For questions on continued fractions.

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8
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2answers
125 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 $p \equiv 3 \pmod 4$ ...
1
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0answers
31 views

Explanations of the Euler's continued fractions to compute exponential

After looking for explanations of the Euler's continued fractions to compute exponential on internet and after reading Euler's explanations about, I still don't understand how Euler find this ...
5
votes
1answer
67 views

Number made from the first digits of $2^n$

Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are : ...
9
votes
3answers
302 views

Value of $f'(0)$ if $f(x)=\frac{x}{1+\frac{x}{1+\frac{x}{1+\ddots}}}$

Consider the function $$f(x)=\cfrac{x}{1+\cfrac{x}{1+\cfrac{x}{1+\ddots}}} $$ Determine the value of $f'(0)$. I tried to differentiate $f(x)$ but it is not subject to chain rule, and now I'm stuck. ...
0
votes
0answers
87 views

Calculate an infinite continued fraction as special function

It is possible to convert this infinite continued fraction $\cfrac{1}{-a+\cfrac{b\;f(0)}{a+\cfrac{b\; f(1)}{-a+\cfrac{b\; f(2)}\ddots}}}$ to a special function ? Please, how do it? where : $(a,b) ...
12
votes
1answer
157 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 ...
1
vote
1answer
70 views

How to prove continued fraction convergents of a number

Let $x=1+\sqrt{3}$. Prove that in pairs the continued fraction convergents of $x$ are $a_n$/$b_n$ < x < $c_n$/$d_n$ where $a_1$ = 2, $b_1$ = 1, $c_1$ = 3, $d_1$ = 1, $a_{n+1}$ = 2$c_n$ + $a_n$, ...
1
vote
2answers
54 views

Short proof of Seidel-Stern theorem on continued fractions

Let $\mathbf{a}=\{a_n:n\ge0\}$ be a sequence of positive real numbers, and consider the formal continued fraction $$K(\mathbf{a})=a_1+\cfrac{1}{a_1+\cfrac{1}{a_2+\ddots.}}$$ Seidel-Stern Theorem. If ...
0
votes
0answers
87 views

Why is this finite continued fraction expression equal to $1$?

How find this value I use computer relsut is $1$, maybe this problem have the general relsut But How can prove this by hand? maybe have nice methods? Thank you
1
vote
1answer
53 views

Finite Continued Fraction Proof

Let $\alpha = \left[a_0, a_1, a_2,\cdots,a_n\right]$ be a finite continued fraction with $a_0 > 0$ and let $C_i = p_i/q_i$ be the convergent of $\alpha$. If $i\ge 1$, prove that ...
7
votes
1answer
128 views

Is this a misuse of the word “evaluate”?

I have found the following use of the word "evaluate" in several math books: "To evaluate the continued fraction, start at the bottom and work your way up:" $\huge \underbrace{2 + ...
1
vote
2answers
58 views

Continued fraction explanation

This is about simple infinite continued fraction. I don't understand the line '...then $C_0 < x < C_1$'. $C_k$ here refers to $C_k=[a_0;a_1,a_2,...,a_k]$ where $1 \leq k \leq n$. $C_o=a_0$. ...
1
vote
1answer
59 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 ...
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) ...
1
vote
1answer
23 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
1
vote
1answer
39 views

Continued fraction with alternative signs

Let $a_n$ be a sequence of real numbers. We can define a formal finite continued fraction as usual ...
0
votes
1answer
70 views

Convergent Sequence from Introduction to Analysis

Consider the sequence of real numbers $$\frac 12, \cfrac 1{2+\cfrac 1 2}, \cfrac 1{2+\cfrac 1{2+\cfrac 12}}, \ldots.$$ Show that this sequence is convergent and find its limit by first ...
1
vote
1answer
30 views

How to restrict the output values of a continued fraction?

I understand that a continued fraction of the form: $g(n_1,n_2,n_3,n_4,n_5,\ldots)= n_1 + \cfrac{1}{n_2 + \cfrac{1}{n_3 + \cfrac{1}{n_4 + \cfrac{1}{n_5+\cdots} } } }$ gives a unique irrational number ...
1
vote
2answers
150 views

Infinite continued fraction expansion

How can we find the first six partial quotients of the infinite continued fraction expansion of $\sqrt[3]2$? I know how to do this by expanding when we have a square root function... but I"m not sure ...
4
votes
1answer
63 views

Expressing $1+a_1(b_1+a_2(b_2+a_3(b_3+a_4(b_4+a_5(\cdots)))))$ as an infinite continued fraction.

