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

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What was Ramanujan's solution to the House Number Problem?

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 problem, Imagine that you are on a street with houses…
Potato
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Is this formula for $\frac{e^2-3}{e^2+1}$ known? How to prove it?

I found an interesting infinite sequence recently in the form of a 'two storey continued fraction' with natural number…
Yuriy S
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How to find continued fraction of pi

I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are: $\pi=[3:7,15,1,292,.....]$ and many others given here. Similarly some nice continued fractions for $e$ and it's derivatives are given…
happymath
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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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Closed form of $\frac{e^{-\frac{\pi}{5}}}{1+\frac{e^{-\pi}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-3\pi}}{1+\ddots}}}}$

It is well known that $$\operatorname{R}(-e^{-\pi})=-\cfrac{e^{-\frac{\pi}{5}}}{1-\cfrac{e^{-\pi}}{1+\cfrac{e^{-2\pi}}{1-\cfrac{e^{-3\pi}}{1+\ddots}}}}=\frac{\sqrt{5}-1}{2}-\sqrt{\frac{5-\sqrt{5}}{2}}$$ where $\operatorname{R}$ is the…
Poder Rac
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A conjectured continued fraction for $\phi^\phi$

As I previously explained, I am a "hobbyist" mathematician (see here or here); I enjoy discovering continued fractions by using various algorithms I have been creating for several years. This morning, I got some special cases for a rather heavy…
Thomas Baruchel
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Intuition behind Khinchin's constant

Khinchin proved that For almost all reals $r$ with continued fraction representation $[a_o; a_1, a_2, \dots ]$ the sequence $K_n = \left(\prod_{i=1}^{n} a_i\right)^{1/n}$ converges to a constant $K$ (Khinchin's constant) independent of…
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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…
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Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime…
Neves
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Continued fraction analog to zeta function - how to properly define it and find its properties?

I do not mean the continued fraction representation of zeta function; I mean the function which has the form: $$f(s)=\cfrac{1}{1^s+\cfrac{1}{2^s+\cfrac{1}{3^s+\cfrac{1}{4^s+\cdots}}}}$$ For some values, we know the function…
Yuriy S
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What does this converge to and why?

What does the below expression converge to and why? $$ \cfrac{2}{3 -\cfrac{2}{3-\cfrac{2}{3-\cfrac2\ddots}}}$$ Setting it equal to $ x $, you can rewrite the above as $ x = \dfrac{2}{3-x} $, which gives the quadratic equation $x^2 - 3x + 2 = 0…
user536852
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a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan's octic continued fraction which I discovered using certain three term relations and algebraic manipulations. Given…
Nicco
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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 to solve it. [paraphrased from Lone Learner] …
GEdgar
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Continued fraction for $\frac{1}{e-2}$

A couple of years ago I found the following continued fraction for $\frac1{e-2}$: $$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$ from fooling around with the well-known continued…
Yonatan N
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Continued fraction involving Fibonacci sequence

What is the limit of the continued fraction: $$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}\ $$ that involves the Fibonacci sequence terms as denominators? I've been looking for this specific continued…
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