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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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The convergent of the continued fraction of $ \sqrt{N} $ modulo $4$.

Consider the continued fraction of $\sqrt{N}$ for an integer $N$. Let $d$ be its period and $ P_n/Q_n $ be its $ n $-th convergent. When $ d $ is odd, Theorem 1 of this article that says that $Q_{d-1}$ is odd. However, I calculated the first few…
Humourprince
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Higher order roots as Continued Fractions

All square roots can be expressed as a regular Continued Fraction. For example: $$\newcommand{\contfrac}{\raise{-0.5ex}\mathop{\Large\mathrm{K}}}\sqrt{2}+1=2+\contfrac_{n = 0}^\infty \frac{1}{2}$$ It is easy to see that comparable regular Continued…
Paul vdVeen
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Continued fraction representation of quadratic irrationals

I just learned Continued Fractions and I was asked to evaluate the simple continued fractions $[\bar{1}]$ , $[\bar{2}]$ , and $[1,\bar{2}]$ so far all I know about Quadratic Irrationalities and Infinite Continued Fractions is this excerpt from…
no lemon no melon
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What is the value of $1+\frac{1}{2+\frac{1}{3+\dots}}$?

This is just a curiousity question. HISTORY:- $Motivation:$ I first started wondering about this question about 3 months ago when I first got interested in continued fractions but I really didn't tried to get its value but when I saw the thumbnail…
Rounak Sarkar
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The average denominator of the continued fraction expansion of $\pi$.

I was interested in the long term behavior of continued fraction denominators, so I plotted the average of the first $n$ terms in the continued fraction expansion of $\pi$ as a function of $n$ and got the following graph: And it turns out that at…
guest196883
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Continued fraction proof from matrix form

By using the definition $$\pmatrix{p_n&p_{n-1}\\q_n&q_{n-1}} = \pmatrix{a_0&1\\1&0} \pmatrix{a_1&1\\1&0} \cdots \pmatrix{a_n&1\\1&0}$$ I need to show that $p_n/q_n$ is the continued fraction $[a_0;a_1, ..., a_n]$
Bill
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Non convergent simple continued fractions?

Let $$ be an infinite sequence of integers such that $$00.$$ For any natural $n$ we know there exists a convergent: one rational number $r_n$ such that it is equal to the simple continued…
Dr Potato
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Continued Fractions periodicity and convolution.

Continued fractions for rationals terminate, for transcendentals like pi, they do not terminate and for irrationals (but non transcendentals) they repeat -- is this correct?
Cris Stringfellow
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Reverse continued fractions

Basic continued fractions arise from recurrence relations such as: $$ n = a + \frac{b}{n}, $$ This gives rise to the continued fraction: $$ n = a + \frac{b}{a+\frac{b}{a+\frac{b}{a+...}}}. $$ What about relations such as: $$ n = a +…
Klangen
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Evaluate $C_f(n)={1\over4n+{1\over12n+{1\over 20n+{1\over 28n+\cdots}}}}$

Does this continued fraction, $C_f(n)$ has a closed form? $$C_f(n)={1\over4n+{1\over12n+{1\over 20n+{1\over 28n+\cdots}}}}\tag1$$ $n\ge1.$ Yes? So I did some mathematical experimental for a while and sort came up with this as a closed form …
gymbvghjkgkjkhgfkl
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What is the value of this infinite fraction, where each successive row counts until a power of two?

Here is the fraction $$\frac{1}{\cfrac{2}{\cfrac{4}{8\cdots+9\cdots}+\cfrac{5}{10\cdots+11\cdots}} + \cfrac{3}{\cfrac{6}{12\cdots+13\cdots}+\cfrac{7}{14\cdots+15\cdots}}}$$ I have tried iterating row by row, seeing if the fraction converges, however…
volcanrb
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Ramanujan's continued fraction for ratio of gamma values

Ramanujan mentioned the following continued fraction formula in his famous letter to G. H. Hardy in 1913: $$\cfrac{4}{x +}\cfrac{1^{2}}{2x +}\cfrac{3^{2}}{2x +}\cdots = \left(\dfrac{\Gamma\left(\dfrac{x + 1}{4}\right)}{\Gamma\left(\dfrac{x +…
Paramanand Singh
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Continued Fraction for $e$

Does anyone know of some nice/simple proofs for the continued fraction of $e$? i.e. $$ e = [2;1,2,1,1,4,1,1,6,...,1,1,2k,1,1,...] $$ I have read a nice method in Cohn, H. "A Short Proof of the Simple Continued Fraction Expansion of e." Amer. Math.…
Hugh Entwistle
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Continued fraction question

I have been given an continued fraction for a number x: $$x = 1+\frac{1}{1+}\frac{1}{1+}\frac{1}{1+}\cdots$$ How can I show that $x = 1 + \frac{1}{x}$? I played around some with the first few convergents of this continued fraction, but I don't get…
Nga
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When $x=2$ in the infinitely continued fraction $x+\frac{1}{x^2+\frac{1}{x^3+\ldots}}$, what algebraic value does it converge to?

Say you have the infinitely continued fraction: $$x+\cfrac{1}{x^2+\cfrac{1}{x^3+\cfrac{1}{x^4+\ddots}}}$$ When $x=1$, you can see that it's $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ which converges upon the golden ratio $\phi =…
Sam
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