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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11
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1answer
572 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}, ...
5
votes
1answer
50 views

Convergence of a sequence to a value and not another.

I was tinkering with the following equation and produced an infinite nested fraction: $$ (x-6)(x-3)=0 $$ $$ x^2-9x+18=0 $$ $$ x=9-\frac{18}{x} $$ $$ x=9-\frac{18}{9-\frac{18}{9-\frac{18}{...}}} $$ ...
1
vote
1answer
121 views

Does the continued fractions $3+\frac{1}{5+\frac{1}{7+\cdots}}$ equal $\pi$?

$3+\frac{1}{5+\frac{1}{7+\cdots}}=\pi$ Is it true? If yes, how to show it? Please help. Thank you.
4
votes
1answer
206 views

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

Let $A = \{(\sqrt{5} + 1)/2\}^{5}$ and let $\alpha,\beta$ be positive reals such that $\alpha\beta = \pi^{2}/5$. Then it is known that $$\left\{A + R^{5}(e^{-2\alpha})\right\}\left\{A + ...
7
votes
3answers
1k views

Computing Infinite Continued Fractions

I am looking for "tricks" used to compute infinite continued fractions. For example, $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$$ is the golden ratio since if we denote it by $x$, then we have ...
0
votes
1answer
45 views

What is the rate of decay of $\min\{k\xi-\lfloor k\xi\rfloor|k\in\{1,\dots,n\}\}$, for irrational $\xi$?

I wish to establish bounds on the sequence of infima of $\{n\xi\}_{n\in\Bbb N}$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part function and $\xi$ is irrational. I can prove that ...
10
votes
0answers
193 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. ...
4
votes
1answer
56 views

After $n$ iterations of the continued fraction algorithm, what kind of rational numbers will have terminated?

For a positive real number $r_0$, we have the continued fraction recursive algorithm: \begin{align} &r_n\in\mathbb{Z}\implies\text{terminate the algorithm}\\ &\text{else } r_{n+1} = ...
3
votes
1answer
113 views

Inequality related to the continued fraction expansion of sqrt(3)

I am working on a problem related to the continued fraction expansion of $\sqrt3$. If $p_k$ and $q_k$ denote the numerator and denominator, respectively, of the $k$th convergent, I should show that ...
7
votes
1answer
104 views

On the Pell-like $Ax^2-By^2 = 1$

This is connected to the post, Mere coincidence? (prime factors). I was looking at NeuroFuzzy's dataset and noticed the line, {{{1, {4, 2}}, {1, 4, 2, 4, 2}, 23762}} It seems this could be ...
1
vote
0answers
105 views

Continued fraction of $\gamma+1$ using recursion

Number $\gamma,$ the Euler-Mascheroni constant, is defined as the value of $$\gamma = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} - \ln(n).$$ We know that $$\lim_{n\to\infty} ...
3
votes
1answer
56 views

Proving that $\sqrt{n^2+1}-n = F(n), n \in \mathbb{N}_{>0}$

Let $F(n)$ denote a infinite continued fraction of form such that: $$F(n) = \cfrac{1}{2n + \cfrac{1}{2n + \cfrac{1}{2n + \cfrac{1}{2n + \cfrac{1}{\dots}}}}}$$ Consider the following equation: ...
2
votes
0answers
104 views

Lower bound for the length of continued fraction

Define $\mathscr L: \mathbb Q \mapsto \mathbb N$ as the minimal number of terms in the continued fraction of a rational number. Example: the continued fraction of $\frac{5}{8}$ is ...
0
votes
2answers
105 views

Closed form for this continued fraction [closed]

Is there a closed form for this continued fraction? $$x+\frac{1}{x+\frac{1}{x+\frac{1}{...}}}$$
1
vote
1answer
133 views

find the value of 1/(2+1/(4+1/(4+1/(…))))

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$. I have totally no clue; thanks for the help! How am I suppose to solve ...
6
votes
1answer
171 views

Are there any real (especially irrational) numbers whose decimal expansion and continued fraction are the same?

If a number with more than one digit occurs in the fraction, it should be expanded to as many digits in the expansion. I will be even more impressed, however, if the fraction consists entirely of ...
3
votes
1answer
121 views

History Question on Continued Fractions

I worked out the periodicity of some infinite continued fractions last night by hand. (Don't ask me why)For example, $\sqrt{13}= [3,1,1,1,1,6,1,1,1,1,6,\ldots]$. Last night I worked out the first ...
8
votes
2answers
136 views

Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
4
votes
1answer
645 views

Finding the value of a continued fraction?

