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

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123 views

Calculating Closed form of Basel type problem.

I want to find the sum of $\displaystyle\sum_{n=1}^\infty \frac{1}{n^{\phi(n)}}$. where $\phi$ is Euler's totient. So for example $\frac{1}{1^1} + \frac{1}{2^1} + \frac{1}{3^2} +\frac{1}{4^2} + ...
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3answers
38 views

Can you have many integrations for a single function? (sorry if the terminology is wrong)

Forgive me for the amateur question (first time here), but it's bugging me! OK, for example, integrating 1/5x would yield what? ln(5x)/5 +C OR ln(x)/5 +C ? Both of these when differentiated form ...
4
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0answers
53 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
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2answers
72 views

Find the limit $\lim_{h\to 0}\frac{ \sqrt{7(a+h)}-\sqrt{7a} }{h} $ in terms of $a$ .

Find the limit in terms of the constant $a$ : $$ \lim_{h \to 0}\frac{\sqrt{7(a+h)}-\sqrt{7a}}{h} $$ I have tried to solve this by multiplying the square roots to both sides but i simply can't solve it ...
3
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1answer
66 views

Does Euler-Mascheroni constant belong to the ring of periods?

I wonder whether $\gamma$ belongs to the ring of periods? UPDATE Well now I know it should not. But $e^{-\gamma}$ should.
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1answer
20 views

Moment generating function of a constant

This might be trivial, but can you elaborate why moment generating function for a constant $c$ is $e^{cX}$, where $X$ is a random variable.
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1answer
21 views

Solve series with two constants and opposite exponents

How can I find the general formula for the sum of this series? $$ \sum_{i=0}^n a^ib^{n-i} $$ Where $a$ and $b$ are unrelated constants? I don't think you can split it into $ \sum_{i=0}^n a^i $ and ...
5
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1answer
61 views

Proving $\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$

How to prove that $$\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$$ Where $F_{n}$ is the Fibonacci Number.
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1answer
46 views

Sum of infinite series - Catalan constant

Why is this identity true? $$\sum_{k=1}^{\infty} \frac{sin(k\pi/2)}{k^2} = C$$ where $C$ is Catalan constant.
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1answer
456 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 whose author has passed away. ...
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1answer
84 views

What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?

The Euler constant is defined as $$\gamma = \lim_{n \to \infty}{\sum_{k=1}^n\frac{1}{k}-\ln{n}}$$ Let $$q = \lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$$ I managed to prove that ...
32
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3answers
470 views

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 ...
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1answer
118 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
2
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1answer
59 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for ...
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2answers
63 views

Is that a known “constant”?

I noticed, using a calculator, that the following operation: $$Y = \sin(\cos(\tan(\log(n))))$$ for values of $n$ from $0.00000001$ to $99999999$, I obtain always quite the same number, id est $$Y = ...
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2answers
40 views

Another way to express $\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$?

I believe that the sum $$\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$$ converges and it is about $1.85193$. Is there another way that this ...
0
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1answer
24 views

Taking derivatives of constants and variables

When implicitly differentiate a function, for example, $f(x)=(G)(x)$, where G is a constant, is it possible to differentiate it such that we can treat G as a variable? From my understanding it is ...
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0answers
41 views

Show that $\lim_{n \to \infty} p(n) = 1-\frac{1}{e}$

Let $p(n)$ denote the probability that a randomly chosen one-to-one function $f:\lbrace 1,2,3...,n \rbrace \rightarrow \lbrace 1,2,3...,n \rbrace$ has a least one fixed point. (least one integer $k$ ...
2
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0answers
51 views

Irrationality of $\pi+c$

How to prove that $\pi+c$ is irrational? where $c$ is the Champernowne Constant.
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0answers
15 views

Finding the constant in a joint density function

I have been having a bit of difficulty with the following example, whenever i integrate the terms vanish. Any help is appreciated! The joint density function of X and Y is given by $$f(x , y) = ...
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1answer
37 views

Find the constants to make it continuous.

I am having trouble starting this question. Find the constants (a, b or c) to make the function continuous. f(x)= { ...
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1answer
44 views

Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...
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0answers
18 views

Use of the constant of integration

I have a differential equations question: Given the differential equation $$t(t+1) \frac{ds}{dt} = sln(s)$$ it can be trivially solved to be $$ln ln s = ln(t) - ln(t+1) +C$$ As known, C is the ...
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1answer
24 views

General question regarding constants and variables

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to relations and functions. To be specific, I'm stuck on ...
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0answers
26 views

Max n value for $\frac{1}{x^x}=y^y+n$?

I was looking at the graph of $$\frac{1}{x^x}=y^y+n$$ and found that it disappears at n greater than about $0.752$. What is the exact constant definition for n and why does the graph disappear ...
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2answers
58 views

What's the value of $\sum_{k=1}^{\infty}(\zeta(2k+1)-\zeta(2k+2))$?

