Questions tagged [continued-fractions]

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Problem with "OverBar"

In my equation, I would like to add a bar to the second fraction, but the "OverBar" does not work. I would appreciate it if you could tell me what the ...
Ali Bizhanpour's user avatar
2 votes
1 answer
73 views

Computing symbolic continued fractions for rational functions with respect to a variable

There are quite a few questions here about continued fractions, so this might be a duplicate, but I honestly could not find what I want. What I want is, having two polynomials ...
მამუკა ჯიბლაძე's user avatar
2 votes
3 answers
185 views

How can I prove that the sequence converges to $\log(2)$?

There are two rational sequences. One of them describes convergents of the continued fraction for $\log(2)$. Its Mathematica code is as follows: ...
Alex's user avatar
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1 vote
0 answers
64 views

Transform a continued fraction to an ordinary fraction

I need to calculate (tranform to a 1-level fraction) the following continued fraction: ...
kekcikon's user avatar
4 votes
4 answers
179 views

Evaluate the following simple continued fractions

A continued fraction is a fraction whose numerator is an integer and whose denominator is an integer added to a fraction whose numerator is an integer and whose denominator is an integer added to a ...
Athanasios Paraskevopoulos's user avatar
3 votes
2 answers
60 views

Is it possible to display numbers in the form of infinitely repeating decimal fractions $0. \bar{1}$ as ticks in a plot?

I want to show the infinitely repeating decimal fractions in these explicit forms $0. \bar{1}$ and $0. \bar{2}$ as Thicks in a plot; is it possible to do that? If ...
math2021's user avatar
  • 749
0 votes
0 answers
101 views

What does [W_Wad] mean?

Found this one in closed forms of a fraction. Wolfram only gives a way to copy as plain text: (3830 W_Wad)/1177≈0.97621070518266779949 And then it won't recognise ...
Alexandr Dorofeev's user avatar
3 votes
3 answers
235 views

Generating a list of continued fractions

Given positive integers $ k $ and $ n $, I would like to generate a list of all the finite continued fractions of length $ k $ in the interval $ [0,1] $ whose partial quotients are bounded above by $ ...
Daniel's user avatar
  • 143
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0 answers
86 views

Why my code give empty plots? AND When I increase the number of mx , mt my code is running for a long time

In my code, two graphs are not displayed. what is the problem?Also when I increase the number of mx and mt, the code enters the running for a long time. The regularization problem is described as $D^{...
user68119's user avatar
2 votes
2 answers
169 views

Representing a general continued fraction in a list for ContinuedFraction

The initial mathematical problem is to solve for $x$ in: $$x = \sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{x}}}}}$$ (Notice the $x$ in ...
David G. Stork's user avatar
0 votes
0 answers
101 views

How to implement statistics for the length of continued fractions of a result on MATHEMATICA

After successfully generating the reduced fraction of two coprime in interval (0,1] with the following; ...
Ekene's user avatar
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2 votes
1 answer
122 views

Deriving formulas for continued fraction expansions

A version of a continued fraction expansion of a rational number $r\in \mathbb Q$ is defined as \begin{align} r =[a_0,a_1,a_2,\ldots,a_k]= a_0 - \frac{1}{a_1 - \frac{1}{a_2 - \dots - \tfrac{1}{a_k}}} \...
Marc Kegel's user avatar
3 votes
1 answer
236 views

Simplify multiple layers fractions

I have a complicated fraction ...
Leoalan.Huang's user avatar
-1 votes
2 answers
71 views

How can we generate this more general form of the continued fraction? [duplicate]

How can I write Mathematica code for this continued fraction with alternating terms? How can we generate this more general form of the above-continued fraction? $$x+\cfrac{a}{y+\cfrac{a^2}{x+\cfrac{...
Shivam K's user avatar
  • 301
8 votes
5 answers
780 views

How can I write Mathematica code for this continued fraction with alternating terms?

$$\varphi+\cfrac{1}{\varphi^{-1}+\cfrac{1}{\varphi+\cfrac{1}{\varphi^{-1}+\cfrac{1}{\varphi+\cdots}}}}$$ I saw this continued fraction on Facebook. I need the Mathematica code for this using ...
Shivam K's user avatar
  • 301
1 vote
1 answer
103 views

difference equation and continued fractions

I'm interested in solving the following difference equation: $x[k-1]+(k^2+k+a)/x[k]=b$, $k=1,2,\ldots$, where $a,b$ are fixed positive numbers; let's say $x[1]=c>0$. Mathematica's ...
Alex's user avatar
  • 755
3 votes
1 answer
133 views

Can Mathematica evaluate a continued fraction using Gauss's $K$ operator?

