For questions on continued fractions.
2
votes
1answer
260 views
Convergent of continued fractions the best rational approximation of a number? [duplicate]
Possible Duplicate:
A nicer proof of Lagrange's 'best approximations' law?
I was reading through the wikipedia article on continued fractions, and they state, essentially, that ...
17
votes
1answer
451 views
A continued fraction involving prime numbers
What is the limit of the continued fraction
$$\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{11+\cfrac{1}{13+\cdots}}}}}}\ ?$$
Is the limit algebraic, or expressible in terms of e or ...
4
votes
1answer
193 views
Is it right that the fundamental recurrence of an arbitrary continued fraction cannot be proved without induction?
Let $\dfrac{A_{n}}{B_{n}}$ be the $n^{th}$ convergent (approximant)
$$
\frac{A_{n}}{B_{n}}=b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{\begin{array}{c}
b_{3}+ \\
\\
\end{array}
...
4
votes
4answers
228 views
Relationship between degrees of continued fractions
I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, e and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
3
votes
1answer
235 views
How to express an irrational as a continued fraction in computer with high precision?
Background
I'm writing a C++ library for continued fraction using MPIR (Multiple Precision Integers and Rationals) library http://www.mpir.org/ due to the limitation of built-in ...
11
votes
2answers
668 views
Continued fraction for $\frac{1}{e-2}$
A couple of years ago I found the following continued fraction for $\frac1{e-2}$:
$$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$
...
6
votes
2answers
188 views
A question on continued fraction
Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive
convergents of its simple continued fraction, where $k\ge 1$.
Is it possible that both ...
4
votes
0answers
209 views
Evaluating matrix-continued fractions?
I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity
$A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...
3
votes
0answers
151 views
Approximation of a real number as a linear combination of two reals with coprime integral coefficients
Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
4
votes
3answers
228 views
Is it practical to use infinite continued fraction to generate random numbers?
I observed the pattern of this irrational number:
$$\sqrt{1 + \sqrt{2}}$$
and realized that each element $a_i$ occurred very randomly. For the first 100 elements, this is the result:
...
1
vote
1answer
222 views
Is the Iterated Continued fraction from Convergent​s for Pi/2 exactly 3/2?
Iterated continued fraction from convergents are described at https://oeis.org/wiki/Convergents_constant and https://oeis.org/wiki/Table_of_convergents_constants.
Do you think there is any error in ...
1
vote
1answer
128 views
How to find continued fraction of the form $a\sqrt{b}$?
For the form $\sqrt{b}$, I could just apply the recursive quadratic formula:
$$P_{k+1} = a_kQ_k - P_k$$
$$Q_{k+1} = \dfrac{d - P^2_{k+1}}{Q_k}$$
$$\alpha_k = \dfrac{P_k + \sqrt{d}}{Q_k}$$
...
-1
votes
3answers
1k views
Extract a Pattern of Iterated continued fractions from convergents
I have been working on an article at
https://oeis.org/wiki/Table_of_convergents_constants
where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
7
votes
1answer
213 views
What causes the convergence of Iterated continued fractions from convergents?
Here is a small discovery I stumbled across a few weeks ago. I hope at least one person will find it interesting enough to help me.
The iterated continued fractions from convergents (or convergents ...
4
votes
0answers
198 views
How to find the number of continued fraction from a periodic representation?
Problem
Find the number that represented by $[2,2,2 \ldots]$
I know it wasn't difficult, but I was absent the last two classes. So I just want to make sure that I got it right.
My attempt was,
...
11
votes
1answer
325 views
Deriving a trivial continued fraction for the exponential
Lately, I learned about the following continued fraction for the exponential function:
$$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$
I thought it was ...
10
votes
2answers
829 views
How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?
Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions.
Description of the Bauer-Muir transformation
...
10
votes
3answers
474 views
Proving the continued fraction representation of $\sqrt{2}$
There's a question in Spivak's Calculus (I don't happen to have the question number in front of me in the 2nd Edition, it's Chapter 21, Problem 7) that develops the concept of continued fraction, ...
2
votes
1answer
230 views
Solving equations using continued fractions?
We solve the pell equation using the continued fraction for square root of 2. What equations can we solve using the continued fraction of cube roots (and other numbers too)?
6
votes
0answers
388 views
What can Euler's identity teach us about (generalised) continued fractions?
We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ ...
5
votes
1answer
217 views
What is the length of a continued fraction expansion of a rational number?
I was reviewing quantum factorization and am slightly unclear on a classical detail of order-finding.
Given a (suitably nice) periodic function $f$ with unknown period $r$ and a power of two $N > ...
10
votes
1answer
383 views
Continued Fraction expansion of tan(1)
Prove that continued fraction of tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a ...
4
votes
2answers
252 views
Adding integers to an infinite continued fraction expansion doesn't change the value?
I'm learning about continued fractions, and I've enjoyed them so far, but I'm unsure if I've done the following correctly. I have no real experience with analysis, so I'm not sure if my reasoning is ...
