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

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23
votes
2answers
315 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 ...
20
votes
2answers
931 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 ...
16
votes
4answers
2k views

I can't remember a fallacious proof involving integrals and trigonometric identities.

My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it. Here's what I know about it: The end result was some variation of 0=1 or 1=2. It involved ...
14
votes
2answers
312 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
433 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 ...
13
votes
2answers
481 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
495 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?
12
votes
4answers
278 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
329 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 ...
11
votes
1answer
244 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
680 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 ...
9
votes
3answers
826 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
297 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
536 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
115 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 prove that these two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and $\frac{d}{dx}\ln x$; some use the former ...
5
votes
2answers
114 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
3answers
125 views

a simple formula linking the value of $e$ to the Golden Ratio $\phi$

These last days, I was looking for an approximation formula to $\pi$. But, surprisingly, the formulas led to this other one: $$ e = \left (\frac {\phi} {\phi - 1} \right)^{\frac {1} ...
5
votes
1answer
74 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
159 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
360 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. ...
4
votes
0answers
110 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
125 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
159 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
2answers
40 views

Can I say that a fixed constant is less or equal infinity?

Mathematically speaking, given $c\in\mathbb{R}$, can I say that: $c\leq\infty$? E.g., is $10 \leq \infty$ a correct mathematical statement? I know this comparison is true in computer arithmetic, ...
3
votes
1answer
38 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
1answer
112 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
44 views

Why does the tribonacci constant have a trilogarithm ladder?

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

Does and where to does $\lim_{n\to\infty}\sum_{m} \prod_k \frac{1}{\lambda_{k,m}!}$ converge?

Given $n$ you get a number of partitions of $n$ and let's denote $\lambda_{k,m}$ to be the $k$th part of the $m$th partition. Now I built the following sum, that stimulated the following question: $$ ...
2
votes
1answer
49 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
47 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
155 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
402 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
29 views

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

I wonder whether $\gamma$ belongs to the ring of periods?
2
votes
2answers
74 views

Methods for calculating $\pi$ that use the sphere?

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as ...
2
votes
1answer
129 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
1answer
72 views

Does this have a name? (Regarding ways to calculate e)

Just wondering...came across this relationship regarding Euler's number in my math tinkerings, but I'm unaware if this particular relationship has a specific name or not: ...
2
votes
1answer
64 views

What is the algebraic role of the mathematical constant $\gamma$?

Mathematical constants $\pi$, $e$, $i$ have a lot of algebraic roles. They appear as identity elements, idempotents, invariant elements etc against various operations and sets. This is illustrated by ...
2
votes
0answers
109 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
47 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
1answer
50 views

Terms that cannot be solved for a variable

Yesterday our analysis professor told us you cannot solve $$ y = e^x+2/(1+x^2) $$ for x, but you have the option to approximate this numerically. He did not prove that, he just noted it. I can't ...
2
votes
4answers
202 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
115 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
3answers
125 views

How to solve following limit

I've been struggeling a bit with the following limit: $\lim\limits_{x \to 0} \frac{a- \sqrt{a^2 - x^2}}{x^2}$ The solution is: If a < 0 then -$\infty$ . If a > 0 then $\frac{1}{2a}$ But I don't ...
1
vote
2answers
153 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
90 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
66 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
436 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
99 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
56 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
88 views

What's the value of tau?

I've seen $\tau$ on a title of a YouTube video and I need help knowing what the value is. I'm serious. I've never heard of the value. So, what is it? Also, is it rational or irrational (this part ...