I don't know a whole lot about Mathematica, and this is a fairly uncommon notation, so here goes: How do I tell Mathematica to evaluate this? $$1+\underset{i=1}{\overset{\infty}{K}}\frac{(-1)^{i-1}}{...
Micah Windsor's user avatar
5 votes
2 answers
238 views

How to calculate this function

Please, I would like to calculate this function which contains an infinite continued fraction ...
Majorana's user avatar
2 votes
1 answer
88 views

Understanding FromContinuedFraction

In the detail section of the documentation page for FromContinuedFraction we read that: FromContinuedFraction[{a1,a2,…,{b1,b2,…}}] returns the exact number whose ...
mattiav27's user avatar
  • 6,687
1 vote
1 answer
420 views

Writing code to produce continued fractions

I have been trying to create a continuing fraction to help prove the theory about any rational number: a/b being able to be written as a continued fraction where the remainders found in the Euclidean ...
Timothy Norris II's user avatar
1 vote
2 answers
194 views

Computing continued fraction

I want to build this infinite continued fraction $$F_{n}(x)= \cfrac{1}{1-x\cfrac{(n+1)^2}{4(n+1)^2-1}F_{n+1}(x)} $$ which gives for $n=0$ $$F_{0}(x)=\cfrac{1}{1-\cfrac{(1/3)x}{1-\cfrac{(4/15)x}{1-\...
Gallagher's user avatar
  • 763
3 votes
3 answers
95 views

Potential bug : missing terms in ContinuedFraction

If I use 0.23 instead of 23/100, the last continued fraction coefficient (7) is not given: ...
mathheadinclouds's user avatar
3 votes
0 answers
61 views

ContinuedFraction: different result with different representation of argument

Why if I write: In[7]:= ContinuedFraction[3.15] FromContinuedFraction[%] N[%] I get: ...
Roberto Terenzi's user avatar
3 votes
1 answer
59 views

Unexpected behavior of ContinuedFractionK with function defined by SetDelayed

I cannot explain the following results: First, I define a function f. Then ContinuedFractionK imvolving ...
Andreas Rychen's user avatar
8 votes
5 answers
1k views

Continued fraction expansion of a rational function

Maybe I'm missing something, but what would be a good way to calculate the continued fraction expansion of a rational function $Z(s) = N(s)/D(s)$? The built-in function ...
pollux314's user avatar
4 votes
2 answers
227 views

Problems with ContinuedFractionK

I am interested in calculating the following series $$ \sum_{k=0}^{+\infty}{ \frac{ x^k }{\, a (a+1) \cdots (a+k) \,} } $$ using this continued fraction (I expect) equivalent form: $$ \cfrac{1}{a + ...
Vicent's user avatar
  • 1,101
2 votes
1 answer
404 views

Gauss Continued Fraction for Hypergeometric Functions

I would like to calculate the Gauss Continued Fraction for this particular Hypergeometric function: \begin{equation} _{2}F_{1}\left( 1-\frac{1}{p}, \frac{1}{p}; 1+\frac{1}{p}; x^p \right) \end{...
Bazinga's user avatar
  • 737
12 votes
1 answer
393 views

Using Mathematica to find an alternative continued fraction for $\zeta(5)$

Given the Riemann zeta function $\zeta(n)$. I. $x=\zeta(3)$ Using Euler's continued fraction formula, we can form $\zeta(3)$'s cfrac as, $$Ax+B = \cfrac{1}{v_1 - \cfrac{1^6}{v_2 - \cfrac{2^6}{...
Tito Piezas III's user avatar
2 votes
3 answers
484 views

Truncate an infinite continued fraction at order 2000

I want to solve an equation which contains an infinite continued fraction $F(n)$. Then I must (obviously) truncate this continued fraction at $n=2000$. The problem here is that Mathematica does not ...
Betatron's user avatar
5 votes
1 answer
1k views

Negative Continued Fraction of a Rational

The $n^{\text{th}}$ negative continued fraction convergent $x_n$ of a positive real $x$ is computed by the nested function \begin{align} x_n = k_1 - \frac{1}{k_2 - \frac{1}{k_3 - \dots - \tfrac{1}{k_n}...
user02138's user avatar
  • 153