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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2
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3answers
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Compare five ways of solving cubic equation by iterations (nested expressions)

Say we have a depressed cubic equation in the general form: $$x^3-bx-c=0$$ There are basically five ways of solving it by iterations. Let's consider them in no particular order (the names are my own)...
6
votes
1answer
115 views

Liouville numbers and continued fractions

First, let me summarize continued fractions and Liouville numbers. Continued fractions. We can represent each irrational number as a (simple) continued fraction by $$[a_0;a_1,a_2,\cdots\ ]=a_0+\...
3
votes
1answer
49 views

How to prove that this infinite product of continued fractions converges to $1-\frac{1}{z}$?

$$\cfrac{z}{1+z} \cdot \cfrac{z}{1+z-\cfrac{z}{1+z}} \cdot \cfrac{z}{1+z-\cfrac{z}{1+z-\cfrac{z}{1+z}}} \cdots= 1-\frac{1}{z}$$ I propose that this works for any $z \in C$ if and only if $|z|>1$. ...
5
votes
1answer
66 views

Nested Radicals and Continued Fractions

Is there some interconnection between these two topics? A sort of classification of the possibile types of nested radicals and maybe some way (hopefully bijective, in some sense) to pass from a ...
1
vote
0answers
57 views

How to rigorously prove the following continued fraction identity?

The following identity seems to be satisfied for any $z \in C$ $$\cfrac{z^2}{1+z^2-\cfrac{z^2}{1+z^2-\cfrac{z^2}{1+z^2-\cfrac{z^2}{1+z^2-\cdots}}}}=\begin{cases}1 & |z| \geq 1\\z^2 & |z| \leq ...
21
votes
1answer
229 views

Continued fraction analog to zeta function - how to properly define it and find its properties?

I do not mean the continued fraction representation of zeta function; I mean the function which has the form: $$f(s)=\cfrac{1}{1^s+\cfrac{1}{2^s+\cfrac{1}{3^s+\cfrac{1}{4^s+\cdots}}}}$$ For some ...
3
votes
0answers
61 views

Euler exponential continued fraction to compute the trigonometric functions and the golden ratio

Using the Euler continued fraction for the exponent, which is convergent everywhere on the complex plane: $$e^{-z}=1-\cfrac{z}{1+z-\cfrac{z}{2+z-\cfrac{2z}{3+z-\cfrac{3z}{4+z-\cdots}}}}$$ We can ...
0
votes
0answers
24 views

Computation of general continued fractions by $2 \times 2$ matrix multiplication - is it the best way?

There are two main ways to compute a continued fraction (or its $n$th convergent). Let's say we have a general fraction: $$ x= a_0 + \dfrac{b_1}{a_1+\dfrac{b_2}{a_2+\dfrac{b_3}{a_3+\dfrac{b_4}{a_4+......
1
vote
0answers
28 views

Stern-Brocot Tree and sum of coefficients of continued fraction

Suppose we are given a continued fraction $$\frac{p}{q}=a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\cdots}}}$$ I am trying to find an expression, possibly asymptotic, for the sum of the $a_i$'...
1
vote
1answer
50 views

Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE K}\limits_{i=1}^{2n}...
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vote
0answers
40 views

Recommended reading for continued fractions? And some results

First of all, I apologize for my amateurness and inexperience. Although I always enjoyed math, only two years ago I started experimenting with continued fractions and gained a deep reverence for them. ...
1
vote
5answers
96 views

Limit of the sequence defined by a recurrence

Given a recurrence formula for an arithmetic sequence, $$U_{n} = \frac{1}{2+U_{n-1}}$$ Show that$$\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+ ...}}}} = (SomeGivenValue)$$ How can we solve questions ...
1
vote
0answers
80 views

Continued fraction and order of a real number

If $\alpha$ is a irrational number prove that $\operatorname{ord}\alpha \geq 1+\exp(\limsup \log(\log(a_n+1))/n)$ I tried the simple things like use the well known formula $\operatorname {ord}\alpha ...
1
vote
0answers
15 views

Continued fraction expansion of $1-x$ for $x\in [0,1[$.

If $\alpha$ is a real number in $[0,1[$ that can be written as $[a_0;a_1,a_2,\ldots]$, what can I say about the continued fraction expansion of $1-\alpha$?
3
votes
0answers
36 views

How to write this function in a “well-formed” way

Given an input $0 \lt x \lt 1$, find $x$'s Nearest Integer Continued Fraction with structure $$x = a_0 \pm \cfrac{1}{a_1 \pm \cfrac{1}{a_2 \pm \cdots}}.$$ Then $$f(c) = a_0 + 1 \mp \cfrac{1}{a_1 + ...
0
votes
2answers
34 views

Sum and Product of continued fraction expansion?

