# 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.

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### A pattern that leads to regular continued fractions of quadratic irrationals [closed]

The following expression can be obtained by converting the continued fraction of quadratic irrationals to single fraction. $$\sqrt{N} = \frac{b\sqrt{N}+aN}{a\sqrt{N}+b}$$ The equation holds for any ...
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### Find an expansion into an arithmetic continued fraction of $\sqrt{x}$, where $x=((4a^2+1)b+a)^2+4ab+1$, $a,b \in \mathbb{N}$

I am trying to find an expansion into an arithmetic continued fraction of $\sqrt{x}$, where $x=((4a^2+1)b+a)^2+4ab+1$, $a,b \in \mathbb{N}$. So far I have: Clearly $x<((4a^2+1)b+a+1)^2$, so the ...
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### why is $2 = \frac{5}{1+\frac{8}{4+\frac{11}{7 + \frac{14}{10 + \dots}}} }$

Why is $2 = \cfrac{5}{1+\cfrac{8}{4+\cfrac{11}{7 + \cfrac{14}{10 + \ddots}}} }$ where the sequences $5,8,11,14,\dots$ and $1,4,7,10,\dots$ are of the form $5 + 3 n$ and $1 + 3n$. (This converges on ...
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### Does this continued fraction have a name? [duplicate]

$1/(2+1/(3+1/(5+...) = [0; 2,3,5,...p_{\infty}] \approx 0.432332$ Does this constant have a name? What is it called? It does appear to converge from my initial calculations, and I'm surprised that I ...
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### Sum of an infinite series with continued fractions

I'm trying to determine if it is possible to sum an infinite series of the form: \frac{1}{x}+\frac{a}{x^2(x-\frac{a}{x})}+\frac{a^2}{x^2(x-\frac{a}{x})^2(x-\frac{a}{x-\frac{a}{x}})}+\...
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