Questions tagged [constants]

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

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Is there a name for this constant and its value where $\alpha = \sum_{p}\frac{\log \left({p}\right)}{p \left({p-1}\right)}$

The constant $$\alpha = \sum_{p} \frac{\log \left({p}\right)}{p \left({p-1}\right)}$$ comes from the calculation $$\sum_{p=2}^{x} \frac{\log \left({p}\right)}{p-1} = \sum_{p=2}^{x} \frac{\log \left({p}...
Lorenz H Menke's user avatar
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Proof of $\lim_{n\to \infty}(1 + \frac{b}{n})^n = e^b$ [duplicate]

I found one definition of $e$ to be $\lim_{n\to \infty}(1 + \frac{1}{n})^n = e$ and I checked it with a graphing calculator to be true. However the $1$ in the numerator is a special case, where it ...
Gustamons's user avatar
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0 answers
92 views

Searching for a paper on "$e$ is not expressible as a simple integral"

I have seen a paper whose main aim was to explain a conjecture that $e$ is not expressible as the integral of a "simple function" over a "simple" region. In contrast to $\pi$ ...
Maesumi's user avatar
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-1 votes
1 answer
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Constant Higher-order derivatives

Given the equation $$\frac{d^ky}{dx^k}=c$$ where $k$ represents the number of higher-order derivatives and $c$ represents any constant real number except $0$. Eg : $$\frac{d^{11}y}{dx^{11}}=100$$ Can ...
Holasoy Mas's user avatar
3 votes
1 answer
105 views

What is the meaning behind "Let" in this instance?

If we were given a few conditions on a complex $z$, say "The modulus of $z$ is $20$" and "the argument of $z$ is $\pi/3$", its clear that $z$ is unique. To find $z$ we might start ...
Nav Bhatthal's user avatar
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1 vote
1 answer
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Family Members Birthday Dates all different, but our birthdays will fall on same day, even Leap Years. There is a total of 9 in this Birthday Club. .

I can compile a list if needed and post, but I noticed this over 50 years ago, My Father, My Brother and Myself our Birthdays fall on the same day of the week every year. Even Leap years, that does ...
David Rinkes Pastor David's user avatar
2 votes
0 answers
34 views

How do I find the constants in this differential equation that describes the time taken for a bubble to rise to the surface.

I'm trying to model the behaviour of bubbles in a water column as accurately as I'm able to, and I modelled a Differential equation using F = ma and a free body diagram. The equation of motion I ...
Movin Jayasinha's user avatar
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0 answers
24 views

Pull out constant from Laplace operator with rescale factor

Say I have a 2 dimensional cartesian laplace operator for a function $\Delta f = \frac{\partial^2 f}{\partial X^2} + \frac{\partial^2 f}{\partial Y^2}$ Part of the exercise is to scale the coordinates ...
reklem2's user avatar
11 votes
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356 views

Asymptotics of sequence of rational numbers

There is a simple sequence of rational numbers. It starts from $a_1=1$, and then $$ a_{n}=\begin{cases} a_{n-1} &\text{for even }n \\ a_{n-1}-\frac1n a_{\frac{n-1}2} &\text{for odd }...
AAK's user avatar
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1 vote
2 answers
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real numbers, constant symbols in first order logic

I have this exercise in my book on first order logic: Let's work out a language for elementary trigonometry. To get you started, let us suggest that you start off with lots of constant symbols- one ...
user394334's user avatar
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0 votes
1 answer
58 views

Declaring constants in Sagemath

I want to find the Groebner base of a ideal,the ideal is generated by some polynomials with constant coefficients, but they do not have numerical values. ...
johnyy's user avatar
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0 answers
41 views

Identities with symmetric constants $d_{abc}$ of $SU(N)$

I am searching for some $SU(N)$ identities to help simplify an expression built out of many of the totally-symmetric constants of $SU(N)$, defined as $d_{abc}=2{\rm Tr}[(T^aT^b+T^bT^a)T^c]$. The ...
mkn's user avatar
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23 views

Can we showed the existence of $a_i,b_i$ such that $\zeta(3)(2C-1)=1+\sum_{i=1}^{\infty}(-1)^{b_i}(\zeta(a_i)-1)$

It's a follow up of Show that : $\frac{1}{\zeta(3)}<2C-1$ I showed by hand that : $$\frac{1}{\zeta(3)}<2C-1$$ Using classical continued fraction . Now I want to go further and a conjecture : ...
Erik Satie's user avatar
  • 3,705
7 votes
6 answers
463 views

Show that : $\frac{1}{\zeta(3)}<2C-1$

Problem : Show that : $$\frac{1}{\zeta(3)}<2C-1$$ Where we can see the zeta function and the Catalan's constant . It recall me the Faulhaber problem which relate the case $n=1,3$ with a square . ...
Erik Satie's user avatar
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0 answers
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Algorithm trial error for two value $x,y$ roots .

