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

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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 ...
0
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0answers
27 views

Is Landau-Ramanujan constant irrational? [closed]

Is Landau-Ramanujan constant irrational? How to prove that?
0
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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 ...
0
votes
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.
1
vote
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
votes
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.
0
votes
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.
4
votes
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 ...
7
votes
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
votes
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 ...
0
votes
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 = ...
1
vote
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
votes
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 ...
0
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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
votes
0answers
51 views

Irrationality of $\pi+c$

How to prove that $\pi+c$ is irrational? where $c$ is the Champernowne Constant.
0
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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) = ...
0
votes
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)= { ...
3
votes
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 ...
1
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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 ...
0
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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 ...
0
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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 ...
1
vote
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
votes
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
votes
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: ...
5
votes
3answers
59 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
votes
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 ...
2
votes
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
votes
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 $a_i$ ...
1
vote
1answer
93 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.
0
votes
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 ...
3
votes
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 ...
4
votes
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 ...
0
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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
votes
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
votes
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 ...
-2
votes
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 ...
0
votes
1answer
115 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 ...
-1
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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
votes
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 ...
1
vote
1answer
26 views

can a constant in a max be taken outside

A really silly question, but can I do: $$\max_x \left( c \cdot f(x) \right) = c \cdot \max_x f(x)$$ It seems that way, since I'm just interested in the maximum value of $f(x)$ which is not influenced ...
3
votes
1answer
127 views

Sequence with Prime Numbers

I was looking a question in a calculus book which used the following steps to show that following sequence has a limit (called Euler's constant $\gamma$): $$t_n = \sum_{i=1}^n\left(\frac{1}{n}\right) ...
0
votes
3answers
53 views

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$?

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$? I know it is true when $S$ is open convex, or open connected, but what about any arbitrary $S$?
0
votes
1answer
20 views

$\sum_{n=2}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{p_{n+1}} = C$?

With $p_n$ prime, does the constant, $C$, exist and have a name? $$\sum_{n=1}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{p_n} = C$$ If not,how about a constant and function ...
0
votes
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
votes
0answers
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} + ...
1
vote
0answers
42 views

The probabilistic interpretation of ramanujan's constant $ e^{\pi\sqrt{163}}$

Some of the mathematical constant has an interesting probabilistic interpretation. For example, "$\large\pi$". Suppose two integers are chosen at random. What is the probability that they are ...
1
vote
1answer
61 views

What is the value of a + b + s + t

I'm confused as to how to approach this. I've tried to foil it out. $$x^3(3x-1)=a+bx+sx^2+tx^3$$ The equation above is true for all values of $x$, where $a$, $b$, $s$, and $t$ are constants. What ...