Mathematics Stack Exchange
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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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Solution to an equation involving continued fractions and decimal expansion.

Does there exist a positive real number $x$, whose decimal expansion is non-terminating, such that: $\large \displaystyle x = \overline{a_0.a_1a_2a_3a_4a_5\cdots} = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 +…
Jack Tiger Lam
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Finding the value of a continued fraction?

I know how to calculate the exact value for continued fractions such as $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}=\frac{1+\sqrt{5}}{2}$$ However, is it possible to find the value of continued…
user67253
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Are all numbers that have a non-repeating, non-terminating continued fraction sequence transcendental?

(By continued fraction sequence, I'm specifically talking about the one kind where the numerator of every fraction is 1.) As a kid in middle school, I learned that all irrational numbers have non-repeating, non-terminating positional notation…
Joe Z.
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How to find a continued fraction

How does one find the value of $$1+\cfrac 1 {2+\cfrac 2 {3+\cfrac{3}{4+\cfrac{4}{5+\cfrac{5}{\ddots}}}}} \ \text{ or }\ 1+\cfrac{2}{3+\cfrac{4}{5+\cfrac{6}{7+\cfrac{8}{9+\cfrac{10}{\ddots}}}}}$$ Is there any way to find a continued fraction that…
user301661
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Continued Fraction Identity Problem

Question: What went wrong in my work to$$\ln(1-x)=-\cfrac x{1-\cfrac x{2+x-\cfrac {2x}{3+2x-\cfrac {3x}{4+3x-\ddots}}}}\tag{1}$$ I started with the expansion$$\begin{align*}\ln(1-x) & =-x-\dfrac {x^2}2-\dfrac {x^3}3-\dfrac {x^4}4-\&\text{c}.\\ &…
Frank
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Continued fraction for $\int_{0}^{\infty}(e^{-xt}/\cosh t)\,dt$

In one of the comments to a question I posted on MSE, I got this wonderful continued fraction $$\int_{0}^{\infty}\frac{e^{-xt}}{\cosh t}\,dt =…
Paramanand Singh
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Calculating the continued fraction of $\sqrt{47}$ using a different result

I have calculated the continued fraction of $\alpha=\frac{6+\sqrt{47}}{11}$ which equals $\overline{[1,5,1,12]}$. Now I am asked to calculated the cont. fraction of $\sqrt{47}$ using this result. I am not sure whether there is a simple formula to…
Hku
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How find this Continued fraction $[1,3,5,7,9,11,\cdots]$ value.

show this: $$\alpha=[1,3,5,7,9,11,\cdots]=1+\dfrac{1}{3+\dfrac{1}{5+\dfrac{1}{7+\dfrac{1}{\cdots}}}}=\dfrac{e^2+1}{e^2-1}$$ I found wiki Continued fraction also not have this problem,maybe this problem can't have simple closed form? Thank you for…
math110
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Continued fraction: Show $\sqrt{n^2+2n}=[n; \overline{1,2n}]$

I have to show the following identity ($n \in \mathbb{N}$): $$\sqrt{n^2+2n}=[n; \overline{1,2n}]$$ I had a look about the procedure for $\sqrt{n}$ on Wiki, but I don't know how to transform it to $\sqrt{n^2-2n}$. Any help is appreciated. EDIT: I…
ulead86
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Taking the negative of a continued fraction

If I have a continued fraction for an irrational number $z= \langle a_0;a_1,a_2,a_3,\ldots\rangle$ it seems that $(-1)*z = \langle-a_0;-a_1,-a_2,-a_3,\ldots\rangle$. Is this true? In general, if you have the continued fraction representation for $y$…
Sara
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Short proof of Seidel-Stern theorem on continued fractions

Let $\mathbf{a}=\{a_n:n\ge0\}$ be a sequence of positive real numbers, and consider the formal continued fraction $$K(\mathbf{a})=a_1+\cfrac{1}{a_1+\cfrac{1}{a_2+\ddots.}}$$ Seidel-Stern Theorem. If $\sum_{n\ge0} a_n=\infty$, then the formal…
Pengfei
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Expressing $1+a_1(b_1+a_2(b_2+a_3(b_3+a_4(b_4+a_5(\cdots)))))$ as an infinite continued fraction.

Euler derived the following identity $$ 1+a_{1}+a_{1}a_{2}+a_{1}a_{2}a_{3}+\cdots= \cfrac{1}{ 1- \cfrac{a_{1}}{ 1+a_{1}- \cfrac{a_{2}}{ 1+a_{2}- \cfrac{a_{3}}{ 1+a_{3} - \ddots}}}}\;\;\;\;\;\;\;(1) $$ where…
Neves
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Evaluation of Rogers-Ramanujan continued fraction $R(e^{-2\pi/5})$

Let $A = \{(\sqrt{5} + 1)/2\}^{5}$ and let $\alpha,\beta$ be positive reals such that $\alpha\beta = \pi^{2}/5$. Then it is known that $$\left\{A + R^{5}(e^{-2\alpha})\right\}\left\{A + R^{5}(e^{-2\beta})\right\} = 5\sqrt{5}A$$ where $$R(q) =…
Paramanand Singh
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Continued Fraction $1+\frac{1}{\frac{1}{2}+\frac{1}{3+\frac{1}{\frac{1}{4}+\frac{1}{5+...}}}}$

I arrived at this while solving for a Physics Problem. $$1+\frac{1}{\frac{1}{2}+\frac{1}{3+\frac{1}{\frac{1}{4}+\frac{1}{5+\frac{1}{\frac{1}{6}+\frac{1}{7+\frac{1}{\frac{1}{8}+\frac{1}{9+\frac{1}{\frac{1}{10}+...}}}}}}}}}$$ I was wondering whether…
Miracle Invoker
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Evaluating the continued fraction $ 1 + \frac{1}{1 + \frac2{1 + \frac3{1 +\cdots}}}$

How to evaluate this infinitely nested fraction? $$ 1 + \frac{1}{1 + \frac2{1 + \frac3{1 +\cdots}}}$$ I tried to define some $g(x)$ such that $$g(x) = 1 + \frac{x}{1+\frac{x+1}{1+\frac{x+2}{1+\cdots}}}$$ and solve the equation $$g(x) = 1 +…
Deadly Chuck314
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