The motivation was to find a path to evaluate the miminum of the factorial for $x>0$. To start I introduce the equation $ax=x!$ a good value is $a=2$ .The second idea is to introduce the inversed ...
Erik Satie's user avatar
  • 3,705
0 votes
0 answers
26 views

Minimising MAPE for a constant predictor

I have a constant predictor $f(x_i) = C$, and a set of true labels $y_i \geq 1$ How to find explicit formula for the C which minimises Mean Absolute Percentage Error (MAPE), that is: $$ \sum_{i=1}^n \...
Vadim Artemov's user avatar
7 votes
3 answers
461 views

Show by hand $\int_{0}^{\sqrt{5}}x^{x}dx>4$

It's a very Challenging question perhaps a kind of olympiad question : Prove that : $$\int_{0}^{\sqrt{5}}x^{x}dx>4$$ It's pretty sharp since the left hand side is almost $4.0005$ . I recall the ...
Erik Satie's user avatar
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2 votes
1 answer
68 views

Difficult limit $\lim_{x\to 0}f(x)=^?e$

Well I continue my previous question with : Define : $$f(x)=\left(\left|\int_{0}^{x}\prod_{n=1}^{\infty}\left(1-e^{2-t-n}\right)dt-x-1\right|\right)^{\frac{1}{x}}$$ Does we have : $$\lim_{x\to 0}f(x)\...
Erik Satie's user avatar
  • 3,705
2 votes
3 answers
125 views

What is $\lim_{x\to \infty}f(x)=\lim_{x\to \infty}\int_{0}^{x}\prod_{n=1}^{\infty}\left(1-e^{-t-n}\right)dt-x+1$

Working on the Gamma function I found the following: Let $$f(x)=\int_{0}^{x}\prod_{n=1}^{\infty}\left(1-e^{-t-n}\right)dt-x+1.$$ What is the limit $$\lim_{x\to \infty}f(x)=\,?$$ As possible guess it's ...
Erik Satie's user avatar
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2 votes
0 answers
109 views

Catalan's constant unexpected closeness

These days I usually wander around on WolframAlpha to experiment and discover many trivial but curious calculations or mathematical relations. Recently, I have randomly discovered a strange closeness ...
γ-stupid-like-irrationality's user avatar
0 votes
2 answers
93 views

Minimum of the Gamma/factorial function using dichotomy and Lagrange inversion theorem .

I want to evaluate the minimum of the Gamma's function via a kind of dichotomy . We have to start the value : $$\left(\frac{1}{\sqrt{5}}\right)!\simeq 0.8856$$ Solving $$x!=\left(\frac{1}{\sqrt{5}}\...
Erik Satie's user avatar
  • 3,705
1 vote
1 answer
94 views

Continued fraction inequality with well know constant as $\pi$ and golden ratio.

I found this inequality beautiful let me share it : Let : $$d=\frac{1}{1+\frac{a}{1+\frac{2a^{2}}{1+\frac{3a^{3}}{1+\cdot\cdot\cdot}}}}+\frac{1}{1+\frac{b}{1+\frac{2b^{2}}{1+\frac{3b^{3}}{1+\cdot\...
Erik Satie's user avatar
  • 3,705
0 votes
2 answers
82 views

Conjecture about the minimum of the Gamma function

Problem/Conjecture: Let the function : $$f(x)=\frac{((x+x_{\min})!-(x_{\min})!)^{\frac{1}{x}}}{x^{\frac{1}{x^2}}}$$ Where $x_\min$ denotes the minimum abscissa of the Gamma function near by $0.4616$ ...
Erik Satie's user avatar
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4 votes
0 answers
121 views

On the irrationality of $\zeta(\frac{3}{2})$

It is known that $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$Where $\zeta$ is riemann's zeta function. Usually people make $s$ an integer. But I thought of non integer values of $s$ and started with $s=...
Kamal Saleh's user avatar
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1 vote
0 answers
33 views

What's the arithmetic form of the value $k$ in this graph?

https://www.desmos.com/calculator/m1kaifseq0 in order to get the control points to line up properly, I had to use trial and error to determine the value for this strange constant $k$ at the bottom of ...
Maurdekye's user avatar
  • 297
2 votes
1 answer
59 views

About an odd constant in a longest arrangement problem

In my previous question about the longest chain of $n$-digit square numbers where last digit equal the first digit of next, the nice solution given by Misha Lavrov took me to consider the ratio of the ...
user967210's user avatar
2 votes
1 answer
169 views

Which number is greater A or B?