Give the continued fraction expansion of two real numbers $a,b \in \mathbb R$, is there an "easy" way to get the continued fraction expansion of $a+b$ or $a\cdot b$? If $a,b$ are rational it is easy ...
6
votes
2answers
77 views

Specification of Hurwitz's Theorem

Hurwitz's Theorem in Number Theory states that for every irrational number $\xi$, the equation $$\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$$ has infinitely many solutions $(p,q)$ ...
0
votes
1answer
62 views

determine the continued fraction of $\sqrt{n^2 + 2}$ for $n \in \mathbb{N}$

determine the continued fraction of $\sqrt{n^2 + 2}$ for $n \in \mathbb{N}$. For rationals it is rather easy to do this and i know the algorithm, i only get stuck a lot by irrational number such as ...
2
votes
2answers
82 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for $e$...
1
vote
0answers
25 views

Convergents of continued fractions

Let $d$ and $m$ be positive integers such that $d$ is not a square and such that $m\leq\sqrt{d}$. I want to prove that if $x$ and $y$ are positive integers stafisfying $x^2-dy^2=m$ then $x/y$ is a ...
1
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1answer
38 views

How to calculate remainder value of a fraction

Question: Four brothers split a sum of money between them. The first brother received 50% of the total, the second received 25% of the total, the third received 20% of the total, and the fourth ...
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votes
5answers
56 views

Find $x$ defined as a continued fraction [duplicate]

I have solved the above using the below method. $$x= 12 + \frac{1}{2+\left(\frac{1}{2}+x\right)}$$ After solving for $x$, I got the answer as $11.7515$ and $-1.41824$ So what is the real value of $...
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votes
1answer
59 views

Find the real number $x$ represented by continued fraction $[12;2,2,12,2,2,12,2,2,12\dots]$

I need to fins the real vlaue of x for the continued fraction (Image attached) I have tried partial coefficient method, but didn't get the exact answer. I there any way where we can identify the ...
2
votes
1answer
38 views

Find the sum of the integers in the continued fraction

Find the sum of integers $a,b,c,d,$ and $e$ if $\dfrac{2011}{1990} = a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d+\dfrac{1}{e}}}}$. I could simplify the big fraction on the RHS, but I don't see how that ...
2
votes
0answers
65 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity $\eta(-\frac{...
0
votes
0answers
63 views

integrating continued fractions

How do you integrate: $$\int \underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_jx^j}{b_j}dx=\int x\cfrac{a_1}{b_1+\cfrac{a_2x}{b_2+\cfrac{a_3x^2}{b_3+\ddots}}}dx$$ Can you use closed form?
1
vote
1answer
26 views

Evaluating a continued fraction

I have a continued fraction in List form: $[0;1,2,1,2...]$ (the $1,2$ are repeated). Evaluating this should give $-1 + \sqrt3$. I am not sure how to get that answer. I am aware of to express $\...
26
votes
1answer
564 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
1
vote
1answer
65 views

a conjecture of two equivalent q-continued fractions related to the reciprocal of the Göllnitz-Gordon continued fraction A111374-OEIS

Given the square of the nome $q=e^{2i\pi\tau}$ and ramanujan theta function $f(a,b)=\sum_{k=-\infty}^{\infty}a^{k(k+1)/2}b^{k(k-1)/2}$ with $|q|\lt1$, define, $$\begin{aligned}M(q)=\cfrac{1-q^3}{1-q^...
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0answers
29 views

Textbooks on transcendence theory

Is there a nice, modern textbook (some lecture notes or survey would do, too) that covers the main results and methods from transcendence theory? Ideally, it should also have some good exercises. So ...
0
votes
1answer
45 views

How to find cases where $m^2$ is near to $2^A$?

In another problem here in MSE I ran into the question how I can (practically, in a program) find (positive) integer $m$ such that they are "near" to perfect powers of $2$, so $$ (0 \lt ) \qquad d_m ...
3
votes
1answer
135 views

Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$

I have been playing around with Mathematica and continued fractions and I noticed something. ContinuedFractionK[n, n + x, {n, 1, Infinity}] ==-x + 1/(E Gamma[1 + x] - E Gamma[1 + x, 1])==-x + 1/(E ...
5
votes
1answer
72 views

Two similar integration about continued fractions

Prove that \begin{align*} \int_0^{+\infty} \cfrac{\sin nx}{x + \cfrac{1}{x + \cfrac{2}{x + \cfrac{3}{x + \cdots}}}} \, dx &= \cfrac{\sqrt {\cfrac{\pi }{2}} }{n + \cfrac{1}{n + \cfrac{2}{n + \cfrac{...
1
vote
2answers
52 views

Are the numerator and the denominator of a convergent of a continued fraction always coprime?

