Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [continued-fractions]

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.

In mathematics, 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 as the sum of its integer part and another reciprocal, and so on.

In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ai are called the coefficients or terms of the continued fraction.

Links:

Continued Fraction at Wolfram MathWorld

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Properties of continuity

Let $f,g :[a,b]\to\mathbb{R}$ be continuous functions such that $$\int\limits_c^df(x)\leq \int\limits_c^dg(x)dx$$ whenever a$\leq$c$<$d$\leq$b. I need to show that $f(x)\leq g(x)$. I have the idea of using proof by contradiction supposing that…
user189013
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Find an expansion into an arithmetic continued fraction of $\sqrt{x}$, where $x=((4a^2+1)b+a)^2+4ab+1$, $a,b \in \mathbb{N}$

I am trying to find an expansion into an arithmetic continued fraction of $\sqrt{x}$, where $x=((4a^2+1)b+a)^2+4ab+1$, $a,b \in \mathbb{N}$. So far I have: Clearly $x<((4a^2+1)b+a+1)^2$, so the integral part of $\sqrt{x}$, $I=(4a^2+1)b+a$, which…
Clyde Kertzer
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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) for a few numbers. It would be nice to support the…
Marvin Ray Burns
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Solving equation involving finite continued fraction

I'm interested in solving the equation $$ \color{red}{x}=1+\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\ddots \cfrac{a_n}{b_n+\color{red}{x}}}}, $$ where $a_i,b_i$ are positive real numbers. Is there a formula to simplify this continued fraction? Any help will…
Wang
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Continued fractions, why the reciprocal?

Why is the reciprocal of the fractional part (vs the integer part) taken at each step? Perhaps this is obvious, if so I don't see it. Appreciate your guidance. Example from Wiki showing reciprocal taken at each step:
Nick
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$1+\frac{3+\frac{5+\frac{⋰}{6}}{4}}{2}$

Suppose we have a recursively defined sequence $T_n$ such that $T_n=n+\frac{T_{n+2}}{n+1}$ and so $T_1$ looks like this: $$1+\cfrac{3+\cfrac{5+\cfrac{7+\cfrac{9+\cfrac{⋰}{10}}{8}}{6}}{4}}{2}$$ How do I solve this? (PostScriptum. I called it T for…
cmarangu
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Algorithm to find floors of multiples of the golden ratio

What is an algorithm to calculate $\lfloor n\phi \rfloor$ given some integer $n$, where $\phi$ is the golden ratio? I am thinking the easiest way will involve calculating multiples of its continued fraction representation, since the golden ratio has…
PyRulez
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description in mathematics

I have a general question. I would like to describe the different ways of representing euler's number. My question is how to describe something in mathematics. The Euler number can be represented either as the limit value, continued fraction and…
Luca
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Why can the transfer operator of the Gauss (continued fraction) map be used to prove the Riemann hypothesis but not the map for the Engel expansion?

Recently there was a paper connecting the zeros of the GKW operator to the zeros of zeta, this is due to the fact that the Mellin transform of the Gauss map is a linear function of zeta . Transfer operator for the Gauss' continued fraction map. I.…
crow
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Solve this equation on x (x is a real number)

How does anyone solve this equation on $x$ ($x$ is a real number)? $$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\cdots}}}}=x+\cfrac{x}{x+\cfrac{x}{x+\cfrac{x}{\cdots}}}$$ Thank you in advance.
Gooble
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If $ b_{n}=2+\frac{1}{b_{n-1}} , \ \ b_{0}=2$ and $\lim_{n \rightarrow \infty} b_n=1+\sqrt 2 $

If $ b_{n}=2+\frac{1}{b_{n-1}} , \ \ b_{0}=2$ and $\lim_{n \rightarrow \infty} b_{n}=1+\sqrt 2 $ , then find $ \ \ \lim_{n \rightarrow \infty} \left|\frac{b_{n+1}-(1+\sqrt 2)}{b_n-(1+\sqrt 2)}\right| $ . Answer: I think $ \ \ \lim_{n \rightarrow…
MAS
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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 solution for a positive continued fraction? If no,…
arnabanimesh
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Find $x$ defined as a continued fraction

I have solved the above using the below method. $$x= 12 + \frac{1}{2+\left(\frac{1}{2}+x\right)}$$ After solving for $x$, I got the answer as $11.7515$ and $-1.41824$ So what is the real value of $x$, it should be one value for the expression. Why…
Roberta Blaize
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Find the real number $x$ represented by continued fraction $[12;2,2,12,2,2,12,2,2,12\dots]$

I need to fins the real vlaue of x for the continued fraction (Image attached) I have tried partial coefficient method, but didn't get the exact answer. I there any way where we can identify the value of x algebrically?
Roberta Blaize
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