Let : $$I_k=\int_{0}^{1}\left(\prod_{n=1}^{k}\left(1+\arctan\left(\left(\frac{y}{4n^{2}}\right)\right)\right)\right)dy$$ And : $$h\left(x\right)=\int_{0}^{1}\left(\prod_{n=1}^{\operatorname{floor}\...
Erik Satie's user avatar
  • 3,705
0 votes
1 answer
89 views

Other closed forms of $\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)$

It is known that $$\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)=\beta$$converges. I wonder if there are any other closed forms for this limit. At first I ...
Kamal Saleh's user avatar
  • 4,490
1 vote
3 answers
179 views

$\lim\limits_{x\to \infty}[f(x)-f(x-1)]\overset{?}{=}e$

Let : $$f\left(x\right)=\int_{0}^{\lfloor x\rfloor}\prod_{n=1}^{\lfloor x\rfloor}\frac{\left(y+2n\right)\ln\left(y+2n-1\right)}{\left(y+2n-1\right)\ln\left(y+2n\right)}dy$$ Conjecture: $$\lim_{x\to \...
Erik Satie's user avatar
  • 3,705
4 votes
1 answer
191 views

Euler-Mascheroni Constant in the Cosine Integral?

I came across this integral when doing calculus homework (Integration by parts) $$\int \frac{\cos x}x \, dx $$ It turned out in the end that there was a typo in the original question and this integral ...
ThreeMilks's user avatar
4 votes
0 answers
74 views

Prove that any continuous function $f:S_\Omega \rightarrow \mathbb{R}$ is eventually constant.

Let $\Omega$ be the first non-numerable ordinal number ( $\aleph_1$ is the first cardinal number greater than $\aleph_0$ when treated as an ordinal number is denoted by $\Omega$ ) and let $[0,\Omega)$ ...
Marco Roys's user avatar
0 votes
1 answer
124 views

How to find $C$ (improving a result due to user Clement.C)?

I improve the result see (Elegant) proof of : $x \log_2\frac{1}{x}+(1-x) \log_2\frac{1}{1-x} \geq 1- (1-\frac{x}{1-x})^2$ : What is the best constant $C$ such that $x\in(0,0.5]$: $\frac{1}{\ln\left(2\...
Erik Satie's user avatar
  • 3,705
4 votes
1 answer
98 views

Prove $f(z) = cz$ for all complex numbers and some $c$

Suppose $f$ is entire and $|f(z)|\geqslant |z|$ $\forall z \in \mathbb{C}$, prove there exists $c \in \mathbb{C} $ such that $f(z) = cz $ $\forall z \in \mathbb{C}$. I want to use Liouville’s theorem ...
Pegi's user avatar
  • 540
0 votes
0 answers
86 views

Finger's Constant

A fingers'constant is defined as to have as first decimal : $$1.2345...$$ I ask for the most emblematic example . For example : $$2-\prod_{k=1}^{\infty}\left(1-\frac{1}{\left(k+1\right)e^{k}}\right)$$ ...
Erik Satie's user avatar
  • 3,705
5 votes
2 answers
462 views

Show that $\int_{0}^{\infty}x^{-x}dx<\pi-\ln\pi$

I ask for an inequality which is a follow up of this question: Prove that $\int_0^\infty\frac1{x^x}\, dx<2$ : $$\int_{0}^{\infty}x^{-x}dx<\pi-\ln\pi$$ You can find a nice proof @RiverLi among ...
Erik Satie's user avatar
  • 3,705
0 votes
1 answer
183 views

Conjecture about the representation of a constant $C=0.6516...$

It's a follow up of my previous question How to find the constant $C$ such that $f(x)\geq Cx$ : We start with : $$\left|\exp\left(1-\prod_{k=1}^{1000}\left(1-\frac{1}{2^{k}(k+1)}\right)\right)-\sqrt{2}...
Erik Satie's user avatar
  • 3,705
0 votes
1 answer
262 views

Find $a_n$ such that the formula is equal to $\zeta(2)$

My question start with the observation : $$\sqrt{e}\simeq \frac{\pi^2}{6}$$ At first glance it's not really convincing but after some work I found : $$\sqrt{e-5\left(\frac{1}{\pi}-\frac{1}{e}\right)^{...
Erik Satie's user avatar
  • 3,705
0 votes
1 answer
165 views

Why do I get this value?

