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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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Divergent continued fractions?

The solutions to $$ x^2-6x+10=0 \tag 1 $$ are $$ x = 3\pm i\tag2. $$ Rearranging $(1)$ just a bit, we get $$ x = 6 -\frac{10}x \tag3 $$ and then substituting the right side of $(3)$ for $x$ within the right side we get $$ x=6 -…
Michael Hardy
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Deriving a trivial continued fraction for the exponential

Lately, I learned about the following continued fraction for the exponential function: $$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$ I thought it was something new, but evaluating the successive…
gorilla
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Are there simple algebraic operations for continued fractions?

I thought about continued fractions as a cool way to represent numbers, but also about the fact they are difficult to treat because standard algebraic operations like addition and multiplication don't work on them in a simple way. My question is: do…
Blex
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Bi-linear relation between two continued fractions

We know that any positive real number $x$ can be represented as a simple continued fraction $$x = a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{a_{3} + \cdots}}} = [a_{0}, a_{1}, a_{2}, a_{3}, \ldots]$$ where $a_{0}$ is a non-negative…
Paramanand Singh
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Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$

I'm currently a high school junior enrolling in AP Calculus, I found this website that's full of "math geeks" and I hope you can give me some clues on how to solve this problem. I'm pretty desperate for this since I'm only about $0.4%$ to an A- and…
Samuel S
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Multiply all terms in continued fraction by a constant

I noticed that continued the fraction for $\sqrt{12}$ is $3;2,6,2,6,2,\ldots$ and the continued fraction for $\sqrt{7\times12}$ is $9;6,18,6,18,6,\ldots$ all the terms in the continued fraction are multiplied by $3$. Is this just "coincidence"? In…
dspyz
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How to prove that $\frac{\pi}{2}=\left[1, 1, \tfrac{1}{2},\tfrac{1}{3},\tfrac{1}{4},\ldots\right]$?

I don't know how to prove the generalized continued fraction $\pi/2=[1;1/1,1/2,1/3,1/4,...]=1+\cfrac{1}{1/1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\ddots}}}}$ It appears on wikipedia without proof, it contains the terms of the harmonic series but…
Dabed
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How does one come up with a Continued Fraction?

All over the place on Wikipedia, I see a bunch of identities related to continued fractions, like $$\arctan…
Franklin Pezzuti Dyer
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Is there any progress in evaluating the Continued Fraction with squares?

The following infinite Continued Fractions have a couple of characteristics in…
Paul vdVeen
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Continued fractions with a bounded sequence of terms

Notation: By $[a_0;a_1,a_2,\ldots]$ I mean the continued fraction $$a_0+\frac1{a_1+\dfrac1{a_2+\dfrac1{a_3+\ddots}}}$$ where $a_n$ is a positive integer. Context: Let $\alpha$ be an irrational number and $[a_0;a_1,a_2,\ldots]$ its continued…
ajotatxe
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What is the length of a continued fraction expansion of a rational number?

I was reviewing quantum factorization and am slightly unclear on a classical detail of order-finding. Given a (suitably nice) periodic function $f$ with unknown period $r$ and a power of two $N > r^2$, the quantum subroutine yields (with bounded…
S Huntsman
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Continued fractions help

I'm trying to learn how to express a square root as continued fraction, but I can't get one thing. The following example of $\sqrt{14}$ is from this page (click the image to see it at full size): In the 2nd row of the table, can anyone please tell…
xylon97
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Continued fraction of $\sqrt{67} - 4$

Find the continued fraction of $ \sqrt{67}-4 $ . $$ $$ We Know that if $ N $ is not a perfect square and if continued fraction of $ \sqrt N $ is $ \sqrt N = [a_{1} , \overline {a_{2},a_{3} , \ldots , 2a_{1}} ]$ , then the continued fraction of…
MAS
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Mistake in Khinchin's "Continued Fractions"

I am reading Khinchin's Continued Fractions page 10. $\lbrack a_1;a_2,a_3\ldots\rbrack$ is a continued fraction and $q_k$ is given by $q_k=a_kq_{k-1}+q_{k-2}$. Suppose $\sum_{n=1}^{\infty}a_n$ converges so that there is a $k_0$ for which $k\ge k_0$…
The Substitute
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Calculate the continued fraction of square root

I was having difficulty understanding the algorithm to calculate Continued fraction expansion of square root. I know the process is about extracting the integer part in repeat and maintaining the quadratic irrational $\frac{m_n + \sqrt{S}}{d_n}$.…
chyx
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