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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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Infinite continued fractions

I am fascinated by the fact that "important" irrational numbers like the golden ratio, base of the natural exponent, pi, square roots have a "regular" representation as an infinite continued fraction. Is there some correspondence between "having a…
vasily
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Question on continued fraction?

I have seen in wikipedia that irrational numbers have infinite continued fraction but I also found $$1=\frac{2}{3-\frac{2}{3-\ddots}}$$ so my question is that does that mean $1$ is irrational because it can be written as an infinite continued…
Souvik
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Convergence of a finite continued fraction for $b_i \in [-1,0]$ and $a_i=1$

based on wiki page here, finite continued fraction is as follows: $$a_0+\cfrac{b_0}{a_1+\cfrac{b_1}{\ddots+\cfrac{\ddots}{a_{n-1}+\cfrac{b_{n-1}}{...}}}}$$ I want to find the limit of finite continued fraction for $b_i\in [-1,0]$ and…
Optimized Life
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Continuous fractions theorem

$$k_n h_{n-1} - k_{n-1} h_n = (-1)^n$$ Can you guys explain to me about the continued fractions and about this particular theorem? What does this mean? (I have some general info about this, but it is kind of vague for me)
math boy
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How to show that the continued fraction of $\sqrt{a^2+1}$ is $[a,2a,2a,2a,\dots]$?

I thought about the following, as $a^2\leq a^2+1 \leq (a+1)^2$, then we have $a\leq \sqrt{a^2+1}\leq a+1$ so $\lfloor \sqrt{a^2+1}\rfloor=a$ but I am having trouble doing the same for the $2a,2a,\dots$ and even If I am able to do that, I don't know…
Red Banana
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continued fractions estimate

I am trying to do some estimates concerning continued fractions and according to my teacher I should get $\left\lvert \alpha - \frac{p_n}{q_n}\right\lvert > \frac{1}{q_n+q_{n+1}} > \frac{1}{2q_{n+1}}$. where $\frac{p_n}{q_n} = [a_0,a_1,...,a_n]$ is…
L.Z.
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Any way to natively calculate a decimal fraction from a continued fraction?

I know that a continued fraction can be converted to a decimal fraction by condensing it to a simple fraction and then performing long division to produce a decimal fraction. I am wondering how the decimal fraction can be discovered "natively,"…
Tyler Durden
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Solving quadratics using continued fractions - 2 puzzling cases for me.

Suppose you have $x^2 - 2x -1 = 0$, as in the related question (Solving Quadratics Using Continued Fractions/Nonsimple to Simple Continued Fractions). Although that question did not go through the steps used to calculate $[2; \overline{2}]$, there…
user77970
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my question is about recursion of continued fraction

1 is not equals 2, then whats wrong with the following... 1=2/(3-1) and if we replace (can we or cannot?) 1 on right hand side by 1=2/(3-1) that is 1=2/(3-2/(3-1)) and if we continue replacing 1 on right hand side that…
Intellectual_
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Prove the continued fraction of $\sqrt{n}$ has a period.

I want to know the proof that the continued fraction of $\sqrt{n}$ has a period. This question and answer prove that when the continued fraction has a period, it can be represented by quadratic form. However, it doesn't prove that the continued…
ueir
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Symmetric continued fraction

Let (a_i)_{1<=i<=n} be a sequence of positive integers and c be a positive integer such that c divides a_1. Suppose that we have this symmetric of continued fraction [a_n, a_n-1,...,a_1]=c^{-1}[a_1, a_2, ...., a_n]. Can we assume that n is even and…
Oussema
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How to Compute Infinite Continued Factions

I'm supposed to find the value of the infinite continued fracton $[2;1,3,1,3,1,3,1,3...]$. How would I go about doing this?
Drew Weisserman
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The continued Fraction Algorithm Proofs

I am trying to prove that for any positive integer $\sqrt{n^{2}+1}= [n; \overline{2n}]$, where $\overline{2n}$ is infinitely repeating. I think the best way to do this is to use the continued fraction algorithm? but I can seem to get it at all. I…
Joanna M
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forming the continued fraction of the euler number

Good afternoon The euler number is a irrational number. And you can have a infinite continued fraction of euler number. But how can you form the coninued fraction of euler number?
Gianna
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Continued fractions formula

How would I find the continued fraction of and number $a/b$? For example, $5/8$ = $1/(1+3/5)$ I tried using its decimal expansion but couldn’t find anything, and I want to be able to describe the golden ratio as a fraction.
Jon due
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