For questions on continued fractions.

learn more… | top users | synonyms

1
vote
1answer
46 views

Approximating $\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$

What is a good approximation for $$\omega=\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$$ This will be used to find $$T=\frac{t}{1-\omega}$$ Without using Lambert's continued fraction ...
2
votes
1answer
141 views

New mathematical constant formed by continued fraction with prime numbers?

Notational convention: $$\bigoplus_{k=0}^{\infty}a_k=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}$$ Let $$ P:=\bigoplus_{k=1}^{\infty}p_k$$ where $p_k$ is the k-th prime ...
0
votes
0answers
9 views

Name/properties of a difference of continuants

Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, \ldots, q_s]$ the numerator (in lowest terms) of the rational number represented by the continued fraction $$ ...
1
vote
0answers
31 views

Sums of nested radicals

Is there a known example of an infinite sum of finitely nested radicals that evaluates to a given value? Or an infinite sum of convergents of an infinite continued fraction? The finitely nested ...
7
votes
0answers
152 views

Continued fraction and double series.

From Euler's continued fraction formula, we have $$x = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}\,$$ and $$x = 1 + \sum_{i=1}^\infty r_1r_2\cdots r_i = ...
1
vote
0answers
30 views

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 ...
3
votes
1answer
70 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
votes
1answer
188 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 } ...
2
votes
1answer
53 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 ...
6
votes
3answers
195 views

Where did the negative answer come from in $1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots }}}$?

In this question we increased solution domain by squaring both sides of equation but what about this one ? Here the question is to evaluate ...
0
votes
0answers
30 views

Fact similar to Ostrowski numeration for reals

I have to prove this fact (found in an article without proof). Let $\alpha \in \mathbb{R}$ be an irrational number. Let $\alpha = [a_0;a_1,a_2,\ldots]$ be the continued fraction expansion. We call ...
1
vote
1answer
32 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 $a(n) \cdot a(n-1)=3a(n-1)+1$ ...
0
votes
1answer
29 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 ...
1
vote
1answer
50 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 ...
1
vote
1answer
40 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, ...
2
votes
1answer
28 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, ...
2
votes
0answers
45 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 ...
2
votes
1answer
73 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 ...
3
votes
2answers
68 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 ...
12
votes
1answer
354 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\\ ...
1
vote
1answer
65 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 ...
0
votes
1answer
24 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?
2
votes
2answers
64 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 ...
1
vote
1answer
28 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
66 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
votes
0answers
20 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
votes
0answers
38 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
votes
0answers
39 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
45 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
68 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
votes
2answers
47 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
vote
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. ...
16
votes
1answer
198 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
votes
0answers
23 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
10 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
181 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
votes
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
votes
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
12 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
75 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
68 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
vote
0answers
12 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
32 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
72 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
28 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
41 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
166 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
votes
0answers
63 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
votes
0answers
59 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 ...