**279 constants questions.**

In the solution of my homework there's this step that I don't understand:
$$\begin{align}\mathsf {Var}[X] & = \sum_{k=1}^{n}\dfrac{n(k-1)}{(n-k+1)^2}\\ &= n \sum_{k=1}^{n}\dfrac{(n-k)}{k^2}\\&...

I am having a problem understanding,
Whether to keep a constant (arbitrary or fixed) in the solution of a differential equation
Figuring which is an arbitrary constant and which a fixed constant
...

I'm struggling with the concept of constants in the context of derivatives.
For example;
$$f(\theta)=r(\cos\theta-1)$$ where $r$ is a constant.
$$f'(\theta)=-r\sin\theta$$
Why is the product rule ...

I found a proof here for a measurable function (instead of probability theory's random variable) being constant if and only if the sigma-algebra generated by it is the trivia sigma-algebra, I think (...

This is a follow up on: Is a random variable constant iff it is trivial sigma-algebra-measurable?. I wanted to comment on the solution, but, unfortunately, this is only possible from 50 reputation.
I ...

I have this equation, and I've tried checking which is bigger,
$ e ^ {3 \pi} \space ? \space 3 ^ {e \pi}$
What I've tried:
$ e ^ {3 \pi} \space ? \space 3 ^ {\pi e} / \sqrt[\pi]{} $
$ e ^ {3} \...

Given X, a continuous random variable and the density:
$$
f(x) =
\begin{cases}
x^2, & \text{if $x$ $\in$ [-1,1] ,} \\
c \cdot \frac{1}{|x|^k}, & \text{else}
\end{cases}
$$
Where $c\in \...

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 "...

Right now I'm calling a convergent number based on the Collatz conjecture the "Collatz constant". I'm wondering if it have an ACTUAL name? And if it actually converges?
Details
The Collatz ...

I'm trying to make sense of a solution to a separation of variables question. Namely where it goes from:
$$y = \exp(-\cos x + C)$$
To:
$$y = A\cdot \exp(-\cos x)$$
I understand the constant of ...

Looking for a finite recursive formula for the constant e preferably using standard operators (ones a calculator could carry out)
i.e. a formula of the form $ x_{n+1}=f(x_n)$ where $\lim_{n\...

How do I plot the cumulative distribution function and probability mass function of the constant random variable $X(\omega)=2$ for all $\omega$?

As Wikipedia explains in number theory, Mills' constant is defined as:
"The smallest positive real number $A$ such that the floor function of the double exponential function $\lfloor A^{3^{n}}\...

For example, I could write this:
$$(e+i\pi)^0=1$$
It has all the five constants and all the addition, multiplication, and exponentiation operators.

With the help of Wolfram Alpha online calculator I know good approximations of the real part of the integral $$\int_0^1 \log \left(\zeta(x)\right)dx,$$
where $\zeta(x)$ denotes the Riemann zeta ...

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 ...

The question comes from the signal processing, but I have problems with the math itself.
I have a finite sum of partial fractions for a transfer function of a system:
$$T(s) = \sum_{j=1}^n \frac{\...

Infinity while being an infinite set or limit is always defined as +∞ or -∞ which can literally be any subset of numbers: ...

The closed-forms of the first three are well-known,
$$x_1=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}\tag1$$
$$x_2=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\dots}}}}\tag2$$
$$x_3=\sqrt{1+2\sqrt{1+3\...

Given the limit
$$\lim_{n \to \infty}{\ln n\over e^n}\int_{0}^{n}{\sinh{x}\over \ln(1+x)}\mathrm dx={1\over 2}\tag1$$
How did I came across this integral $(1)$?
I was observing this integral $(2)$
$...

I'm looking for the answer to this question. But I could not find the "satisfactory" answer.
This is obvious,
$$\lim_{n\to \infty} \left(1+\frac 1n \right)=1+0=1$$
and
$$\lim_{n\to \infty} \...

We can be sure, that
$$\log(1+i)=\frac{\pi i}{4}+\frac{\log(2)}{2}$$
then if we take
$$\log(1+k)=\frac{\pi}{4}+\frac{\log(2)}{2}$$
so sign-alternating is $++--$.
Is there a constant $k$ which $k^{4n+1}...

Forgive me ignorance in advance for this question as I'm not a mathematician in any respect. I can't even pretend to give correct terminology so I'll just give an example:
We're a a retail company. ...

Today, the 14th of March, is $\pi$ day (in America the date is 3-14 - in the rest of the world today's date is 14-3).
We define $\pi=\frac{C}{d}$. Yet, that seems to be the last time we use the ...

Well, I've the following system of equations:
$$
\begin{cases}
\text{f}=\frac{\text{R}_3+\text{R}_4}{\text{R}_3}\\
\\
\text{d}=\pi\cdot\text{h}\cdot\left\{\text{C}_1\cdot\left(\text{R}_1+\text{R}_2\...

