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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71 views

Continued fraction $1 + \frac 2{3 + \frac 4 {5 + \cdots}} = \frac 1 {\sqrt{e} - 1}$?

I saw this link (written in Japanese) and found an interesting problem: Calculate $1 + \frac 2{3 + \frac 4 {5 + \cdots}}$. The link provides the answer ($\frac 1 {\sqrt e - 1}$) and a hint that one ...
2
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1answer
42 views

Proving a particular infinite continued-fraction identity

By iterating the basic relation $$ \forall z \in \mathbb{C} \setminus \{ -1 \}: \quad z = 1 + \frac{z^{2} - 1}{z + 1}, $$ one obtains the following finite continued-fraction identities: \begin{alignat}...
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1answer
53 views

Proof of $\frac{q_n}{q_{n-1}} = [a_n,a_{n-1},a_{n-2},…,a_2,a_1]$?

Proof of continued fractions axiom. Let $c=[a_0,a_1,a_2,\dots,a_n,\dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \ddots}}$ be a continued fraction which could be finite or infinite. By $\frac{p_n}{...
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0answers
19 views

Interpretation of $ \tau $ in the Stephen Astels paper '' Cantor set and numbers with restricted partial quotients?

I am trying to read Stephen Astels paper 'Cantor sets and numbers with restricted partial quotients'. Visit http://www.ams.org/journals/tran/2000-352-01/S0002-9947-99-02272-2 In this he directly ...
8
votes
3answers
85 views

Continued fraction for $c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $ - is there a systematic expression?

I want to use the convergents of the continued fraction for $$c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $$ - but of course a numeric software is very limited here, so I hope there exists a systematic ...
5
votes
1answer
125 views

Does this integral $\int_0^\infty \frac{dx}{(1+e^x)(a+x)}$ have a closed form?

Note that $a>0$, thus I'm not sure if we can apply residues here. (For $a=0$ the integral doesn't converge). $$\int_0^\infty \frac{dx}{(1+e^x)(a+x)}$$ Despite the simple expression under the ...
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30 views

Recurrence relation connected with the continued fraction of the Exponential Integral.

I've been trying to solve this non-homogeneous recurrence relation with no luck. $$k_n=a_nk_{n-1}+k_{n-2}$$ $$a_n= \begin{cases} x, & \text{if $n$ is even}\\ -2/(n+1), & \text{if $n$ is odd} \...
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0answers
11 views

Prove that these continued fractions are equal.

The wikipedia page on Generalized Continued Fractions gives this equation. $$ \log \left( 1+\frac{x}{y} \right) = \cfrac{x} {y+\cfrac{1x} {2+\cfrac{1x} {3y+\cfrac{2x} {2+\cfrac{2x} {5y+\cfrac{3x} {2+\...
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18 views

Transformation of this log-type continued fraction

I've learned that $$\ln\left(1+\cfrac{x}{y}\right)=\frac{x}{y+\cfrac{1x}{2+\cfrac{1x}{3y+\cfrac{2x}{2+\cfrac{2x}{5y+\frac{3x}{2+\cdots}}}}}}=\cfrac{2x}{2y+x-\cfrac{(1x)^2}{3(2y+x)-\cfrac{(2x)^2}{5(2y+...
1
vote
1answer
100 views

continued fraction $F(x)$ that is a generating function of central binomial coefficients

Given the following continued fraction $$F(x) =\cfrac{1}{x+\cfrac{2^2(2^2-1)}{6x+\cfrac{3^2(3^2-1)}{12x+\cfrac{4^2(4^2-1)}{20x+\cfrac{5^2(5^2-1)}{30x+\ddots}}}}}=\frac{1}{\sqrt{x^2+4}}$$ Then $$\...
1
vote
1answer
46 views

Estimation on the accuracy of the convergents of $\sqrt{n}$

I have noticed that the accuracy of the best rational approximations to $\sqrt{n}$ given by his continued fraction expansion, when the numerator and deniminator are large numbers, is approximately ...
1
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0answers
27 views

On a problem of Erdős about continued fractions and Liouville numbers

In 1938, Erdős and Mahler raised the following question: Let $\xi$ be a real number such that $(p_n/q_n)_n$ are convergents of its continued fraction. If there exists a subsequence $(p_{n_j}/q_{...
4
votes
1answer
51 views

How to prove this continued fraction connection between $\gamma$ and $e$?

There is apparently a curious connection between Euler-Mascheroni constant $\gamma$ and $e$ in the form of an infinite series and continued fraction: $$e \gamma=e \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}...
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43 views

On a particular continued fraction related to $\pi$.

