Mathematics Stack Exchange
Mathematics Stack Exchange

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.

In mathematics, 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 as the sum of its integer part and another reciprocal, and so on.

In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ai are called the coefficients or terms of the continued fraction.

Links:

Continued Fraction at Wolfram MathWorld

1069 questions
615
votes
8 answers

Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?

So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x = 1 + \dfrac{1}{x}$, solving the resulting…
Martin Ender
  • 5,710
  • 4
  • 17
  • 28
89
votes
7 answers

$1=2$ | Continued fraction fallacy

It's easy to check that for any natural $n$ $$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$ Now,…
Mher
  • 4,911
  • 1
  • 18
  • 35
65
votes
3 answers

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Let $$ \text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction} $$ So for example, the terms of the sum $\text{S}_6$…
Simon Parker
  • 4,083
  • 2
  • 15
  • 23
56
votes
2 answers

Crazy pattern in the simple continued fraction for $\sum_{k=1}^\infty \frac{1}{(2^k)!}$

The continued fraction of this series exhibits a truly crazy pattern and I found no reference for it so far. We have: $$\sum_{k=1}^\infty \frac{1}{(2^k)!}=0.5416914682540160487415778421$$ But the continued fraction is just beautiful: [1, 1, 5, 2,…
Yuriy S
  • 30,220
  • 5
  • 48
  • 168
54
votes
1 answer

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 sequence of $u_n$, starting with $n = 1$, is given by…
Tito Piezas III
  • 47,981
  • 5
  • 96
  • 237
45
votes
2 answers

An infinite series plus a continued fraction by Ramanujan

Here is a famous problem posed by Ramanujan Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + \left(\cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}\right) = \sqrt{\frac{\pi e}{2}}$$ The…
Paramanand Singh
  • 79,254
  • 12
  • 118
  • 274
45
votes
2 answers

a conjectured continued fraction for $\tan\left(\frac{z\pi}{4z+2n}\right)$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$ is…
Nicco
  • 2,763
  • 14
  • 32
44
votes
5 answers

The monster continued fraction

My title may have come off as informal or nonspecific. But, in fact, my title could not be more specific. Define a sequence with initial term: $$S_0=1+\frac{1}{1+\frac{1}{1+\frac1{\ddots}}}$$ It is well-known that $S_0=\frac{1+\sqrt{5}}{2}$. I do…
Descartes Before the Horse
  • 6,500
  • 3
  • 23
  • 65
42
votes
2 answers

Weird large K symbol

Take a look at this symbol: $$ \pi=3 + \underset{k=1}{\overset{\infty}{\large{\mathrm K}}} \frac{(2k-1)^2} 6 $$ Does it look familiar to you? If so please help me!
VJZ
  • 521
  • 5
  • 11
38
votes
10 answers

What are the applications of continued fractions?

What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?
user92877
  • 859
  • 2
  • 8
  • 13
38
votes
7 answers

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 algorithm for computing it? I tried to read the wiki…
user51819
  • 1,061
  • 1
  • 8
  • 14
36
votes
0 answers

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. This leads me to wonder to what value this…
G Tony Jacobs
  • 29,851
  • 4
  • 46
  • 100
35
votes
1 answer

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is wrong in my question. I enjoy doing numerical…
Thomas Baruchel
  • 973
  • 7
  • 22
34
votes
1 answer

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 process in reasonable time with numerical libraries…
Vladimir Reshetnikov
  • 45,303
  • 7
  • 151
  • 282
30
votes
4 answers

Can anything interesting be said about this fake proof?

The Facebook account called BestTheorems has posted the following. Can anything of interest be said about it that a casual reader might miss? Note that \begin{align} \small 2 & = \frac 2{3-2} = \cfrac 2 {3-\cfrac2 {3-2}} = \cfrac 2 {3 - \cfrac 2 {3…
Michael Hardy
  • 1
  • 30
  • 276
  • 565
1
2 3
…
71 72