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.

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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
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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…
sova
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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
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$\pi^4 + \pi^5 \approx e^6$ 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…
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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…
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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…
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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
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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
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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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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, $$\lim_{n\to\infty} \frac{L_{n+1}}{L_n}=\lambda\tag{1}$$ where…
Tito Piezas III
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Integral over domain of infinite tetration of x over extended domain from 0 to $\sqrt[e]e$. Possible $\int_{e^{-e}}^{e^\frac1e} x^{x^{…}}dx$ solution.

I have been trying to find an interesting constant over the domain of the infinite tetration of x and have just almost figured out the area with a non integral infinite sum representation. Just one constant is in my way. D denotes the domain. This…
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Prove that $\lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!} = e$

Define $e, e'$ by $$e: =\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{n} \quad \text{and} \quad e' := \lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!}$$ Prove that $e' \in \mathbb R$ and $e' = e$. Could you please verify whether…
Akira
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Integral Representation of the Dottie Number

I noticed that a lot of commonly-used mathematical constants that can't be expressed in closed-form can be expressed by integrals, such as $$\pi=\int_{-\infty}^\infty \frac{dx}{x^2+1}$$ and $$\frac{1}{1+\Omega}=\int_{-\infty}^\infty…
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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
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