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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

1191 questions
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Continued Fraction Counting Problem

The house of my friend is in a long street, numbered on this side one, two, three, and so on. All the numbers on one side of him added up exactly the same as all the numbers on the other side of him. There is more than fifty houses on that side of…
Beans on Toast
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Can the number of digits in the denominator of a node in Stern-Brocot-Tree decrease in its children?

The Stern-Brocot-Tree looks like this: (image source files) It is an infinite binary tree that contains every positive rational number as exactly one node. The children of a node $\frac{a}{b}$ are: Left: Search the first left parent $\frac{x}{y}$,…
Martin Thoma
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An infinite continued fraction of $x^n$

Below this answer @WeiZhong commented that it is wrong, I also think it is wrong. Reducing each numerator to…
hbghlyj
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Closed form for this continued fraction

Is there a closed form for this continued fraction? $$x+\frac{1}{x+\frac{1}{x+\frac{1}{...}}}$$
Ken
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Mirrored Continued fractions

I know how to evaluate the Continued Fraction $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\ddots}}}$ and its dual version: $1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\cfrac{4}{1+\ddots}}}}$ I also understand…
Paul vdVeen
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continuous fraction for $ 30/\pi^2 $

I read that $ 30/\pi^2 $ can be represented as a continued fraction of the form $$ 3 + \cfrac{1}{25 + \cfrac{16}{69 + \cfrac{81}{\ddots}}} $$ over here: http://www.ramanujanmachine.com/wp-content/uploads/2020/06/pi_square.pdf where can I find the…
Vanessa
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What is the minimum difference between decimal and continued fraction formed by its digits?

How close can an infinite decimal (with no zeroes except the leading one, and between 0 and 1) get to its continued fraction formed by using its digits after the decimal point? Example : difference between 0.3219… and continued fraction…
user575696
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Calculate the continued fraction

Find the limit $$ \frac{\color{red}{1+\cfrac{3}{4+\cfrac{7}{8+\cfrac{11}{12+\dots}}}}}{\color{blue}{2+\cfrac{5}{6+\cfrac{9}{10+\cfrac{13}{14+\dots}}}}}. $$ By direct calculation I have got that it is about 0.59*** but I hope there exists an exact…
Leox
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Continued fraction for $\sqrt{D}$ upper bound of cycle length

Is there a formula for determining the upper bound for the length of the cycle for the continued fraction expansion of $\sqrt{D}$? I am attempting to expand for $D=4097280036$ and have tried up to 3500 loops but no luck. Does the repetitive cycle…
Maestro13
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Prove that these continued fractions are equal.

The wikipedia page on Generalized Continued Fractions gives this equation. $$ \log \left( 1+\frac{x}{y} \right) = \cfrac{x} {y+\cfrac{1x} {2+\cfrac{1x} {3y+\cfrac{2x} {2+\cfrac{2x} {5y+\cfrac{3x} {2+\ddots}}}}}} = \cfrac{2x} {2y+x-\cfrac{(1x)^2}…
tyobrien
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What do High-Water Marks in Continued Fractions mean?

While reading through several articles concerned with mathematical constants, I kept on finding things like this: The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...\right]$. The…
draks ...
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Unsure about infinite continued fraction

How do you/is it possible to express $a=\cfrac{1}{2+\cfrac{3}{4+\cfrac{5}{6+\cdots}}}$ in the form $\frac{p}{q}(k+\sqrt{n})$? I'm still in high school, so I'm not familiar with especially sophisticated approaches for evaluating infinitely continued…
clueless
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Great Common Division with Continued Fractions

If I have this GCD equation: $$89=16\cdot5+9\\ 16=9\cdot1+7\\ 9=7\cdot1+2\\ 7=2\cdot3+1\\ 2=1\cdot2+0$$ Then my continued fraction will be: $[5: 1, 1, 3, 2]$ But if I will have this GCD equation: $$300=99\cdot3+3\\ 99=3\cdot33+0$$ Then my continued…
user165488
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How to find the terms of the continued fraction representation for $e^\pi$

The question is - Find the first ten terms of the continued fraction representation for $e^ \pi $
S.Dan
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Proof of continued fraction convergence theorem

How does one prove that $$|\alpha - \frac{p_n}{q_n}| < |\alpha - \frac{p_m}{q_m}|$$ for all $n>m$? I know that the left side is less than $\frac{1}{2q_n^2}$ and the right side is less than $\frac{1}{2q_m^2}$, but how do we know they are not equal?…
Elliot Gorokhovsky
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