I'm confused about this. I have this expression $$ \frac{1}{2}=\sum_{k=1}^{\infty}(\zeta(2k)-\zeta(2k+1)) $$ Now if I want claculate $\zeta(2)$ I'll do the apropriate manipulations to get $$ ...
4
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0answers
76 views

Find a constant $C_p$ that satisfies $|f(x+p)-f(y+p)|\le C_p|f(x)-f(y)|$

Let $B^n$ be the unit open ball in $\mathbb{R}^n$, $p\in \mathbb{R}^n$ and $f\colon \mathbb{R}^n\to B^n$ defined as $f(x)=\frac{1}{1+|x|}x$. I believe there are constants $C_p>0$ such that ...
3
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1answer
114 views

Hypergeometric function values and the Baxter constant

While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant: ...
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1answer
47 views

calculating the position of a given digit in a constant (e.g. $\pi$)

I'm aware that there are a lot BBP type formulas out there which extract the n-th digit of the observed constant. I'm asking for the reverse action, namely, is it possible to find the first ...
2
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3answers
387 views

How to rewrite $\frac{d}{d(x+c)}$? [closed]

I would like to know how to rewrite the following equations: $$ \frac{d (f(x))}{d(x+c)} =0\\ \frac{d^2 (f(x))}{d(x+c)^2} =0\\ $$ Here $x$ is a variable, $c$ is a constant and $f(x)$ is a function of ...
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3answers
60 views

Logs - Simplifying with arbitrary constant

I've tried simplifying my answer, which has a constant in it. I would like to know if I am on the right track: $$ \ln(y) = -{x^2\over 2y^2} + C $$ C can be considered as an Arbitrary Constant. From ...
2
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1answer
58 views

Proving the Fibonacci sum $\sum_{n=1}^{\infty}\left(\frac{F_{n+2}}{F_{n+1}}-\frac{F_{n+3}}{F_{n+2}}\right) = \frac{1}{\phi^2}$ and its friends

In this article, (eq.92) has, $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_{n+1}F_{n+2}} = \frac{1}{\phi^2}\tag1$$ and I wondered if this could be generalized to the tribonacci numbers. It seems it can ...
8
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1answer
105 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
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1answer
94 views

When is $\zeta(s)=0$?

At what real constant does $\zeta(s)=0$? Does that constant have any significance? Thank you very much for any help provided.
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1answer
27 views

What is the summatory function of the number of (not necessarily distinct) prime factors?

In the Math World article on Merten's Constant, a related constant $B_2$ is mentioned which "appears in the summatory function of the number of (not necessarily distinct) prime factors." I am very ...
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2answers
96 views

How is the Twin Primes Constant useful? What value does it provide over Brun's Constant?

The Twin Primes Constant is: $$\prod_{p > 2 \text{ and a prime }}\left(1 - \frac{1}{(p-1)^2}\right) = 0.6601618158\ldots$$ It appears that in this case $p$ does not have to be a prime. But if ...
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0answers
34 views

clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
0
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2answers
45 views

When do I have to respect the $C$ constant and when can I combine?

Question Verify that the given two-parameter family of functions is the general solution of the non-homogeneous differential equation on the indicated interval. $$ y''-4y'+4y = 2e^{2x}+4x-12 $$ $$ ...
3
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1answer
94 views

Mathematical importance of the golden ratio [duplicate]

I know the golden ratio is the limit of the ratios of consecutive Fibonacci numbers and that it appears when studying many related combinatorial objects (such as the sequences of zeros and ones with ...
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1answer
119 views

How values of the constants are derived mathematically? [closed]

As said by Jan regarding constant value $\pi$ ,Imagine you have a circle and you are able to measure its circumference "c". Then, you can also find out what its diameter "d" is. When you divide ...
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1answer
116 views

What is the most intuitive explanation for euler's identity? [duplicate]

Is there any intuitive explanation for: $$e^{i\pi} + 1 = 0$$ About whether this question is a duplicate, what is asked for is not a proof but an explanation that helps with the not-so-intuitive ...
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1answer
92 views

What is $1 \over \infty$ really? [closed]

Let's define infinity ($\infty$) as the number larger than any finite number. Also, let's define the infinitesimal constant ($\epsilon$) as the smallest number greater than zero. What is ...
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2answers
77 views

Online encyclopedia of maths and physics constants?

The OEIS is amazing, I used to spend hours creating sequences using arbitrary algorithms and look it up to see if it has an alternate definition and what's the meaning and the history of this ...
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0answers
17 views

How to find the constant of an expanded binomial expression?

Can somebody please explain how to find the Constant Term in an Expanded Binomial Expression? I have looked online and a lot of the explanations have confused me. The problem i'm currently on is ...
0
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3answers
1k views

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 ...
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0answers
83 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
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1answer
300 views

What is a “constant fraction” of a total?

What is it mean to say that some quantity is a "constant fraction" of another quantity?
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0answers
44 views

What is he easiest way to approximate γ as a decimal number?

What is the easiest way to give the numerical value of the Euler-Mascheroni constant? The mathmetical way to give that value? Thanks lot!
0
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1answer
42 views

Find A,B such that the given function is a solution to the given differential equation

The equation is: $x(t)=A\sin(t)+B\cos(t), \; x' - 3x = \frac{1}{2}\cos(t)$ I'm honestly just lost from where to start. I'd really appreciate any help. Answer: $A = \frac{1}{20}, \; B= ...
2
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0answers
62 views

Are there famous complex constants?

Are there any famous constants (like $\pi$ and $e$) that are complex? More specifically, to rule out trivial complex numbers, are there any famous constants of the form $a+bi$ with $a \neq 0 \wedge b ...