Is this continued fraction well studied? $$\Theta(m)=m+\cfrac{1^2}{2m+\cfrac{3^2} {2m+\cfrac{5^2}{2m +\cfrac{7^2}{2m+\cfrac{9^2}{2m+\ddots}}}}}$$ Note $\Theta(1)=\frac4\pi$. Denote $\Theta(m)=\...
3
votes
2answers
224 views

a general continued fraction satisfying $\frac{(i+\Theta\sqrt{z})^m}{(i-\Theta\sqrt{z})^m}=\frac{(ik+\sqrt{z})^{m+1}}{(ik-\sqrt{z})^{m+1}}$

Given any natural number $m\gt2$, let $z$,$k$ be complex numbers, where $i=\sqrt{-1}$ and consider the general continued fraction $$\Theta(k,z,m)=\cfrac{(m+1)}{km+\cfrac{z(0m-1)(2m+1)} {3km+\cfrac{z(...
3
votes
2answers
198 views

A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$

Similar to the cfracs in this post, define the two complementary continued fractions, $$x=\cfrac{-(m+1)}{km\color{blue}+\cfrac{(-1)(2m+1)} {3km\color{blue}+\cfrac{(m-1)(3m+1)}{5km\color{blue} +\cfrac{...
3
votes
0answers
93 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
4
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1answer
209 views

a continued fraction related to pythagoras theorem $a^2+b^2=c^2$

For our purpose,let $a,b,c$ and $x\gt2$ be natural numbers such that the positive integers $a,b$ and $c$ form a special pythagorean triple $(a,b,c)$,then it is conjectured that the following is true $...
0
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0answers
27 views

Periodicity of the continued fraction of a square root

Writing $\sqrt{n}=[a_0; a_1, a_2, \dots ]$, at which $a_i$ does the period start? Is it $a_1$? I just put "for some $n\ge 1$, where $a_{n-1}=a_i$", is that a good enough answer?
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2answers
52 views

Can I make an infinte sum using rational numbers that makes an irrational but not transcendental number?

I looked a lot on the internet for examples and I tried to do it myself, but I haven't seen any infinite sums of rational numbers that equal for example something like square root of 10 or cube root ...
0
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0answers
44 views

Is there a Similar type for Fibonacci numbers?

The sequence of Lucas numbers is: $2,1,3,4,7,11,18,\ldots$ $L_0=2$, $L_1=1$; $L_{n+1}=L_n+L_{n-1}$ $\phi=\frac{1+\sqrt5}{2}$ is the golden ratio Ramanujan's continued fraction $$\frac{5}{1+[5^{0....
1
vote
3answers
72 views

Only valid for Pythagoraean triples $\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2\cdots}}}=\sqrt{c+a}$?

$$\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2\cdots}}}=\sqrt{c+a}$$ Where (a,b,c) are the Pythagoraean Triples and are satisfy by the Pythagoras theorem $a^2+b^2=c^2$ An example of ...
0
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2answers
27 views

Calculate a quadratic irrational from its periodic continued fraction

I have a periodic continued fraction [2; 1, 3] and I want to convert it into a quadratic irrational. Any helps?
2
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1answer
183 views

Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$…

Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$. So far I have: $\sqrt{27323} = 1 + (\sqrt{27323} - 1)$ which gives... $= 1 + \frac{1}{(\frac{1}{164.2967029})}$ I'm ...
1
vote
1answer
31 views

Find the infinite simple continued fractions for …

Find the infin ite simple continued fractions for $\sqrt{2};\sqrt{5};\sqrt{6};\sqrt{7};\sqrt{8}$. I have solved similar equations for continued fractions but only using a fraction, if someone could ...
2
votes
1answer
95 views

Is $ 2.7182818281828…$ a semiconvergent of e?

Euler's number $e=2.71828 18284 59045... $ can be approximated by the rational number: $$ x=\frac{271,828-27}{100,000-10}= \frac{271,801}{99,990} =2.7182818281... $$ Also, $e$ has the well-known ...
2
votes
1answer
48 views

Which irrational number represents the infinite simple continued fraction [0;7]?

Which irrational number represents the infinite simple continued fraction [0;7]? -So from my current understanding [o;7] can be represented as the following: $ = \frac{1}{7 + \frac{1}{7 + \frac{1}{7 ...
1
vote
1answer
36 views

Find the simple continued fractions for both $\pm \frac{39}{25}$…

Find the simple continued fractions for both $\pm \frac{39}{25}$? So far for $\frac{39}{25}$ I have: $39 = 1 \times 25 + 14 $ $ 25 = 1\times 14 + 11 $ $14 = 1 \times 11 + 3$ $11 = 3 \times 3 +...
3
votes
0answers
63 views

Is this well known? [duplicate]

How to prove $$1+\cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cfrac{e^{-6\pi}}{\cdots}}}} = \left(\sqrt{5\phi}-\phi\right) e^{2\pi/5}.$$ i dont know how to do it. like if there were repeating ...
5
votes
1answer
114 views

Link between the negative pell equation $x^2-dy^2=-1$ and a certain continued fraction

Consider the generalized continued fraction $$F(x)=(x-1)-\cfrac{(x+1)}{x+\cfrac{(-1)(5)} {3x+\cfrac{(1)(7)}{5x+\cfrac{(3)(9)}{7x+\cfrac{(5)(11)}{9x+\ddots}}}}}$$ I experimentally discovered that at ...
0
votes
1answer
35 views

Any good books for studying Continued Fraction?

