For questions on continued fractions.
3
votes
0answers
44 views
Eigenvalues of a tridiagonal trigonometric matrix
Let $A$ be the diagonal matrix w/alternating in sign diagonal entries:
$$ A =
\begin{pmatrix}
\pm \tan(\frac{\pi}{2n+1}) & 0 & 0 & \ldots & 0 \\
0 & \mp ...
0
votes
0answers
23 views
how we can find the continued fraction of incomplete Gamma function
the continued fraction is a beautiful mathematical tool
when i read about incomplete Gamma function in wikipedia I saw the continued fraction of it
i have some information about find the continued ...
4
votes
0answers
46 views
A numerical coincidence with continued fractions
My brother built a garage that measures 45 feet by 30 feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches,
$$
...
0
votes
0answers
35 views
continued fraction
$[a_0,a_1,a_2,\cdots,a_n]:=1/(a_0 + 1/(a_1 + 1/(a_2 + \cdots + 1/(a_n)\cdots )))$
I am so curious that what is the shape of $ f_n(x) $ such that
$$ \sum_{n \geq 0} f_n(x) y^n = [-y,1] +[-y,y,1]x + ...
2
votes
1answer
43 views
A trigonometric identity for special angles
Prove that for a natural number $n$,
$$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$
2
votes
0answers
33 views
Cantor set as a set of continued fractions?
Does the classical cantor set have a nice description as a set of continued fractions?
I made a (superficial) search and didn’t find anything, but I’m very tired right now, so please forgive me that ...
2
votes
1answer
82 views
Limit of a continued fraction
Given the continued fraction:
$$f(x,N)=\left[2,3,4,...N,x\right]$$
$$f(x,N)=\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{...+\cfrac{1}{x}}}}}$$
is it possible to find an expression for the integral:
...
8
votes
5answers
517 views
Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$
I'm currently a high school junior enrolling in AP Calculus, I found this website that's full of "math geeks" and I hope you can give me some clues on how to solve this problem. I'm pretty desperate ...
3
votes
1answer
67 views
Continued fraction expansion related to exponential generating function
A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series:
$$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
2
votes
2answers
81 views
Continued fraction for $\sqrt{14}$
I'm referencing this page: An Introduction to the Continued Fraction, where they explain the algebraic method of solving the square root of $14$.
$$\sqrt{14} = 3 + \frac1x$$
So, $x_0 = 3$, Solving ...
5
votes
1answer
61 views
For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?
Inspired by this question I ask this. For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?
The original question concerned $a=e$, the usual ...
24
votes
1answer
382 views
How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$
I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
5
votes
0answers
49 views
Evaluation of a slow continued fraction
Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
6
votes
1answer
125 views
Finding near-integers in a range
I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
1
vote
1answer
31 views
Defining piecewise summation of continued fractions and rationality of sums
Let $a=[a_1,a_2\dots]$ and $b=[b_1,b_2\dots]$ be two real numbers and their continued fraction representations. They may be infinite or finite.
Let us define a thing $+^*$ so that ...
2
votes
1answer
48 views
How to deal with infinite continued fractions in formal language?
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 ...
13
votes
2answers
203 views
Evaluation of a continued fraction
Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
1
vote
0answers
68 views
By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?
this is a question from a book I'm struggling with, please could you provide a clear proof
Consider the sequence of rational numbers
$a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1
...
2
votes
3answers
44 views
A question about $[c_0,c_1,\ldots,c_n]$ notation for continued fractions
I try to understand why by definition
$[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also
$[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ .
Those are continued fractions, and ...
2
votes
2answers
146 views
Calculate an infinite continued fraction
Is there a way to algebraically determine the closed form of any infinite continued fraction with a particular pattern? For example, how would you determine the value of ...
3
votes
0answers
126 views
Algorithm For Continued Fraction of $\pi$ without error.
Is there an algorithm that will output the numbers in the continued fraction of $\pi$ without error? If one uses the usual method of calculating continued fractions, an approximation of $\pi$ is ...
1
vote
2answers
56 views
Nth number of continued fraction
Given a real number $r$ and a non-negative integer $n$, is there a way to accurately find the $n^{th}$ (with the integer being the $0^{th}$ number in the continued fraction. If this can not be done ...
2
votes
2answers
67 views
Project Euler Problem 65
I am working on solving Project Euler problem #65 and run upon the following statement:
What is most surprising is that the important mathematical constant,
e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , ...
2
votes
2answers
70 views
The average denominator of the continued fraction expansion of $\pi$.
I was interested in the long term behavior of continued fraction denominators, so I plotted the average of the first $n$ terms in the continued fraction expansion of $\pi$ as a function of $n$ and got ...
5
votes
1answer
50 views
Are all numbers that have a non-repeating, non-terminating continued fraction sequence transcendental? [duplicate]
(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 ...
7
votes
3answers
150 views
Continued fractions for $\sqrt{x} $ and beyond, valid formula?
For $x > 0$, is this trick valid?
I use $$ ( \sqrt{x}-1)(\sqrt{x}+1)=x-1 $$
then $$ \sqrt{x}+1 = \frac{x-1}{\sqrt{x}+1-2} $$
so I can use iterations to get the rational approximant
$$ \sqrt{x} ...
1
vote
1answer
93 views
Continued fractions with $n$ layers
Solve the equation
$$x=2+\dfrac1{2+\dfrac1{...2+\dfrac1{2+\dfrac1x}}}$$
Where there are n layers in the fraction
5
votes
5answers
141 views
Continued Fraction [1,1,1,…]
If the continued fractional representation of an irrational number $\alpha$ is given by [1,1,1,...], I can compute that $\alpha = \frac{1+\sqrt{5}}{2}$ by solving the equation $\alpha = 1+ ...
