Questions tagged [continued-fractions]

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2
votes
1answer
88 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}}} \...
3
votes
1answer
56 views

Simplify multiple layers fractions

I have a complicated fraction ...
-1
votes
2answers
60 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{...
7
votes
5answers
694 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 ...
1
vote
1answer
54 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 ...
2
votes
1answer
52 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}}{...
5
votes
2answers
172 views

How to calculate this function

Please, I would like to calculate this function which contains an infinite continued fraction ...
1
vote
1answer
89 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 ...
1
vote
2answers
139 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-\...
3
votes
3answers
76 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: ...
3
votes
0answers
47 views

ContinuedFraction: different result with different representation of argument

Why if I write: In[7]:= ContinuedFraction[3.15] FromContinuedFraction[%] N[%] I get: ...
3
votes
1answer
46 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 ...
3
votes
2answers
161 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 + ...
2
votes
1answer
264 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{...
8
votes
1answer
281 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}{...
2
votes
3answers
394 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 ...
5
votes
1answer
986 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}...