Is it true that for a convergent $A_k = p_k/q_k$ of the continued fraction expansion $[a_0;a_1,a_2,\dots]$ the numerator $p_k$ and the denominator $q_k$ are always coprime? If yes, how would one show ...
0
votes
0answers
43 views

Prove that $p_nq_{n-1}$ - $p_{n-1}q_n=(-1)^{n-1}$ for $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$

Let $p_n$/$q_n$ for $n=0,1,2,..$ be the convergents of $a∈ R$ $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$ $p_n= a_np_{n-1}+p_{n-2}$ $q_n= a_nq_{n-1}+q_{n-2}$ I need to prove that $p_nq_{n-1} - p_{...
2
votes
1answer
48 views

Square root of continued fraction

Assuming I've been given a number in the form of a continued fraction. Is there a general way to write the square root of that number as continued question? For example, consider $$1+\sqrt{2} = 2+\...
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0answers
17 views

Q-D scheme, continued fractions

What is a Q-D scheme for a continued fraction? I am reading this text on numerical evaluation of the H-function and the author suggests using continued fractions as done by many other special ...
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vote
1answer
41 views

Convert partial fraction to continued fraction?

Lets say you have a partial fraction of the form: $$ f(x) = a_0 + \sum_{n=0}^{\infty} \frac{a_n}{\lambda_n + x} $$ Can anyone explain to me, in mildly plain English, how to convert this partial ...
3
votes
0answers
111 views

Asymptotic solutions of a sparsely perturbed recurrence relation

Recurrence relation I am trying to find approximate solutions $T(n)$ of the recurrence relation $$ p\ T(n-1) - \left[p+q+\overline{S} + \varepsilon \tilde{S}(n)\right]T(n) + q\ T(n+1) = 0,\\ \text{...
0
votes
1answer
21 views

Show simple continued fraction with Euclid's Algorithm

I have this math problem, I have to show a simple continued fraction from equations. Here's the question. Use these equations: $$397 = 204(1) + 193$$$$204 = 193(1) + 11$$$$193 = 11(17) + 6$$$$11 =...
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0answers
16 views

Rapidly converging regular infinite continued fractions - anything special about them?

Say we have a simple continued fraction with rapidly increasing terms. Then it obviously converges very quickly and has a very good rational approximation. But is there anything special or ...
16
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1answer
370 views

Nested solutions of a quadratic equation.

A quadratic equation of the form $x^2+bx+c=0$ can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by ...
0
votes
3answers
65 views

How Can I calculate this expression?

I have this repeating expression $5+\dfrac {6} {5+\dfrac {6} {5+..}}$ I saw a solution on a book. which is: $5+\dfrac {6} {5+\dfrac {6} {5+..}}=x$ $5+\dfrac {6} {x}=x$ $x^2-5x-6=0$ $x=6 $ or $x=...
12
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1answer
1k views

a conjectured continued-fraction for $\displaystyle\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\cot\left(...
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0answers
475 views

a conjectured continued fraction for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\tan\left(...
5
votes
1answer
77 views

Where to learn about continued fractions?

I sought a convergent subsequence of $\sin n$, and I met a proof that utilizes continued fractions. I always stumble upon things related to them as well. So, I would like to learn about them. What is ...
5
votes
2answers
43 views

Can we define sum and product of two irrational numbers using Cauchy sequences of their simple continued fraction convergents?

There is a lot of questions about sum and product of irrationals here, so I hope you'll bear with me. Simple continued fraction is a very convenient way to represent any number since every real ...
0
votes
0answers
71 views

a continued fraction related to the exponential function $e^x$

Given a natural number $n$,with $|x|\lt1$ define the following conjectured identity $$G(n,x)=\begin{aligned}\cfrac{-n}{1-x-\cfrac{(1+n)(1-x^2)}{1-x^3-\cfrac{x^2(1-x)(1-x^3)}{1-x^5-\cfrac{x^3(1-x^2)(1-...
4
votes
1answer
60 views

While resolving a continued fraction by simplifying it into a quadratic equation, does the extraneous root have any significance?

Suppose we want to find the value of the the following expression, with infinite terms: $$ y = {1\over 1+ {1\over 1+ {1\over 1+ {1\over \ddots}}}} $$ To solve this, we follow the following procedure, ...
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vote
0answers
38 views

What kinds of functions can be expanded into a form of an infinite continued fraction?

In this 1943 paper by Hudson on the theory of elastic waves in beams the author offers an interesting way to calculate a function defined as $$ \theta_n (z)=\frac{z J'_n (z)}{J_n (z)} $$ with $z$ ...