For questions about mathematical constants, that are "significantly interesting in some way".

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12
votes
4answers
473 views

Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?
11
votes
1answer
328 views

Proving that $\frac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\frac{(-1)^{(p_{k}-1)/2}}{p_{k}} \right )^{-1}$ an identity of Euler's.

This is another identity of Euler's relating $\pi$ to the prime numbers, available here \begin{align*} \dfrac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\dfrac{(-1)^{\dfrac{p_{{k}}-1}{2}}}{p_{k}} \right ...
13
votes
6answers
430 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
3
votes
1answer
155 views

How to derive the Golden mean by using properties of Gamma function?

The Golden mean known as $\frac{1+\sqrt{5}}{2}$. How could one show the Golden mean can be expressed as $$ \frac{2\cdot 3\cdot 7\cdot 8\cdot 12\cdot 13\cdots}{1\cdot 4\cdot 6\cdot 9\cdot 11\cdot ...
9
votes
3answers
774 views

On the “Look-and-Say” sequence and Conway's constant

The look-and-say sequence starting with $S_1=1$ is, $$S_n = 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,\dots$$ If $L_n$ is the number of digits of the $n$th term then, ...
3
votes
0answers
333 views

MRB constant proofs wanted

This article has been edited for a bounty. $C$ MRB, the MRB constant, is defined at http://mathworld.wolfram.com/MRBConstant.html . There is an excellent 56 page paper who's author has passed away. ...
0
votes
3answers
43 views

How to prove that $\frac{d}{dx}\epsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant

How to prove that $\frac{d}{dx}\varepsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant? Thanks in advance for your immense help.