Euler derived the following identity $$ 1+a_{1}+a_{1}a_{2}+a_{1}a_{2}a_{3}+\cdots= \cfrac{1}{ 1- \cfrac{a_{1}}{ 1+a_{1}- \cfrac{a_{2}}{ 1+a_{2}- \cfrac{a_{3}}{ ...
1
vote
1answer
291 views

Pell's Equation through Continued Fractions

Use continued fractions to find the minimal solution to $x^2-11y^2=1$. I know that $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+...}}}$ I took $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\sqrt{11}}}$ and I ...
2
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0answers
81 views

Interesting Recursive Continued Fraction Limit

I was messing around with recursive functions the other day and came up with something that could be interesting: Definition of $\bar{\Xi}(n)$:\ Let $\Xi ...
0
votes
1answer
70 views

Continued fraction : How to find the first 3 terms

I can't calculate the exact first tree terms $F_0$, $F_1$ and $F_2$ of this continued fraction : $$F_n=\cfrac{1}{-\text{i$\omega $}\,+A\,\cfrac{(n+1)^2}{{4 (n+1)^2-1}}F_{n+1}}$$ $A$ and ...
9
votes
0answers
162 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. ...
3
votes
1answer
169 views

Definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$

Is there a definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$? I am somewhat familiar with continued fractions; that is, I am aware that their convergence depends on whether ...
2
votes
1answer
95 views

About continued fractions as best rational approximations

I'm reading this notes about continued fractions: http://www.math.jacobs-university.de/timorin/PM/continued_fractions.pdf I had no problems understanding everything there, except one thing that has ...
5
votes
1answer
85 views

Is the infinite continued fraction $[0;0,0,\ldots]=0$?

Wolfram|Alpha states that the infinite continued fraction $$\cfrac{1}{0+\cfrac{1}{0+\cdots}}=0.$$ Assuming $[0;0,0,\ldots]$ exists implies that the continued fraction is $1$, since ...
3
votes
1answer
98 views

Are continued fractions a mere curiosity?

Does algebraic geometry have a good understanding of continued fractions? What kind of geometric or arithmetic information does a continued fraction expansion contain, if any? Are there rings of ...
3
votes
1answer
75 views

Can the number of digits in the denominator of a node in Stern-Brocot-Tree decrease in its children?

The Stern-Brocot-Tree looks like this: (image source files) It is an infinite binary tree that contains every positive rational number as exactly one node. The children of a node $\frac{a}{b}$ ...
10
votes
1answer
429 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}, ...
3
votes
1answer
48 views

Why does this pattern fail (sometimes) for the continued fraction convergents of $\sqrt{2}$?

This is connected to my post on the continued fraction convergents of pi. Motivated by Calvin Lin's comment whether a similar pattern exists for other constants, I checked $\sqrt{2}$. Its convergents ...
5
votes
1answer
49 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
117 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.
-1
votes
1answer
520 views

Relation between e and pi [closed]

I found the following relation $\pi=3+\frac{1}{5+\frac{1}{7+\frac{1}{9+\dotsb}}}$ known and $e=3-\frac{1}{5-\frac{1}{7-\frac{1}{9-\dotsb}}}$ Can we relate these directly?
4
votes
1answer
159 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 + ...
6
votes
1answer
164 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 ...
0
votes
1answer
41 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 ...
4
votes
1answer
50 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
105 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
94 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 ...
0
votes
0answers
91 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
54 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
95 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 ...
1
vote
1answer
124 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 ...
3
votes
1answer
113 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 ...
4
votes
1answer
478 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 ...
16
votes
3answers
940 views

Motivation behind this eccentric Ramanujan Identity

I just visited the MathJaX page due to the Math.SE website showing some problems while loading the page. I saw some demo math equations samples at this page, when this identity actually caught my ...
6
votes
1answer
145 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 $\sum_{n=1}^{\infty}a_n$ ...
2
votes
1answer
135 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? ...
7
votes
5answers
402 views

How are continued fractions useful?

On Wolfram Alpha, I see continued fractions being listed in the results. Although I understand continued fractions, and how they can be used for approximations, what is a better approximation than a ...