Mathematics Stack Exchange
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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

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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
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$1=2$ | 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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Are these continued fractions of integrals known?

Crossposted on MathOverflow Motivated by Bowman and McLaughlin (2000) on polynomial continued fractions, I considered an extension to functional continued fractions, where the numerator of each fraction is the integral of the previous…
TheSimpliFire
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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 $\pi$? What is the fastest way to approximate the…
Angela Pretorius
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Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$ Prove it converges and, evaluate the series. For the first part of the question, I prove it converges by considering the irrationality measure, $$|\sin…
Tianlalu
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