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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4
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2answers
149 views

a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
3
votes
1answer
166 views

The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ ...
1
vote
0answers
50 views

Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
3
votes
0answers
50 views

A $q$-continued fraction connected to the divisor function?

In this post, the following two continued fractions discussed by Nicco are given, $$A(q)= \left(\frac{\vartheta_2(0,q)}{2\,q^{1/4}}\right)^2= ...
1
vote
1answer
35 views

Why is this continued fraction expansion what it is?

We have to find the continued fraction expansion of the roots of $1553 t^2 + 6014 t + 5820 = 0$, that is, $(\sqrt{14356} - 6014) / 3106$ Simplifying, $(\sqrt{3589}- 3007) / 1553$ The continued ...
3
votes
1answer
170 views

Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
16
votes
1answer
371 views

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction of the square of ramanujan's octic continued fraction which I discovered using certain three term ...
8
votes
1answer
89 views

are these two continued fractions equivalent?

I would like to pose the following conjecture.Given $$\phi(q) =\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q^3(1-q^2)^2}{1-q^5+\cfrac{q^5(1-q^3)^2}{1-q^7+\ddots}}}}$$ and ...
3
votes
1answer
71 views

an interesting q-series and a certain continued-fraction

My aim is to find a rigorous proof of the following conjectured identity.Given ...
0
votes
0answers
76 views

Does there exist a continued-fraction for geometric series

I would like to know if there exists a continued fraction representation of a geometric series.Motivated by the fact that,many elementary functions in math can be represented as such,I wondered if ...
1
vote
2answers
119 views

a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
4
votes
1answer
146 views
12
votes
0answers
118 views

Bi-linear relation between two continued fractions

We know that any positive real number $x$ can be represented as a simple continued fraction $$x = a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{a_{3} + \cdots}}} = [a_{0}, a_{1}, a_{2}, a_{3}, ...
53
votes
3answers
607 views

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Let $$ \text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction} $$ So for example, the terms of the sum $\text{S}_6$ ...
6
votes
3answers
91 views

Find the value of this infinite term

goes on till infinity. I get two solutions by rewriting the term in the form of the equation $x = 3-(2/x)$, which are $1$ and $2$. But in my opinion this term should have only one possible value. ...
2
votes
0answers
35 views

Riemann Zeta continued fraction approximants

In the paper Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms by Djurdje Cvijovic and Jacek Klinowski, there is a claim that I cannot reproduce. In the abstract they ...
4
votes
1answer
84 views

When $x=2$ in the infinitely continued fraction $x+\frac{1}{x^2+\frac{1}{x^3+\frac{1}{x^4+…}}}$, what algebraic value does it converge to?

Say you have the infinitely continued fraction: $$x+\frac{1}{x^2+\frac{1}{x^3+\frac{1}{x^4+...}}}$$ When $x=1$, you can see that it's $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}$$ which converges ...
3
votes
2answers
79 views

Why am I getting two answers for 8th root of continued fraction

Find value of $x$: $x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$ On solving ,we have $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ ...
7
votes
4answers
162 views

how to solve $3 - \frac{2}{3 - \frac {2}{3 - \frac {2}{3 - \frac {2}{…}}}}$

$$ A = 3 - \frac{2}{3 - \frac {2}{3 - \frac {2}{3 - \frac {2}{...}}}} $$ My answer is: $$ A = 3 - \frac {2}{A} $$ $$ \frac {A^2-3A+2}{A}=0 $$ $$ A^2-3A+2=0 $$ $$ (A-1)\cdot(A-2)=0 $$ $$ A=1 $$ ...
2
votes
2answers
45 views

Continued Fraction Counting Problem

The house of my friend is in a long street, numbered on this side one, two, three, and so on. All the numbers on one side of him added up exactly the same as all the numbers on the other side of him. ...
3
votes
0answers
60 views

Is there an advantage in using continued fractions for Catalan or Fibonacci-Lucas primality tests?

I am studying the basic theory about continued fractions and also reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I found references to the Fibonacci ...
2
votes
0answers
129 views

a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
11
votes
1answer
88 views

Continued fraction estimation of error in Leibniz series for $\pi$.

The following arctan formula for $\pi$ is quite well known $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\tag{1}$$ and bears the name of Madhava-Gregory-Leibniz series after ...
3
votes
0answers
95 views

Continued fraction for $\int_{0}^{\infty}(e^{-xt}/\cosh t)\,dt$

In one of the comments to a question I posted on MSE, I got this wonderful continued fraction $$\int_{0}^{\infty}\frac{e^{-xt}}{\cosh t}\,dt = \frac{1}{x +}\frac{1^{2}}{x +}\frac{2^{2}}{x ...
8
votes
1answer
92 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
5
votes
2answers
301 views

What is the geometric average of 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10…?

It is known that the Khinchin constant is not the geometric mean of the first $n$ coeffecients, as $n$ approaches infinity, of the continued fraction of e, which is $$[2; 1, 2, 1, 1, 4, 1,1, 6, 1, 1, ...
0
votes
2answers
58 views

Visualisation of the reciprocal of an continued fraction?

