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

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22
votes
2answers
293 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 ...
17
votes
2answers
803 views

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 ...
14
votes
2answers
297 views

A new constant?

I was experimenting in Wolfram Alpha the answer to the equation $\int_0^k x^x dx=1$ And I got about 1.19... But, What is this number k (and could you calculate it to more decimal places?) And is it ...
13
votes
6answers
419 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
12
votes
2answers
436 views

Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

Long time ago I've been playng with formulas for $\pi$ and found that one above in the title which can also be expressed as \begin{align*} ...
12
votes
4answers
276 views

Are there two $\pi$s?

The mathematical constant $\pi$ occurs in the formula for the area of a circle, $A=\pi r^2$, and in the formula for the circumference of a circle, $C= 2\pi r$. How does one prove that these constants ...
11
votes
1answer
321 views

Proving that $\frac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\frac{(-1)^{(p_{k}-1)/2}}{p_{k}} \right )^{-1}$ an identity of Euler's.

This is another identity of Euler's relating $\pi$ to the prime numbers, available here \begin{align*} \dfrac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\dfrac{(-1)^{\dfrac{p_{{k}}-1}{2}}}{p_{k}} \right ...
10
votes
4answers
381 views

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?
10
votes
1answer
232 views

Is there an efficient method for the calculation of $e^{1/e}$?

(I wonder whether this is appropriate for the Math StackExchange or whether it'd be better on Stack Overflow as it deals with computing, but I'm asking about mathematical details, not about ...
9
votes
8answers
672 views

“How I wish I could calculate pi” analogs…

You might know the mnemonic for $\pi$ in the title or even this more elaborated one: Sir, I bear a rhyme excelling In mystic force, and magic spelling Celestial sprites elucidate All my own ...
8
votes
3answers
624 views

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, ...
8
votes
0answers
265 views

$\pi^4 + \pi^5 \approx e^6$ 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 ...
6
votes
2answers
445 views

Two questions about Euler's number $e$

I am on derivatives at the moment and I just bumped into this number $e$, "Euler's number" . I am told that this number is special especially when I take the derivative of $e^x$ , because its slope of ...
6
votes
2answers
102 views

How to proof that $\lim_{h \to 0}\frac{e^h-1}{h} = 1$ using the definition $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$?

In other words, how I can proof that the two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and ${d\over dx}\ln x$, some uses the former ...
5
votes
2answers
103 views

If $\pi $ is a normal number, is $\tau $ one?

If $\pi$ is a normal number, would that imply that $\tau =2\pi $ is also a normal number? If so, why? Something tells me that it should be, but I have no idea how to prove it. If all digits of $\pi$ ...
5
votes
1answer
68 views

Is $e$ involved in some geometric figure in any way?

Let's take some popular numbers in math: $\pi$, $e$, $\sqrt{2}$ and $\phi$. The number $\pi$ is the ratio between the circumference and the diameter of a circle; $\sqrt{2}$ is the length of a diagonal ...
5
votes
1answer
118 views

On a constant defined by Ramanujan.

In the second letter to Hardy Ramanujan writes about the number of prime numbers less than $n$ there he writes. Here this constant $\mu$ facinated me . What is its closed form? and How to compute ...
4
votes
0answers
102 views

The Tribonacci constant and the Dragon

Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation, $$4^x(2^x-1)=(2^x+1)$$ Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
3
votes
1answer
123 views

Why $\operatorname e$ was named e ? What is the history and reason behind it?

Why the constant that Euler discovered has symbol $\operatorname e$ and not other symbols? What is the history and reason behind it? Thanks.
3
votes
1answer
151 views

How to derive the Golden mean by using properties of Gamma function?

The Golden mean known as $\frac{1+\sqrt{5}}{2}$. How could one show the Golden mean can be expressed as $$ \frac{2\cdot 3\cdot 7\cdot 8\cdot 12\cdot 13\cdots}{1\cdot 4\cdot 6\cdot 9\cdot 11\cdot ...
3
votes
1answer
35 views

A misconception about arbitary constant

Given a function $f(x)$ from $\mathbb R$ to $\mathbb R$, If $f'(x)=0$ $\text{ for all } x\in \mathbb R$. Then $f(x)=C$.(This is my understanding) Question: I think that $C$ has to remain constant ...
3
votes
0answers
80 views

Finding relatives of the series $\varphi =\frac{3}{2}+\sum_{k=0}^{\infty}(-1)^{k}\frac{(2k)!}{(k+1)!k!2^{4k+3}}$.

Consider $\varphi=\frac{1+\sqrt{5}}{2}$, the golden ratio. Bellow are series $(3)$ and $(6)$ that represent $\varphi$ $$ \begin{align*} \varphi &=\frac{1}{1}+\sum_{k=0}^{\infty}\cdots&(1)\\ ...
3
votes
0answers
320 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 who's author has passed away. ...
2
votes
1answer
47 views

Do I need different constant names for different levels of integration?

I'm just getting my feet wet in integration, so pardon me if I misuse a term. Let's take the anti-triple-derivative (I'm not sure if that's what it's actually called) of $8x$. $y'''=8x$ ...
2
votes
1answer
44 views

Antiderivative where resulting constant depends on x?

Everything was going really well until one week before the exam when the teacher gave us this problem: $\int \frac{2x^2+13x+19}{x^2+5x+6} dx$ For which I and Wolfram Alpha finds the solution: ...
2
votes
1answer
92 views

Express the mathematical constant $e$ in terms of a limit that goes to zero.

