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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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Nested solutions of a quadratic equation.

A quadratic equation of the form $x^2+bx+c=0$ can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by means of continued fraction. In this case, from the…
Emilio Novati
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Direct proof that for a prime $p$ if $p\equiv 1 \bmod 4$ then $l(\sqrt{p})$ is odd.

Definition: Assume $p$ is a prime. $l(\sqrt{p})=$ length of period in simple continued fraction expansion of $\sqrt{p}$. The standard proof of this uses the following: $p$ is a prime implies $p \equiv 1 \bmod 4$ iff $x^2-py^2=-1$ has integer…
Jason Smith
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Arithmetic of continued fractions, does it exist?

I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider $$ f(x)=\cfrac{f_{0}(x)}{1-\cfrac{f_{1}(x)}{1+f_{1}(x)-\cfrac{f_{2}(x)}{1+f_{2}(x)-\cfrac{f_{3}(x)}{1+f_{3}(x)-\cdots}}}} $$ and…
Neves
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How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in http://arxiv.org/abs/0911.1929. When I try it I get…
user58512
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Will every rational number eventually be in this set?

Let $A_0=\{0\}$. For every $n\ge 0$, let $B_n=\bigcup_{k\ge 0}A_n+k$, and let $A_{n+1}=f(B_n)$, where $f:[0,\infty)\to[0,1):x\mapsto\frac{x}{x+1}$. Let $q\in\Bbb Q\cap [0,1)$. Can we prove that $q\in A_n$ for some $n\ge 0$? Equivalently, I would…
G Tony Jacobs
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Found an recursive identity (involving a continued fraction) for which some simplification is needed.

This is my second question in this forum; as I previously explained it, I am a "hobbyst" mathematician and not a professional one; I apologize by advance if something is wrong in my question. I enjoy doing numerical computations on my leisure time,…
Thomas Baruchel
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Why does Khinchin's constant "work"?

I apologize if I missed an existing question on this, perhaps with a different spelling of Khinchin's name. I feel like I'm missing something basic. From Wikipedia, almost all real numbers have a continued fraction representation whose terms have a…
user3461142
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Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is $2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cdots}}}$. Another thing I…
Angela Pretorius
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Numbers whose decimal digits are the coefficients of its continued fraction form

A curious question recently crossed my mind: can we construct decimal numbers of the form $$\text{"a.bcdefghij…"}$$ where each letter represents a digit $0-9$ (where the number may or may not be rational), so that it is equal to a continued fraction…
tyobrien
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Surprising continued fractions of numbers in the form $\sum_{n=0}^\infty \frac{1}{a^{2^n}}$, including the same pattern for every $a>2$

I've been interested in the numbers of this form because it can be proved that for integer $a \geq 2$ all of them are irrational: $$x_a=\sum_{n=0}^\infty \frac{1}{a^{2^n}}$$ They satisfy the conditions listed in this paper: The Approximation of…
Yuriy S
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conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is…
Nicco
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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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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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Continued fraction for Apéry's constant conjectured by The Ramanujan Machine

Recently the following identity was conjectured by The Ramanujan Machine: $$ \frac{8}{7\zeta(3)}=1-\frac{u_1}{v_1-\frac{u_2}{v_2-\frac{u_3}{v_3-\ddots}}}, $$ where $u_n=n^6$ and $v_n=(2n+1)(3n^2+3n+1)$. It has not yet been proven (authors are not…
Sil
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Motivation behind this eccentric Ramanujan Identity

I just visited the MathJaX page due to the Math.SE website showing some problems while loading the page. I saw some demo math equations samples at this page, when this identity actually caught my attention: $$ \dfrac{1}{\Bigl(\sqrt{\phi…
user9413
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