Does anyone have recommendations for cheap books on Continued Fractions? I do not have much money and so it needs to be a cheap book.
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2answers
19 views

Numerical algorithm: Spectral function -> Continued Fraction

I am trying to code up a numerical algorithm which takes a spectral function of the form $$c(\zeta) = w_0 +\sum_{m=1}^N \frac{w_m}{\lambda_m+\zeta}$$ into a continued fraction of the form $$c(\zeta) = ...
2
votes
1answer
96 views

How to use Euler's continued fraction formula?

I am trying to convert some continued fractions to series by using Euler's continued fraction formula (see the link to Wikipedia). But there is something I obviously misunderstood in it. Take for ...
1
vote
1answer
60 views

Continued Fractions : Under which branch of mathematics do they come?

I wanted to know in which branch of Mathematics do Continued Fraction come? By branch I mean for example Geometry or Differential Equation are a branch of maths so is there any particular branch of ...
3
votes
1answer
214 views

conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
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1answer
41 views

Good book for self study of Continued Fractions

Does anyone have a recommendation for a rigorous while readable book to use for the self study of continued fractions? PS - As examples of "rigorous while readable book" for self-learning, A. ...
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votes
2answers
71 views

Negative solution for a positive continued fraction

$$ x=1+\cfrac{1}{1+\cfrac{1}{1+...}}\implies x=1+\frac{1}{x}\implies x=\frac{1\pm \sqrt{5}}{2} $$ Can the negative solution be considered as a solution? If yes, how is it possible to have a negative ...
8
votes
1answer
82 views

Newton's method for square roots 'jumps' through the continued fraction convergents

I know that Newton's method approximately doubles the number of the correct digits on each step, but I noticed that it also doubles the number of terms in the continued fraction, at least for square ...
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0answers
37 views

How to make continued fractions of any number?

I recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...
2
votes
0answers
55 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
0
votes
1answer
39 views

Approximation of a continued fraction

I'm new to continuous fractions and since I haven't dabbled in mathematics for several years I'm finding it quite difficult to get back on the horse. I'm trying to find e given: $$e = 2 + \frac{1}{1 +...
3
votes
2answers
90 views

Infinite nested radical and infinite continued fractions

If $$a = \sqrt{k_0+\sqrt{k_1+\sqrt{k_2+\sqrt{k_3+\sqrt{\cdots}}}}}$$ and $$b = \cfrac{1}{k_0+\cfrac{1}{k_1+\cfrac{1}{k_2+\cfrac{1}{\cdots}}}}$$ what is the relation between $a$ and $b$. What function ...
9
votes
0answers
121 views

Infinitely nested radical expansions for real numbers

Conjecture. For any real number $x \in (0,1]$ there exists a unique expansion in the form $x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$ with $a_k$ being natural numbers from the set $(2,3,4,5,6)$. ...
15
votes
2answers
257 views

What's the formula for this series for $\pi$?

These continued fractions for $\pi$ were given here, $$\small \pi = \cfrac{4} {1+\cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} {2+\ddots}}}} = \sum_{n=0}^\infty \frac{4(-1)^n}{2n+1} = \frac{4}{1} - \...
3
votes
1answer
92 views

Show that $p_n=q_{n-1}=\frac{\alpha^n-\beta^n}{\alpha-\beta}$

If $\frac{p_n}{q_n}$ be the nth convergent of the simple continued fraction $$\cfrac{1}{a+\cfrac{1}{a+\cfrac{1}{a+\ddots}}}$$ show that $$p_n=q_{n-1}=\frac{\alpha^n-\beta^n}{\alpha-\beta}$$ where, $\...
386
votes
6answers
27k views

Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?

So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $...
2
votes
1answer
38 views

Consecutive smooth number generator recovery

The numbers $n=811150370266636218705704$ and $n+1$ have highest factors 173 and 167, and they happen to be the largest consecutive 173-smooth numbers. They were found via Størmer's theorem and the ...
0
votes
1answer
61 views

Problem on continued fraction of $\frac{\sqrt{5}+1}{2}$ [closed]

If $\frac{p_r}{q_r}$ be the $r^{\text{th}}$ convergent of the continued fraction of $\frac{\sqrt{5}+1}{2}$ then prove that $p_{n+1}=p_{n}+p_{n-1}$ and $p_{2n}=p_{2n-1}+p_{2n-2}$. Attempt: I have ...
2
votes
3answers
78 views

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+\...