
98 views

Different ways of operating an infinite continued fraction

Given the continued fraction below, $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}+\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}} $$ I wanted to know to which number it converged, so I ...
5
votes
1answer
57 views

Number made from the first digits of $2^n$

Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are : ...
0
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0answers
85 views

Calculate an infinite continued fraction as special function

It is possible to convert this infinite continued fraction $\cfrac{1}{-a+\cfrac{b\;f(0)}{a+\cfrac{b\; f(1)}{-a+\cfrac{b\; f(2)}\ddots}}}$ to a special function ? Please, how do it? where : $(a,b) ...
7
votes
3answers
261 views

Value of $f'(0)$ if $f(x)=\frac{x}{1+\frac{x}{1+\frac{x}{1+\ddots}}}$

Consider the function $$f(x)=\cfrac{x}{1+\cfrac{x}{1+\cfrac{x}{1+\ddots}}} $$ Determine the value of $f'(0)$. I tried to differentiate $f(x)$ but it is not subject to chain rule, and now I'm stuck. ...
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vote
1answer
65 views

How to prove continued fraction convergents of a number

Let $x=1+\sqrt{3}$. Prove that in pairs the continued fraction convergents of $x$ are $a_n$/$b_n$ < x < $c_n$/$d_n$ where $a_1$ = 2, $b_1$ = 1, $c_1$ = 3, $d_1$ = 1, $a_{n+1}$ = 2$c_n$ + $a_n$, ...