If: $$a=\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\cdots}}}$$ Then could you help me visualize $1/a$? I really don't understand it. Thank you so much!
3
votes
1answer
141 views

A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
4
votes
0answers
47 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
1
vote
1answer
37 views

Continued fraction manipulation

I have the following continued fraction $$ \frac{1}{a_1x+}\;\;\frac{1}{b_1+}\;\;\frac{1}{a_2x+}\;\;\frac{1}{b_2} $$ The paper I am reading then converts this to the following continued z-fraction ...
3
votes
1answer
118 views

A real number is rational $\iff$ its continued fraction expansion is finite.

I know that if this expansion is finite, then I can go to the lowest denominator in the whole fraction and turn it into a fraction and keep doing so until I get a fraction which means the number is ...
0
votes
2answers
50 views

How can I approximate a decimal with two fractions where denominator is less or equal to $d$

I was looking for a way to approximate a decimal number with a fraction, whose denominator is less or equal to $d$. Basically, having a decimal $X$, I want to find two fractions such that ...
11
votes
1answer
530 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 ...
15
votes
2answers
890 views

Continued fraction for $\frac{1}{e-2}$

A couple of years ago I found the following continued fraction for $\frac1{e-2}$: $$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$ ...
0
votes
0answers
12 views

continued fractions-decreasing series

I am dealing with this problem. Let $\theta=[a_0,...]$ be irrational number. Prove that the following series are strictly decreasing: \begin{eqnarray} \left| \theta-\frac{p_n}{q_n} \right|\\ \left| ...
10
votes
1answer
916 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 ...
1
vote
0answers
38 views

For which prime numbers $p$ there exist $x,y\in \Bbb{Z}$ such that $p=x^2+2y^2$? [duplicate]

For which prime numbers $p$ there exists $x,y\in \Bbb{Z}$ such that $p=x^2+2y^2$? I guess I am to use continued fraction, but I am not sure how. I know how to find solutions for defined numbers but I ...
2
votes
1answer
36 views

Computing periodic continued fractions.

Compute $[1,2,3,\overline{1,4}]$ where $\overline{1,4}$ is the periodic part. I looked into explanations about that, but haven't come by an actual algorithm of computing such a thing. I know it is ...
6
votes
1answer
83 views

Find the derivative of $f$ if it exists, else, prove it doesn't exist

Let $f: \mathbb R^+ \to \mathbb R $ $$f=\mathop{\vcenter{\LARGE\mathrm K}}\limits_{j=1}^{+\infty}\frac{x}{x^j}=\cfrac{x}{x^1+\cfrac{x}{x^2+\cfrac{x}{x^3+\ddots}}}$$ I saw this problem in a math ...
7
votes
3answers
3k 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 ...
6
votes
1answer
96 views

Even Fibonacci Numbers and $\sqrt{5}$

My question is simple, but a mystery to me. Skip to the last paragraph if you're not interested in the story of my exploration. EDIT: I seem to have misinterpreted a key detail regarding how the ...
8
votes
1answer
70 views

Multiply all terms in continued fraction by a constant

I noticed that continued the fraction for $\sqrt{12}$ is $3;2,6,2,6,2,\ldots$ and the continued fraction for $\sqrt{7\times12}$ is $9;6,18,6,18,6,\ldots$ all the terms in the continued fraction are ...
0
votes
1answer
127 views

The radius of image of a circle under mobius transformation

A Mobius transformation of the plane takes $z \mapsto \frac{az+b}{cz+d}$. These are known to take circles to circles, but given an explicit circle, how do we compute the radius. Let's parameterize ...
5
votes
1answer
434 views

Continued fraction for $\tan(nx)$

I found this beautiful continued fraction expansion of $\tan(nx)$, $n$ being a positive integer, online but I don't remember the source now: $\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} ...
1
vote
2answers
95 views

Finding two solutions to $x^2 - 6y^2 = 1$ using continued fractions [closed]

Can anyone show me how to find the solutions to $x^2-6y^2=1$ by using continued fractions? I know how to find the fractions for $\sqrt6$ but do not know how to proceed. THANK YOU!!!
6
votes
2answers
235 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...
3
votes
0answers
92 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x ...
0
votes
1answer
25 views

Farey Sequence implemenatation

I'm trying to use the Farey sequence to get the next lowest reduced fraction in a list. For example, for $n = 8$, we have $\dots, \frac13, \frac38, \frac25, \frac37, \frac12, \dots$ So let's take ...
-1
votes
1answer
33 views

Continued Fraction Expansions Confusion

Let $\theta$ be an irrational number with continued fraction expansion $[a_0; a_1, a_2, \cdots]$. Suppose $P_n/Q_n = [a_0; a_1, \cdots , a_n]$ is the $n^{th}$ convergent. Then how do I show that ...
8
votes
3answers
110 views

Minimal $ab$ for Rational Number $a/b$ in an Interval

Given rational numbers $L$ and $U$, $0<L<U<1$, find rational number $M=a/b$ such that $L \le M<U$ and $(a\times b)$ is as small as possible---$a$ and $b$ are integers. For example, If ...