# 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.

1,006 questions
Filter by
Sorted by
Tagged with
1answer
38 views

### Only one solution proofs [closed]

I have $2$ questions in my homework that I can not figure out how to deal with: $1.$ Let $(a,b,c)\in \mathbb{R}^3$, $a>1$ and $0<b \leq 1$. Prove that $a x=b \sin x+c$ has one solution. For ...
0answers
54 views

### What is the algorithm for performing continued fraction arithmetic

I am trying to write a python package for doing exact arithmetic with continued fractions, I've been looking for a good while now but can't find any good reference anywhere. I've already read gosper's ...
1answer
72 views

### Explanation of continued fraction for Bessel functions

Doing some computational searches, I found some nice continued fractions that could be used to compute Bessel functions of the first kind (I and J): ...
0answers
59 views

### Patterns in convergents of continued fraction of $\sqrt{D}$?

First, to give some background: If $D$ is an integer, then the continued fraction of $\sqrt{D}$ is always periodic. For example, the continued fraction of $\sqrt{7}$ is $[2; \overline{1,1,1,4}]$. Also,...
2answers
121 views

### Infinite fraction's derivative [closed]

$f(x)=x+\dfrac{1}{x+\dfrac{1}{x+\dfrac{1}{x+\ldots}}}$ $f'\left(\dfrac{3}{2}\right)=?$ I tried to make equation like $y^2=xy+1$ but I can't made it clearly.
2answers
92 views

1answer
34 views

### (Exponential) Mixing for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of the mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$ ...
1answer
358 views

3answers
48 views

### Closed Continued Fraction

I am trying to find a closed form of a continued fraction: $$[1,2,3,5,3,5,3,5...]$$ I understand I have to form a quadratic of some sort and factorise and to give me an irrational number but I am not ...
3answers
240 views

### 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 ...
1answer
30 views

### Convergence of a finite continued fraction for $b_i \in [-1,0]$ and $a_i=1$

based on wiki page here, finite continued fraction is as follows: $$a_0+\cfrac{b_0}{a_1+\cfrac{b_1}{\ddots+\cfrac{\ddots}{a_{n-1}+\cfrac{b_{n-1}}{...}}}}$$ I want to find the limit of finite continued ...
1answer
46 views

### How to simplify the finite continued fraction?

I want to simplify the following expression for $q_i$ and $p_i$, where $i \in \{1,2,...,n \}$ and $q_i \in [0,1]$ and $p_i \in [0,1]$ , is there any standard method for it? For example, based on ...
1answer
487 views