Can somebody explain this? Why does this happen? Yesterday I was on a popular chat bot and I asked it to make me a code to generate a sequence of numbers. What I wanted, was a script that given a ...
simogne 's user avatar
1 vote
4 answers
309 views

How to find the constant $C$ such that $f(x)\geq Cx$

Problem : Define for strictly positive $x$ : $$f\left(x\right)=\left(\prod_{k=1}^{\operatorname{floor}\left(x\right)}\left(1+\sum_{n=1}^{k}\frac{1}{k\cdot2^{n}}\right)\right)$$ Does there exists a ...
Erik Satie's user avatar
  • 3,705
0 votes
0 answers
26 views

How many significant figures should constants be used to in a calculation where other values are given to varying significant figures?

I was recently helping someone study and review the results of a homework assessment in which physics calculations were being performed. The question was related to calculating planetary core pressure ...
Polynomial's user avatar
2 votes
3 answers
135 views

On the tribonacci constant with $\cos(2\pi\,k/11)$, plastic constant with $\cos(2\pi\,k/23)$, and others

(This post is indebted to Oscar Lanzi.) Part I. $\color{blue}{p = 11}$ The equation, $$\big(4\sin[3t] - \tan[t]\big)^2 = 11$$ seems to have five solutions, given by $t = \frac{2\pi\,k}{11}$ for $k = 1,...
Tito Piezas III's user avatar
0 votes
3 answers
199 views

Arbitrary Constants in Differential Equations

Question: Trouble understanding which constants are arbitrary and need to be eliminated while forming Differential Equation (DE) from its general solution (In contrast to constants which are fine to ...
JustCurious's user avatar
10 votes
0 answers
212 views

On the solvable octic $x^8-44x-33 = 0$ and the tribonacci constant

I had discussed the solvable octic trinomial, $$x^8-44x-33=0\tag1$$ way back in this old MSE post, but I revisited this inspired by another solvable octic, $$y^8-y^7+29y^2+29=0\tag2$$ which I also ...
Tito Piezas III's user avatar
4 votes
0 answers
177 views

Natural number sequences that sum to mathematical constants

Let $\Sigma_\text{p} : (\mathbb{N} \to \mathbb{C}) \rightharpoonup \mathbb{C}$ where $\Sigma_\text{p}(a)$ is the analytic continuation of the power series $\sum_{n=0}^\infty a_n x^n$ to $x=1$, when ...
user76284's user avatar
  • 5,817
0 votes
2 answers
73 views

Why does $\sum_{n=0}^{\infty} \frac{\pi^n}{n!} = e^\pi$?

Why is it that $$ \sum_{n=0}^{\infty} \frac{\pi^n}{n!} = e^\pi \quad ?$$ I think it has to do with the gamma function, but I'm not sure how that would work.
Math Man's user avatar
  • 112
3 votes
0 answers
267 views

Explanation for repeating numbers $\displaystyle{\sum_{n=1}^{\infty}(-1)^n(n^{1/n}-1)}$ added with smaller and smaller steps of negative powers of 10?

I noticed the following about the original MRB constant sum. Can you offer an explanation of why there are several repeating numbers in the following expansions? $\displaystyle{\sum_{n=1}^{\infty}(-1)^...
Marvin Ray Burns's user avatar
3 votes
1 answer
187 views

What are the odds that De Vries' formula for the fine structure constant $\alpha$ is a numerical coincidence?

The dimensionless fine structure constant $\alpha \approx \frac1{137}$ has intrigued physicists for over a century. Whilst not currently a majority view, there is a school of thought that considers ...
Pallas's user avatar
  • 147
0 votes
1 answer
36 views

Understanding a function $A(W+r)^{b}$.

I am given the following function $$Q = A(W+r)^{a},$$ where $A$ and $r$ are constants. I am asked to do several things with this function: Find $\frac{\partial Q}{\partial W}$. Given constants $a$, $...
aikka3's user avatar
  • 3
0 votes
0 answers
61 views

What can we say about $S=\sum_{n=1}^{\infty}\frac{1}{n^{2}\operatorname{erf}\left(n\right)}$?

I'm really curious If we can find a closed form for this infinite sum : $$S=\sum_{n=1}^{\infty}\frac{1}{n^{2}\operatorname{erf}\left(n\right)}$$ As $f(x)=\operatorname{erf}(x)\leq 1$ for $0<x$ and $...
Erik Satie's user avatar
  • 3,705
0 votes
2 answers
112 views

Show by hand the inequality $\frac{1}{2\ln2}\left(1-\sqrt{\frac{1-\ln2}{1+\ln2}}\right)>\sqrt{2}-1$

Problem : Show that : $$\frac{1}{2\ln2}\left(1-\sqrt{\frac{1-\ln2}{1+\ln2}}\right)>\sqrt{2}-1$$ Using some approximation using itself algoritm found here (https://en.wikipedia.org/wiki/...
Erik Satie's user avatar
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