I know how to calculate the exact value for continued fractions such as $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}=\frac{1+\sqrt{5}}{2}$$ However, is it possible to find the value of continued ...
6
votes
1answer
164 views

Mistake in Khinchin's “Continued Fractions”

I am reading Khinchin's Continued Fractions page 10. $\lbrack a_1;a_2,a_3\ldots\rbrack$ is a continued fraction and $q_k$ is given by $q_k=a_kq_{k-1}+q_{k-2}$. Suppose $\sum_{n=1}^{\infty}a_n$ ...
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} ...
2
votes
1answer
231 views

Continued Fractional representation of irrational numbers [duplicate]

I know that any rational number can be expressed as a continued fraction, but what about irrational numbers? For example, what is the continued fractional representation of Pi, or e for that matter? ...
3
votes
1answer
116 views

Pell-like equations and continued fractions

Why does the continued fraction method work? Could be applied in order to solve, for example, $x^{17}-19y^{17}=1$ ?
14
votes
2answers
315 views

Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
55
votes
6answers
4k 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, ...
4
votes
1answer
235 views

Finding an upper bound on a fraction

$0<\varepsilon <1$. If $n_k$ and $a_k$ are positive integers for which $$n_{k+1}=a_{k+1}n_k+n_{k-1}$$ Let $L\in\mathbb{N}.$ If $L>a_k \ge 3$, what's the smallest upper bound I can place on ...
0
votes
2answers
85 views

Bounded partial quotients set is nowhere dense

I've stumbled upon a claim that the set: $$ B_N = \{[a_0;a_1,a_2,...] | \exists n_0 >0\forall n\geq n_0 a_n<N\} $$ for some $N$, is nowhere dense (and closed). Unfortunately, I have found that ...
2
votes
1answer
108 views

Why are continued fractions for irrational numbers always convergent?

Like in the title: Why are continued fractions for irrational numbers (i.e. infinite fractions) always convergent?
0
votes
1answer
96 views

Non-Recursive Fundamental Recurrence Formulas

Is there a non-recursive version of the fundamental recurrence formulas for continued fractions? I am trying to compute $A_{1000}$, and it is taking me an extremely long time. By the way, I am ...
7
votes
1answer
357 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $A$ be the diagonal matrix w/alternating in sign diagonal entries: $$ A = \begin{pmatrix} (-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\ 0 & ...
4
votes
2answers
141 views

A numerical coincidence with continued fractions

My brother built a garage that measures 45 feet by 30 feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches, $$ ...
0
votes
0answers
112 views

continued fraction

$[a_0,a_1,a_2,\cdots,a_n]:=1/(a_0 + 1/(a_1 + 1/(a_2 + \cdots + 1/(a_n)\cdots )))$ I am so curious that what is the shape of $ f_n(x) $ such that $$ \sum_{n \geq 0} f_n(x) y^n = [-y,1] +[-y,y,1]x + ...
2
votes
1answer
118 views

A trigonometric identity for special angles

Prove that for a natural number $n$, $$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$
2
votes
0answers
105 views

Cantor set as a set of continued fractions?

Does the classical cantor set have a nice description as a set of continued fractions? I made a (superficial) search and didn’t find anything, but I’m very tired right now, so please forgive me that ...
3
votes
1answer
149 views

Limit of a continued fraction

Given the continued fraction: $$f(x,N)=\left[2,3,4,...N,x\right]$$ $$f(x,N)=\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{...+\cfrac{1}{x}}}}}$$ is it possible to find an expression for the integral: ...
10
votes
4answers
769 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 ...
6
votes
2answers
838 views

Continued fraction expansion related to exponential generating function

A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series: $$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
2
votes
2answers
318 views

Continued fraction for $\sqrt{14}$

I'm referencing this page: An Introduction to the Continued Fraction, where they explain the algebraic method of solving the square root of $14$. $$\sqrt{14} = 3 + \frac1x$$ So, $x_0 = 3$, Solving ...
6
votes
1answer
122 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 ...
30
votes
1answer
628 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
12
votes
1answer
170 views

Evaluation of a slow 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 ...
6
votes
1answer
159 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 ...
1
vote
1answer
101 views

Defining piecewise summation of continued fractions and rationality of sums

Let $a=[a_1,a_2\dots]$ and $b=[b_1,b_2\dots]$ be two real numbers and their continued fraction representations. They may be infinite or finite. Let us define a thing $+^*$ so that ...
6
votes
2answers
584 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 ...
2
votes
1answer
91 views

How to deal with infinite continued fractions in formal language?

A continued fraction 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, then writing this other number ...
19
votes
2answers
467 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 ...
0
votes
0answers
89 views

By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?

this is a question from a book I'm struggling with, please could you provide a clear proof Consider the sequence of rational numbers $a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1 ...
2
votes
3answers
57 views

A question about $[c_0,c_1,\ldots,c_n]$ notation for continued fractions

I try to understand why by definition $[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also $[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ . Those are continued fractions, and ...
2
votes
2answers
276 views

Calculate an infinite continued fraction

Is there a way to algebraically determine the closed form of any infinite continued fraction with a particular pattern? For example, how would you determine the value of ...
3
votes
0answers
163 views

Algorithm For Continued Fraction of $\pi$ without error.

Is there an algorithm that will output the numbers in the continued fraction of $\pi$ without error? If one uses the usual method of calculating continued fractions, an approximation of $\pi$ is ...