Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [constants]

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

A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics. Constants such as $e$ and $\pi$ occur in diverse contexts such as geometry, number theory and calculus.

What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste. Some mathematical constants are notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places.

588 questions
83
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2 answers

What does the mysterious constant marked by C on a slide rule indicate?

Years ago, before everyone (or anyone) had electronic calculators, I had a pocket slide rule which I used in secondary school until the first TI-30 cane out. Recently I dug it out. Here's a photo of one end of it. As you can see, there's a number…
timtfj
  • 3,012
59
votes
23 answers

Intuitive Understanding of the constant "$e$"

Potentially-related questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "ending up at") the constant e. How would you explain…
sova
  • 1,797
42
votes
3 answers

On the Paris constant and $\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$?

In 1987, R. Paris proved that the nested radical expression for $\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches $\phi$ at a constant rate. For example, defining $\phi_n$ as using $n = 5, 6, 7$ "ones" respectively,…
Tito Piezas III
  • 60,745
39
votes
1 answer

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can approximate just about any number. In formal…
cactus314
  • 25,084
35
votes
11 answers

List of integrals or series for Gieseking's constant $\rm{Cl}_2\big(\tfrac{\pi}3\big)$?

Catalan's constant $K$ can be defined as, $$K = \text{Cl}_2\big(\tfrac{\pi}2\big) = \Im\, \rm{Li}_2\big(e^{\pi i/2}\big)= \sum_{n=0}^\infty\left(\frac1{(4n+1)^2}-\frac1{(4n+3)^2}\right)=0.91596\dots$$ It seems to have a natural cubic analogue called…
Tito Piezas III
  • 60,745
31
votes
4 answers

Conjecture: $\lim\limits_{x\to 0}(x!\,x!!\,x!!!\,x!!!!\cdots )^{-1/x}\stackrel?=e$

Well, it's a conjecture so let me propose it: $$\lim_{x\to 0}(x!\,x!!\,x!!!\,x!!!!\cdots)^{-1/x}\stackrel?=e$$ Where I use desmos notation and $x!! := ((x!)!,x!!!=(((x!)!)!)$ It seems so hard that I haven't any clue to show it. I already know…
Barackouda
  • 3,879
26
votes
2 answers

Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the Dirichlet Eta function. Could it be proved that…
Neves
  • 5,701
26
votes
2 answers

A magnificent series for $\pi-333/106$

Stated here without proof is the magnificent series $$\frac{48}{371} \sum_{k=0}^\infty \frac{118720 k^2+762311 k+1409424}{(4 k+9) (4 k+11) (4 k+13) (4 k+15) (4 k+17) (4 k+19) (4 k+21) (4 k+23)} \\=\pi-\frac{333}{106},$$ which proves that…
clathratus
  • 18,002
25
votes
3 answers

Proving that the arithmetic-geometric mean of $1$ and $\sqrt{2}$ is $\pi/\varpi$, where $\varpi$ is the lemniscate constant

I have just read on Wikipedia that in 1799 Gauss proved that $$\color {blue}{AGM(1,\sqrt2)=\frac{\pi}{\varpi},}$$ where $AGM$ is the arithmetic–geometric mean (see the previous link) and $\varpi$ is the lemniscate constant, i. e. the ratio of the…
Davide Masi
  • 1,677
25
votes
7 answers

Show by hand : $e^{e^2}>1000\phi$

Problem: Show by hand without any computer assistance: $$e^{e^2}>1000\phi,$$ where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$. I come across this limit showing: $$\lim_{x\to…
Barackouda
  • 3,879
24
votes
2 answers

Is there a non-trivial definite integral that values to $\frac{e}{\pi}$?

I know that both $$\int_{-\infty}^\infty \frac{\cos(x)}{x^2+1} \, \text{d}x = \frac{\pi}{e}$$ and $$\int_{-\infty}^\infty \frac{x\sin(x)}{x^2+1}\text{d}x = \frac{\pi}{e}$$ But what about $\frac{e}{\pi}$? Is there a non-trivial definite integral…
Markus
  • 453
24
votes
5 answers

Difference between variables, parameters and constants

I believe the following 4 questions I have, are all related to eachother. Question 1: Of course I've been using constants, variables and parameters for a long time, but I sometimes get confused with the definition. It seems to me that these terms…
user1534664
  • 1,330
23
votes
10 answers

Relationship between Catalan's constant and $\pi$

How related are $G$ (Catalan's constant) and $\pi$? I seem to encounter $G$ a lot when computing definite integrals involving logarithms and trig functions. Example: It is well known that $$G=\int_0^{\pi/4}\log\cot x\,\mathrm{d}x$$ So we see that…
clathratus
  • 18,002
21
votes
4 answers

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?
Neves
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