For questions on continued fractions.
36
votes
1answer
510 views
A new continued fraction for Apéry's constant, $\zeta(3)$?
As a background, Ramanujan also gave a continued fraction for $\zeta(3)$ as
$\zeta(3) = 1+\cfrac{1}{u_1+\cfrac{1^3}{1+\cfrac{1^3}{u_2+\cfrac{2^3}{1+\cfrac{2^3}{u_3 + \ddots}}}}}\tag{1}$
where the ...
30
votes
5answers
2k views
Continued fraction fallacy: $1=2?$
It's easy to check that for any natural $n$
$$\frac{n+1}{n}=\frac{1}{2-\frac{n+2}{n+1}}.$$
Now
...
24
votes
1answer
399 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 ...
17
votes
1answer
467 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 ...
15
votes
2answers
717 views
Motivation behind this eccentric Ramanujan Identity
I just visited the MathJaX page due to the Math.SE website showing some problems while loading the page. I saw some demo math equations samples at this page, when this identity actually caught my ...
15
votes
1answer
601 views
What was Ramanujan's solution?
The wikipedia entry on Ramanujan contains the following passage:
One of his remarkable capabilities was the rapid solution for
problems. He was sharing a room with P. C. Mahalanobis who had a
...
14
votes
2answers
206 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 ...
13
votes
2answers
207 views
Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$
How can I solve for $x$:
$$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$
Any clues?
13
votes
1answer
422 views
Arithmetic of continued fractions, does it exist?
I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider
$$
...
11
votes
2answers
683 views
Continued fraction for $\frac{1}{e-2}$
A couple of years ago I found the following continued fraction for $\frac1{e-2}$:
$$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$
...
11
votes
2answers
207 views
How to do a very long division: continued fraction for tan
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
11
votes
1answer
329 views
Deriving a trivial continued fraction for the exponential
Lately, I learned about the following continued fraction for the exponential function:
$$\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}}}}$$
I thought it was ...
11
votes
1answer
111 views
Request for a proof of the following continued-fraction identity
I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close):
$$
...
10
votes
3answers
485 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, ...
10
votes
1answer
386 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 ...
10
votes
2answers
837 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
...
9
votes
3answers
338 views
Closed form for a pair of continued fractions
What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ?
What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ?
It does bear some resemblance to the continued fraction for $e$, which is ...
8
votes
5answers
594 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 ...
8
votes
1answer
100 views
optimality of 2 in a continued fraction theorem
I'm giving some lectures on continued fractions to high school and college students, and I discussed the standard theorem that, for a real number $\alpha$ and integers $p$ and $q$ with $q \not= 0$, if ...
8
votes
1answer
83 views
Do best lower approximations of a quadratic irrational always form a linear recurrence sequence?
Let $\theta$ be an irrational number and let
$$
{\cal L}= \bigg\lbrace (a,b) \in {\mathbb Z} \times {\mathbb N}^{*} \bigg| \frac{a}{b} \leq \theta \bigg\rbrace
$$
and
$$
{\cal B}= \bigg\lbrace ...
7
votes
3answers
155 views
Continued fractions for $\sqrt{x} $ and beyond, valid formula?
For $x > 0$, is this trick valid?
I use $$ ( \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} ...
7
votes
2answers
160 views
Continued fractions
I'd really love with concluding that for given integers $a_0,a_1,...a_N$ with $a_i>0$ for $i>0$, representing the continued fraction $[a_0; a_1,....,a_N]$, with the following recursion:
...
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 ...
7
votes
1answer
206 views
Eigenvalues of a tridiagonal trigonometric matrix
Let $A$ be the diagonal matrix w/alternating in sign diagonal entries:
$$ A =
\begin{pmatrix}
(-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\
0 & ...
7
votes
0answers
397 views
What can Euler's identity teach us about (generalised) continued fractions?
We know that $$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+ ...
6
votes
3answers
579 views
Why are some mathematical constants irrational by their continued fraction while others aren't?
Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...
6
votes
4answers
334 views
How to detect when continued fractions period terminates
I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates?
Let me give you an example:
$\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
6
votes
2answers
189 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 ...
6
votes
1answer
127 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 ...
6
votes
1answer
223 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
85 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
...
6
votes
0answers
55 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
0answers
62 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 ...
5
votes
5answers
144 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+ ...
5
votes
1answer
66 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 ...
5
votes
1answer
52 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 ...
5
votes
2answers
64 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):
...
5
votes
1answer
224 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 > ...
5
votes
2answers
149 views
A continued fraction involving composite numbers
What is the limit of the continued fraction whose partial denominators are the composites?
5
votes
1answer
351 views
Continued Fraction of an Infinite Sum
What is the continued fraction for $\displaystyle\sum_{i=1}^n\frac{1}{2^{2^i}}$
It seems to be "almost" periodic, but I can't figure out the exact way to express it.
5
votes
2answers
144 views
Calculating the continued fraction of $\sqrt{47}$ using a different result
I have calculated the continued fraction of $\alpha=\frac{6+\sqrt{47}}{11}$ which equals $\overline{[1,5,1,12]}$. Now I am asked to calculated the cont. fraction of $\sqrt{47}$ using this result. I am ...
5
votes
1answer
216 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,
...
5
votes
0answers
170 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
4
votes
3answers
232 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:
...
4
votes
3answers
200 views
finding the rational number which the continued fraction $[1;1,2,1,1,2,\ldots]$ represents
I'd really love your help with finding the rational number which the continued fraction $[1;1,2,1,1,2,\ldots]$ represents.
With the recursion for continued fraction $( p_0=a_0, q_0=1, p_{-1}=1, ...
4
votes
2answers
255 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 ...
4
votes
4answers
232 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 ...
4
votes
1answer
579 views
A nicer proof of Lagrange's 'best approximations' law?
Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a ...
4
votes
2answers
286 views
Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?
I have been looking at concise ways to represent irrational numbers using only integers.
I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...
4
votes
1answer
169 views
How can I find power series of $f(x)$?
$$f(x)=\dfrac{1}{1+\dfrac{x}{1+\dfrac{x^2}{1+\dfrac{x^3}{1+\dfrac{x^4}{\ddots}}}}}$$
How can a power series be found given the continued fraction $f(x)$?
I'm trying to find $f(x) ...