4
votes
1answer
38 views
Continued fraction proof from matrix form
By using the definition
$$\pmatrix{p_n&p_{n-1}\\q_n&q_{n-1}} = \pmatrix{a_0&1\\1&0} \pmatrix{a_1&1\\1&0} \cdots \pmatrix{a_n&1\\1&0}$$
I need to show that $p_n/q_n$ is ...
5
votes
2answers
62 views
Continued fractions help
I'm trying to learn how to express a square root as continued fraction, but I can't get one thing.
The following example of $\sqrt{14}$ is from this page (click the image to see it at full size):
...
1
vote
2answers
81 views
Can every transcendental number be expressed as an infinite continued fraction?
Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
5
votes
0answers
58 views
Properties of a continued fraction convolution operation
Usually the partial numerators of a continued fraction are all 1s.
Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
3
votes
2answers
66 views
Continued Fractions periodicity and convolution.
Continued fractions for rationals terminate, for transcendentals like pi, they do not terminate and for irrationals (but non transcendentals) they repeat -- is this correct?
1
vote
1answer
43 views
Is it possible to define a zero-set of $X$ to be the zero-set of some $f\in C^{*}(X)$?
It is possible to define a cozero-set of $X$ to be the cozero-set of some $f\in C^{*}(X)$, in fact;
Every cozero-set in $X$ is the cozero-set of a function taking
values in $[0, 1]$.
$proof$: ...
2
votes
1answer
28 views
Applications of hypergeometric continued fractions
http://en.wikipedia.org/wiki/Gauss%27s_continued_fraction
Using a technique due to Gauss a lot of special functions can be expressed as continued fractions.
What applications of this are there ...
10
votes
2answers
200 views
How to do a very long division: continued fraction for tan
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
0
votes
1answer
50 views
Using Maple for continued fraction expansions
I can find the continued fraction expansion of a value using Maple. Is there a simple way for finding the sequence of convergents (approximants) of the continued fraction expansion in Maple? Currently ...
1
vote
1answer
124 views
How do I determine appropriate rational approximations to a sum of square roots in order to bound the error accumulation?
I have two numbers, $A$ and $B$, that are sums of integer multiples of a set of square roots of small primes (and 1) and their products:
$A = a_0 + a_1\sqrt 2 + a_2\sqrt 3 + a_3\sqrt 5 + a_4\sqrt 6 + ...
11
votes
1answer
103 views
Request for a proof of the following continued-fraction identity
I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close):
$$
...
2
votes
4answers
479 views
Continued fraction of a square root
If I want to find the continued fraction of $\sqrt{n}$ how do I know which number to use for $a_0$? Is there a way to do it without using a calculator or anything like that? What's the general ...
6
votes
1answer
198 views
Faster arithmetic with finite continued fractions
I was curious about different representations of rational numbers and came across the finite continued fraction (see wp:Finite_continued_fractions ).
Note: I will refer to traditional rational ...
6
votes
1answer
78 views
General Continued Fractions and Irrationality
A while back I came across a result about non-simple continued fractions that allows proving that some numbers are irrational. The result in modern terminology is:
If, in the continued fraction
...
1
vote
0answers
40 views
A lower bound for continued fraction approximation.
It is known that, for a continued fraction expansion of an irrational $\alpha$ we have that:
$$
\left| \alpha - \frac{p_n}{q_n} \right| = (\alpha_{n+1}q_n^2 + q_nq_{n-1})^{-1}
$$
Show that the ...
1
vote
1answer
124 views
Continued fractions with rational functions
Express the following rational function in continued-fraction form:
$${4x^2+3x-7\over 2x^3+x^2-x+5}$$
The answer is :
$${4 \over 2x- \frac{1}{2}} + { \frac{23}{8} \over ...
1
vote
2answers
107 views
Continued Fractions Approximation
I have come across continued fractions approximation but I am unsure what the steps are.
For example How would you express the following rational function in continued-fraction form:
$${x^2+3x+2 ...
1
vote
0answers
44 views
Proof of a Continued Fraction Identity using basic CF definition.
Two definitions (the first is informal) of continued fraction:
This is the basic Continued Fraction algorithm for real numbers.
Let $\alpha \in \mathbb{R}$. If $[\alpha]=\alpha$, then we are done. ...
3
votes
1answer
216 views
Calculate the continued fraction of square root
I was having difficulty understanding the algorithm to calculate Continued fraction expansion of square root.
I know the process is about extracting the integer part in repeat and maintaining the ...
0
votes
3answers
105 views
Continued fractions proof?
Let $b_1=1$ and $$b_n=1+\frac{1}{1+b_{n-1}}$$ for $n\ge 2$. Note that $b_n \ge 1$ for all $n$ in $\mathbb N$. ($\mathbb N$ represents the positive integers)
Show that $b_{2k-1}^2<2$ for all $k ...
3
votes
1answer
262 views
Approximating $\arctan x$ for large $|x|$
I would like to know if there is reasonably fast converging method for computing large arguments of arctan.
Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
4
votes
1answer
96 views
A question about continued fractions and Gauss map
For $\alpha \in (0,1)$, write $\alpha$ as a continued fraction like $\alpha=[a_1, a_2, \ldots]$ (note that the implicit $0$th coefficient $a_0=0$ has been omitted), and let $\frac{p_n}{q_n}$ be the ...
