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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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Length of continued fraction

Why is the periodic length of simple continued fraction expansion of any quadratic irrational i.e irrational of the form $$\dfrac{P+\sqrt{R}}{Q}$$ is less than $2R$?
SSK
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Proof that e is the following infinite fraction:

How do I show $e = 2+ 1/(1+1/(1+4+1/(1+...$? Where for every 2ones in a row there is an even number (I am sure you all know what I am talking about).
mtheorylord
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Periodic Continued Fraction Formula for $\sqrt{a^2+4}$ for any non-zero integer $a$

If $$x = \sqrt{ a^2 + 4 } $$ then I have conjectured that: $$ x = [ a ; \overline{ \frac{a}{2} , 2a }] \ \ \ \text{ if a is even}$$ and $$ x = [ a ; \overline{ \frac{a}{2}, 1, 1, \frac{a}{2}, 2a } ]\ \ \ \text{ if a is odd} $$ where we…
Hugh Entwistle
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Finding 1/(2+1/(3+1/(4+...)))

I was wondering how to find the value of the following series: $$\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\ddots}}}}}$$ How can I solve this?
AMACB
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Is $ 2.7182818281828...$ a semiconvergent of e?

Euler's number $e=2.71828 18284 59045... $ can be approximated by the rational number: $$ x=\frac{271,828-27}{100,000-10}= \frac{271,801}{99,990} =2.7182818281... $$ Also, $e$ has the well-known continued fraction expansion $$ e =…
pdmclean
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Theorem 1 in Khinchin's "Continued Fractions"

I'm reading an English translation of Khinchin's Continued Fractions and I may have found an error in Theorem 1, page 4. Khinchin observes that if we simplify a finite continued fraction $[a_0; a_1, ... a_n]$ as $p/q$, and the continued fraction…
Jack M
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Continued fraction manipulation

I have the following continued fraction $$ \frac{1}{a_1x+}\;\;\frac{1}{b_1+}\;\;\frac{1}{a_2x+}\;\;\frac{1}{b_2} $$ The paper I am reading then converts this to the following continued z-fraction but does not show any work $$…
Darcy
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Continued Fraction for Root 5

How can I find the continued fraction expansion for the square root of 5. Do this without the use of a calculator and show all the steps.
Jessie
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On the continued fraction of $e$

The following question comes up during analysis of Padé approximants to $e^x$ (see my related question in MathOverflow for more background). Recall that the continued fraction expansion of $e$ is $$ e = [2,1,2,1,1,4,1,1,6,1,1,8,\ldots].…
Yuval Filmus
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Convergents of continued fraction proof

Let $\frac{P_n} {Q_n} and \frac{P_{n+1}} {Q_{n+1}}$ be two consecutive continued fraction convergents for $b$. Then prove that: $$\left|{\frac{P_n} {Q_n}-b}\right|< \frac{1}{2Q_n^2}$$ or $$\left|{\frac{P_{n+1}} {Q_{n+1}}-b}\right|<…
L887
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functions on a continued fraction expansion

Let $x$ be an irrational number with continued fraction expansion $[a_0;a_1,a_2,\ldots]$. Is there an $x$ and a non-identity function $f$ such that $f(x)=[f(a_0);f(a_1),f(a_2),\ldots]$. Given that I don't know too much about continued fractions…
snulty
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Is there a bijection between real numbers and continued fractions?

It is known that there is a bijection between rational numbers and finite continued fractions, so every rational number is uniquelly identified by a finite continued fractions and vice versa. It is also known that for any irrational number, we can…
Emo
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Calculate an infinite continued fraction as special function

It is possible to convert this infinite continued fraction $\cfrac{1}{-a+\cfrac{b\;f(0)}{a+\cfrac{b\; f(1)}{-a+\cfrac{b\; f(2)}\ddots}}}$ to a special function ? Please, how do it? where : $(a,b) >0$ and $f(n)=\cfrac{(n+1)^2}{4(n+1)^2-1}$ , $n \in…
Betatron
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Continued fraction with alternative signs

Let $a_n$ be a sequence of real numbers. We can define a formal finite continued fraction as usual $$[a_0]=a_0,[a_0,a_1]=a_0+\frac{1}{[a_0]},\cdots,[a_0,a_1,\cdots,a_n]=a_0+\frac{1}{[a_1,\cdots,a_n]}.$$ If all $a_n\ge0$, then by Seidel-Stern Theorem…
Pengfei
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Is the Iterated Continued fraction from Convergent​s for Pi/2 exactly 3/2?

Iterated continued fraction from convergents are described at https://oeis.org/wiki/Convergents_constant and https://oeis.org/wiki/Table_of_convergents_constants. Do you think there is any error in the computations, or perhaps in my interpretation,…
Marvin Ray Burns
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