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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7
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4answers
4k views

Continued fraction of a square root

If I want to find the continued fraction of $\sqrt{n}$ how do I know which number to use for $a_0$? Is there a way to do it without using a calculator or anything like that? What's the general ...
8
votes
4answers
704 views

How to detect when continued fractions period terminates

I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates? Let me give you an example: $\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
13
votes
2answers
504 views

How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
11
votes
3answers
510 views

Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is ...
5
votes
1answer
806 views

A nicer proof of Lagrange's 'best approximations' law?

Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a ...
52
votes
6answers
3k views

Continued fraction fallacy: $1=2$

It's easy to check that for any natural $n$ $$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$ Now, ...
0
votes
3answers
1k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
0
votes
2answers
98 views

Closed form for this continued fraction [closed]

Is there a closed form for this continued fraction? $$x+\frac{1}{x+\frac{1}{x+\frac{1}{...}}}$$
9
votes
1answer
198 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 } ...
21
votes
1answer
772 views

A continued fraction involving prime numbers

What is the limit of the continued fraction $$\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{11+\cfrac{1}{13+\cdots}}}}}}\ ?$$ Is the limit algebraic, or expressible in terms of e or ...
15
votes
2answers
865 views

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

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

Baire space homeomorphic to irrationals

I try to show that the Baire space $\Bbb N^{\Bbb N}$, with regular product metric, is homeomorphic to the unit interval of irrationals $(0,1)\setminus\Bbb Q$. I already know that the needed function ...
5
votes
5answers
398 views

Continued Fraction [1,1,1,…]

If the continued fractional representation of an irrational number $\alpha$ is given by [1,1,1,...], I can compute that $\alpha = \frac{1+\sqrt{5}}{2}$ by solving the equation $\alpha = 1+ ...
7
votes
3answers
2k 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 ...
5
votes
1answer
392 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} ...
10
votes
3answers
1k views

Proving the continued fraction representation of $\sqrt{2}$

There's a question in Spivak's Calculus (I don't happen to have the question number in front of me in the 2nd Edition, it's Chapter 21, Problem 7) that develops the concept of continued fraction, ...
6
votes
1answer
118 views

For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?

Inspired by this question I ask this. For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$? The original question concerned $a=e$, the usual ...
4
votes
2answers
175 views

Continued Fractions periodicity and convolution.

Continued fractions for rationals terminate, for transcendentals like pi, they do not terminate and for irrationals (but non transcendentals) they repeat -- is this correct?
3
votes
1answer
881 views

Calculate the continued fraction of square root

I was having difficulty understanding the algorithm to calculate Continued fraction expansion of square root. I know the process is about extracting the integer part in repeat and maintaining the ...
1
vote
3answers
291 views

How do I solve a Continued Fraction of solution to quadratic equation?

I know that it is possible to make a CF (continued fraction) for every number that is a solution of a quadratic equation but I don't know how. The number I'd like to write as a CF is: $$\frac{1 - ...
19
votes
2answers
638 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 ...
10
votes
0answers
178 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
19
votes
2answers
453 views

Evaluation of a continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
14
votes
1answer
805 views

Arithmetic of continued fractions, does it exist?

I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider $$ ...
10
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4answers
762 views

Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$

I'm currently a high school junior enrolling in AP Calculus, I found this website that's full of "math geeks" and I hope you can give me some clues on how to solve this problem. I'm pretty desperate ...
7
votes
2answers
307 views

Continued fractions

I'd really love with concluding that for given integers $a_0,a_1,...a_N$ with $a_i>0$ for $i>0$, representing the continued fraction $[a_0; a_1,....,a_N]$, with the following recursion: ...
6
votes
3answers
304 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 ...
6
votes
1answer
157 views

Finding near-integers in a range

I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
6
votes
1answer
170 views

General Continued Fractions and Irrationality

A while back I came across a result about non-simple continued fractions that allows proving that some numbers are irrational. The result in modern terminology is: If, in the continued fraction ...
3
votes
0answers
58 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 ...
3
votes
0answers
68 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 ...
3
votes
0answers
64 views

Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
2
votes
1answer
111 views

A trigonometric identity for special angles

Prove that for a natural number $n$, $$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$
10
votes
1answer
528 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
6
votes
1answer
262 views

Is it right that the fundamental recurrence of an arbitrary continued fraction cannot be proved without induction?

Let $\dfrac{A_{n}}{B_{n}}$ be the $n^{th}$ convergent (approximant) $$ \frac{A_{n}}{B_{n}}=b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{\begin{array}{c} b_{3}+ \\ \\ \end{array} ...
3
votes
3answers
223 views

Solution to $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$ [duplicate]

Possible Duplicate: Why does this process, when iterated, tend towards a certain number? (the golden ratio?) Please post your favorit solution to the following Compute ...
1
vote
2answers
59 views

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$?

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$ ? $\displaystyle[1;1,2,1,2,\dots]=1+\frac{1}{[1;2,1,2,\dots]}=1+\frac{1}{1+\frac{1}{2+\frac{1}{[1;2,1,2,\dots]}}}$ If I set $x=[1;2,1,2,\dots]$ then; ...
1
vote
2answers
72 views

From the continued fraction

What would be the irrational number $\dfrac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers given by this expression: $$ \left( ...
4
votes
0answers
116 views

Finding a closed expression for a calculated value.

Sometimes, when getting some numerical results when investigating number theory sequences with a computer, I find myself suspecting that a decimal value ($a$) I have found might be a quadratic ...
4
votes
4answers
298 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, e and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
2
votes
0answers
13 views

On the asymptotics of a continued fraction

I know the coefficients of the continued fraction representation of a function that diverges like $O(\sqrt{x})$, where the $a_k$ depend on $x$, $ f(x)=1+a_0/\left(1+\underset{k=1}{\overset{\infty ...
2
votes
1answer
541 views

Convergent of continued fractions the best rational approximation of a number? [duplicate]

Possible Duplicate: A nicer proof of Lagrange's 'best approximations' law? I was reading through the wikipedia article on continued fractions, and they state, essentially, that ...
1
vote
1answer
46 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the ...
1
vote
1answer
61 views

Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show that ...
1
vote
3answers
365 views

Can every transcendental number be expressed as an infinite continued fraction?

Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
1
vote
2answers
233 views

Continued Fractions Approximation

I have come across continued fractions approximation but I am unsure what the steps are. For example How would you express the following rational function in continued-fraction form: $${x^2+3x+2 ...
0
votes
3answers
176 views

Continued fractions proof?

Let $b_1=1$ and $$b_n=1+\frac{1}{1+b_{n-1}}$$ for $n\ge 2$. Note that $b_n \ge 1$ for all $n$ in $\mathbb N$. ($\mathbb N$ represents the positive integers) Show that $b_{2k-1}^2<2$ for all $k ...