The mathematical expression of the mathematical constant $e$ in terms of a limit that goes to infinity is $$e = \lim\limits_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n$$ But can we express the ...
2
votes
3answers
298 views

Positive constant scalar definition

In French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one explain ...
2
votes
1answer
91 views

Can the Landau-Ramanujan Constant be generalized towards cubes?

The Landau-Ramanujan Constant is related to the sum of 2 squares. See : http://en.wikipedia.org/wiki/Landau%E2%80%93Ramanujan_constant Can a similar thing be said for the sum of 4 positive cubes ? Or ...
2
votes
0answers
93 views

In the Hunt for Kaprekar's Constants for more than 4 digits.

Kaprekar's constant is $6174$ . Take any four digit number with at least two different digits; create two four digit numbers by writing the digits in descending order and in ascending order; subtract ...
2
votes
0answers
44 views

Is $e$ uniformly distributed in all bases?

There has been talk of whether or not $\pi$ is normal, i.e. uniformly distributed in all bases $b$ where $b\ge2$. The general response has been that we expect that it is, and have found no obvious ...
2
votes
4answers
186 views

Why does $e^{i\pi}=-1$? [duplicate]

I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are. As far as I know ...
1
vote
5answers
95 views

Limits to infinity Finding Constant Number

Hi I have a question regarding of limits to infinity please help which I need to find the constant number for a and b. Please help! Thank You! The question states the user to find the following ...
1
vote
2answers
149 views

Is there any reason why $4-\pi$ is quite close to $\frac{\sqrt{3}}{2}$?

In this question obviously the error of our "approximation" is $4-\pi=0.858...$ . I tried to reconstruct the false argument with $\tau=2\pi$, and the error in that case would be $8-\tau=1.716...$, ...
1
vote
1answer
75 views

How is PI used to predict weather patterns?

I've heard that using PI to predict weather patterns is possible. I would like verification on this, and how this is possible. I can't seem to find any other sources explaining this concept. My ...
1
vote
1answer
58 views

Why we can't define more mathematical constant?

I would like to know how many mathematical constant are there? I saw this link and I know the names. Who can define a mathematical constant? Someone can say that ...
1
vote
4answers
237 views

Calculus Derivative - Finding unknown constants

Determine the constants $a$, $b$, $c$, and $d$ so that the curve defined by $y = ax^3 + bx^2 + cx + d$ has a local maximum at the point $(2, 4)$ and a point of inflection at the origin. Sketch the ...
1
vote
1answer
92 views

Proving this identity $\gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\zeta(k+1)-1}{k+1}$ where $\gamma$ is the Euler-Masceroni constant

I've seen this identity here $$ \displaystyle \gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\dfrac{\zeta(k+1)-1}{k+1} $$ and I'd like to know how it is deduced. Could anyone help? Thanks. ...
1
vote
1answer
51 views

Mathematical constants

Am I mistaken, or is there a mistake on the mathematical constants Wikipedia page that describes the Conic constant / Schwarzschild constant in terms of Napier's constant?
1
vote
1answer
51 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
1
vote
1answer
65 views

Asymptotic Big Omega Proof

Let $f(n) = 2n^4 − 4n^3 + 16n^2 − 64^n + 3$. (a) Using the constant $c = 1.9$, prove that $f(n) \in \Omega(n^4)$. Just by looking at it it is clear to me that this is true as for $f(n) \in ...
1
vote
3answers
77 views

General form for the series expansion of $e$

I've found a lot of series expansions of the Napier's constant. I was wondering if a general form for this could be devised. They all have a trend and similarities. I've been trying but I've been ...
1
vote
1answer
74 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?
1
vote
0answers
34 views

Quartic Polynomial Manipulation

I have a quartic polynomial in $x$ (too long to write here) $f(x,c_1, c_2, c_2)$ where $c_1, c_2, c_3$ are constants which I have complete freedom over how to fix their values, as long as they are ...
1
vote
0answers
27 views

Champernowne constant - summation and behavior of terms in continued fraction expansion

Is there an infinite summation that gives the Champernowne constant? Wikipedia has one, and so does Wolfram MathWorld. Are they valid? If so, could someone explain why, i.e how do they work? Also, ...
0
votes
3answers
397 views

What are the uses of Euler's number $e$?

People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ...
0
votes
2answers
78 views

Is Planck's constant a mathematical or a physical constant?

Planck's constant $h$ came out of his considerations of black body radiation and features prominently in quantum physics. Recently I came across the statement that $-i\hbar = pq - qp$ for elements ...
0
votes
3answers
38 views

How to prove that $\frac{d}{dx}\epsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant

How to prove that $\frac{d}{dx}\varepsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant? Thanks in advance for your immense help.
0
votes
1answer
37 views

What is the constant $c$ in $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy}$?

In the manner in which $\frac{d}{dx} e^{x} = e^{x}$. What is the value of the constant $c$ for which $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy} $ ?
0
votes
1answer
62 views

Real Analysis constant function

Suppose that $f:(a,b) \to \Bbb R$ satisfies $|f(x)-f(y)|=M|x-y|^\alpha$ for some $\alpha > 1$, some $M \geq 0$, an all $(x,y)$ in $(a,b)$. Prove $f$ is constant on $(a,b)$ I have absolutely no ...
0
votes
1answer
79 views

Closed Forms of Certain Zeta constants?

The Riemann Zeta function $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ converges for $Re(s)>1$. I am interested in some particular "irrational " Values of it. Like $\zeta(\pi)=1.176241738...$ ...