# 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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### arithmetic with continued fraction abbreviated notation?

say we have the string $[4;3,2,2]$ and [3;1,1]. they represents $73/17$ and $7/2$. the product of these numbers is $511/34$, which is $[15;34]$ in continued fraction abbreviated notation form. is ...
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### Gauss measure and continued fraction

For $x \in [0,1)$ then the continued fraction representation of $$x=0 + \cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\cfrac{1}{a_3(x)+\cfrac{1}{\dots}}}}$$ can be written as $[0; a_1(x), a_2(x), a_3(x), \dotsc]$ ...
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### Let $x=1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{…}}}}$; then the value of $(2x-1)^2$ equals…

Let $$x=1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{...}}}};$$ then the value of $(2x-1)^2$ equals... I don't how to start this question. Please help.
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### How to Compute Infinite Continued Factions

I'm supposed to find the value of the infinite continued fracton $[2;1,3,1,3,1,3,1,3...]$. How would I go about doing this?
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### The continued Fraction Algorithm Proofs

I am trying to prove that for any positive integer $\sqrt{n^{2}+1}= [n; \overline{2n}]$, where $\overline{2n}$ is infinitely repeating. I think the best way to do this is to use the continued fraction ...
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### Find the Farey triple given the base vertex of any isosceles triangle in Farey diagram.

There is a visualization of the circular Farey diagram where all triangles are isosceles. Observe that any rational $\frac{a}{b}$ distinct from $\frac{0}{1}$ and $\frac{1}{0}$ is always the vertex ...
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### Algorithm for regular continued fraction of a square root

Say I have a number $n$, and want to find the expression of $\sqrt{n}$ as a regular continued fraction. How would I do such a thing systematically? A naive computer algorithm wouldn't work, due to ...
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### how to convert continued fractions into normal fractions?

i couldnt find anything on google so i just tried opening it normally and recording each step. so i got: [d,c,b,a] = ((((a)*b+1)*c+A)*d+B)/C. [e;d,c,b,a]=(((((a)*b+1)*c+A)*d+B)*e+C)/D. etc.. (X ...
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### Non convergent simple continued fractions?

Let $<a_0;a_1,a_2,\dots>$ be an infinite sequence of integers such that $$0<n\implies a_n>0.$$ For any natural $n$ we know there exists a convergent: one rational number $r_n$ such that ...
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### Sequence Converging to the Square Root of an Integer $S \gt 1$

I noticed this answer to the question $\quad$ Continued fraction of a square root and the comment So I felt obliged to take this on using the theory of sequences. Confession: I find it difficult ...
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### Integral over recurrence relationship

I'm interested in evaluating the following definite integral \begin{equation} I_n = \int_0^{\gamma} F_n(x)\:dx \end{equation} Where $\gamma \gt 0$ and $F_n(x)$ is based on the recurrence ...
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### Proof of this formula for $\sqrt{e\pi/2}$ and similar formulas.

\begin{align} \sqrt{\frac{e\pi}{2}}=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}} \end{align} as seen here. ...
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### Simple continued fraction for irrational numbers.

I read it here that: "What you must have read is that a number with an infinite simple continued fraction expansion is irrational. A continued fraction is "simple" if all the partial numerators are ...
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### I have found a way of computing Euler's number. Is there any possible intuition of how that might be the case?

So a few days ago I just kind of messed around with my calculator, when I had an idea about a new continued fraction. I inputted it, and I found that it converged really quickly, and, quite wondrously,...
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### Continued fraction of $\phi^3$

I found that $$\phi^3=4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ How should I prove this? Attempt: Suppose$$x= 4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ To ...
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### Identifying the space of geodesics on the hyperbolic plane as a topological space.

On pg. 2 of this PDF, the author defines $\mathcal G=(-\infty, 0)\times(0, 1)$ and mentions that $\mathcal G$ can be thought of as "a space of geodesics on the hyperbolic $2$-space $\mathbb H^2$." ...
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### $N \in \mathbb{N}$ is not a square, show that the continued fraction expansion of $\sqrt N/\lfloor\sqrt N\rfloor$ is $[1,\overline{a_1,a_2,\dots,2}]$

Let $N \in \mathbb{N}$ not a square, show that the continued fraction expansion of $\sqrt{N}/\lfloor\sqrt{N}\rfloor$ is $[1,\overline{a_1,a_2,\dots,2}]$. My notations: the fractional part of $a$ is ...
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### Predicting the change in the denominator of a continued fraction when reversing the order of $a_1$ through $a_n$.

When reversing the order of $a_1$ through $a_n$ in a continued/extended fraction, (ie. [$a_1$: $a_2$, ... $a_{n-1}$, $a_n$] becomes [$a_n$: $a_{n-1}$, ... $a_2$, $a_1$]) we see that the denominator ...
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### Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$ Prove it converges and, evaluate the series. For the first part of the question, I prove it ...
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### Representation of Transcendental number via continued fractions

My question is quite simple. As far as I know, rational numbers has finite continued fraction representation. And if for any number we have infinite continued fraction, then it necessarily is ...