I was playing around with numbers like I always do, and I came up with a conjecture: $$\ln(e^{\pi + n} - e^{\pi + n - 1} - \ldots - e^{\pi + 1} - e^{\pi}) - \frac{\pi}{\sqrt{2}} \approx n$$ such that $...

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$).
For irrational ...

(Inspired by this post.) Given the tribonacci constant $\Phi_3$, the tetranacci constant $\Phi_4$, etc. How do prove that,
$$\sum_{n=0}^\infty \binom{4n}{n}\frac{1}{(3n+1)\,2^{4n+1}}=\frac{1}{2}{\;}...

If I want to calculate the sample variance such as below:
Which becomes: $\left(\frac{1}{n}\right)^2 \cdot n(\sigma^2)= \frac{\sigma^2}{n} $...
My question is WHY does it become $$\left(\frac{1}{n}\...

How does one go about finding mathematical constants and patterns...
I enjoy playing around on my graphing calculator, typing random functions in trying to find patterns and such and sometimes I'll ...

While study Numerics and playing with famous constants ($e$, $\pi$, Golden ratio) I came across the following
relation
$$ \color{blue}{1.6^2+2.7^2 = 9.85\approx 3.14^2}$$
This is nothing special ...

While trying to find the tribonacci cousin of this post, I came across this nice short article A Geometric Problem of Omar Khayyam and its Cubic by Wolfdieter Lang. Given the figure,
$\hskip1.7in$
...

Can someone help me to understand what is going on here because I have some misunderstanding about some summation formulas of constant
1) Why we can write this constant summation like below? Some ...

I am trying to solve the following differential equation
$$y''=2yy'$$
and ended up with the answer:
$$\frac{1}{C_1}\arctan\frac{y}{C_1}+C_2=x.$$
However, the answer should be,
$$\arctan\frac{y}{...

I have this differential equation to solve:
$(y^2+1)y''-2y(y')^2=0$ and the answer should be $arctan(y)=Cx$
On the last part of integration, I need to integrate
$\int\frac{dy}{y^2+1}=\int C_1dx$
So,
...

I'm solving $y''=2y(y')$
So using the substitution
$P=y'$ and $P\frac{dP}{dy}=y''$
That yields to a separable equation then by integrating i end up with another separable equation,
$$\frac{dy}{dx}=...

Consider the sequence $f$ :
$$f(1) = 1$$
$$ f(n) = f(n-1) + \frac{1}{n f(n-1)}$$
Now we have for large $n$ :
$$ f(n) = \sqrt {2 \ln(n)} + \frac{1}{19} + C + eps(n) $$
Where $C$ is a constant and $...

In Euler's paper "De Fractionibus Continuis Dissertatio" (English Translation) he proves that the constant $e\approx 2.71828$ is irrational.1 One step in the proof threw me for a loop, though. In the ...

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}{...

If we had two number variables A and B with their product to be a constant C;
A x B = C.
Doesn't that mean that if I increased A by an amount "n" and decreased B with the same amount "n" , then their ...

For a real number $x$,
Floor Function returns the largest integer less than or equal to $x$, denote as $\lfloor x \rfloor$.
Ceiling Function returns the smallest integer larger than or equal to $x$, ...

If:
$t^3+e^t+t^{-1/2}+C_1=f(t)+C_2$
where $C_1$ and $C_2$ are constants
$f(t)$ is a function of $t$ without any constant term
then will $C_1=C_2?$

We can be sure, that
$$\lim\limits_{n\to\infty}n\sin(\frac{\pi}{n})=\lim\limits_{n\to\infty}n\tan(\frac{\pi}{n})=\pi$$
then we have
$$\sum\limits_{n=2}^{\infty}\left(\pi-n\sin(\frac{\pi}{n})\right)=k\...

Given the set of all prime numbers $P$, is there a constant that has an Engel Expansion of $P$?
I coded a script to calculate the number, and it is about $0.705230171791801...$
Mathematically, the ...

I've been struggling a while now trying to find an alternate form for
$$G=\int_0^{\infty}{x^{1/x-x}dx}$$
in function of known constants, (like $e$,$\pi$,etc.)
Having looked at the problem, I don't ...

Suppose I define $G(x)=\int_0^xf(t)dt$ as the area under the function $f(t)$ of which both $0$ and $x$ are within the domain of $f$, and $x$ can take any value within the domain.
Hence we can ...

- sequences-and-series
- calculus
- integration
- limits
- prime-numbers
- pi
- differential-equations
- number-theory
- derivatives
- functions
- exponential-function
- golden-ratio
- transcendental-numbers
- definite-integrals
- summation
- logarithms
- probability
- algebra-precalculus
- geometry
- trigonometry
- soft-question
- approximation
- analytic-number-theory
- math-history
- riemann-zeta

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