Unexpected appearances of $\pi^2 /~6$.
"The number $\frac 16 \pi^2$ turns up surprisingly often and frequently in unexpected places." - Julian Havil, Gamma: Exploring Euler's Constant.It is well-known, especially in 'pop math,'...
View ArticleProof $\pi$ is transcendental without symmetric function theory
Recently for a bonus homework assignment in my algebra class, I was asked to review the literature and write up a proof that $\pi$ is transcendental. Essentially every source I found ("The...
View ArticleLet : $\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$ then the minimum over...
Problem :Let :$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have...
View ArticleIs there a way to calculate a specific digit of PI
Is there any mathematical I could find a specific digit of 𝛑If I had f(x) = ... what would the function to return the x digit of 𝛑e.g.f(1) = 3f(2) = 1f(5) = 5f(12) = 9This is Probably impossible, but I...
View ArticleCompute $\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$
How can I evaluate the following integral?$$I=\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$$ I tried it with Wolfram Alpha, it gave me a numerical solution: $0.785398$.Although I immediately know that it...
View ArticleDoes there exist an irrational number $x$ such that both $x^π$ and $π^x$ are...
Can we find a number, let's say $x$ that's not a simple fraction, and if you raise it to the power of $π$ or raise $π$ to the power of $x$, both results end up being not simple fractions too?
View ArticleParametric representation of the Euler-Beta function, Zeta functions and pi
Using quantum field theory arguments, in https://arxiv.org/pdf/2401.05733.pdf, we found that there is a parametric representation of the (slightly generalised) Euler-Beta...
View Articlewhy is $\sqrt[3]{31}$ so close to $\pi$?
$\sqrt[3]{31}$ is about $3.14138$. Why is this so close to $\pi$?
View ArticleCan an irrational number have a finite number of a certain digit?
This question came up because I was wondering the following: If the digits of PI are placed in ascending order, what is the <insert-large-finite-number-here>th digit?I believe that the answer is...
View ArticleInfinite ways to represent $\pi$ as product of nested square roots of $2$ and...
Polygon inscribed in a circle leads to famous infinite product by Viete's formula.hereOne way to represent that as infinite nested radical of 2 as follows$$\pi = \lim_{n\to\infty}2^n \times...
View ArticleFind $\lim_{n\to\infty}\sqrt{6}^{\...
We have the following representation of pi:$$\pi=\lim_{n\to\infty}2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\dotsb+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}}}}}_{n\text{ square root...
View ArticleHow to calculate the ratio of convergence for Euler's, Gauss' and Viète's...
Let $\sqrt{6\sum_{k=1}^\infty{\frac{1}{k^2}}}$ be Euler's approximation of $\pi$; $\lim_{n\rightarrow\infty}\frac{2}{g_n}$ Gauss approximation of $\pi$; and...
View ArticleMight there be an $n^{\text{th}}$ digit of $\pi$ where the sequence becomes...
Assuming $n>1$, would it be reasonable to think there is an $n^{\text{th}}$ digit of $\pi$ where stopping there would yield a palindromic number $(3.14159...951413)$?Would it be more likely that...
View ArticleWhy no one uses the product formula for sine function to calculate $\pi$?
$$\sin(\pi x)=\pi x \prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)$$$$\pi = \frac{\sin(\pi x)}{x\prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)}$$Let $x=\frac{1}{2}$$$\pi = \frac{2}{\prod_{n \ge...
View ArticleMathematical coincidences concerning the numbers $\pi$, $e$ and $163$
Something similar to this has probably been posted, but since I can't find any at the moment I will post it here.There are many numerical expressions to do with $\pi$, $e$ and $163$ (Wikipedia has many...
View ArticleCould PI have a different value in a different universe?
The value of pi is determined by the circumference of a circle.Why is it any particular constant number? Would a circle as defined as a perfect circle in any universe lead to a different value of...
View ArticleConjecture: The sequence $\frac{2}{n} \sum_{i=1}^n...
I found that the series$$s(n) = \frac{2}{n} \cdot \sum_{i=1}^n \sqrt{\frac{n}{i-\frac{1}{2}}-1}$$converges to $\pi$ as $n \to \infty$.To verify this I have computed some...
View ArticleA new formula for $\pi$ using cheated Bernstein's polynomials with...
Well I give an answer here and I want to know if it's true for a special exampleAnswer :Not an answer just some speculation about gamma function and Bernstein's polynomial which is of independent...
View ArticleHere is another "$e$-$\pi$-without -calculating" comparison
When playing in a Python console I observed$\,\pi^\pi\approx 36.46\,$ and $\,e^e\approx 15.15$, and that their ratio is close to $\dfrac{12}{5} =$ a fraction with small numerator and denominator, hence...
View ArticleApproximation of $\pi$ by integral and rational number
Via WolframAlpha, I observed that$$\int_0^1\frac{x^{4n}(1-x^2)}{1+x^2}dx=\frac\pi2-\frac pq\approx 0$$when the integer $n\to\infty$. This gives an approximation of $\pi$ by a rational number. It is not...
View ArticleEfficient division by 99 by hand to compute digits of $\pi$
I want to compute some digits of $\pi$ by hand. There are several formula which have been used in pre-computer time, among...
View ArticleWhy is $3 + \sin(3) + \sin(3+\sin(3))$ near $\pi$?
$3 + \sin(3) + \sin(3+\sin(3)) = 3.1415926535721955587...$$3 + \sin(3) + \sin(3+\sin(3)) + \sin(3 + \sin(3) + \sin(3+\sin(3)) ) = 3.1415926535897932384626433832795019...$$\pi =...
View ArticleProof of an approximation of $\pi$
Here is an approximation of $\pi$:$$\prod_{k=1}^{+\infty} \frac{4k^2}{4k^2-1} = \frac{\pi}{2}$$Proof$$\prod_{k=1}^{+\infty} \frac{4k^2}{4k^2-1} =...
View ArticleWhy is the definition of $\pi$ as integral by Weierstrass "inverted"?
Reading https://en.wikipedia.org/wiki/Pi#Definition I stumpled upon the following definition as an integral, presumably given by Weierstrass:$$\pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}$$However I don't...
View ArticleProve that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} =...
Prove that $$\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$$I tried to look at $$ f_n(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^n $$And maybe taking it's derivative...
View ArticlePI Approximation With Arcsine Infinite Series
While Playing with numbers last night I stumbled on an approximation of pi that was quite exciting. I'm sure it has been discovered before, but it was a fun journey nevertheless and I wanted to share...
View ArticleHow did Ramanujan find $\sum_{n=0}^\infty...
The formula$$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$(in older notation) appears as eq. 38 in Ramanujan's paper Modular equations and...
View ArticleWhat is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$ Does...
What is the result of the following limit?$$ \frac{1\times3\times5\times\cdots}{2\times4\times6\times8\times\cdots} = \lim_{n \rightarrow \infty}\prod_{i=1}^{n}\frac{(2i-1)}{2i}$$If I remember...
View ArticleHow to Evaluate $\sum_{n=0}^{\infty}\frac{(-1)^n(4n+1)(2n)!^3}{2^{6n}n!^6}$
I want to Evaluate$\sum_{n=0}^{\infty}\frac{(-1)^n(4n+1)(2n)!^3}{2^{6n}n!^6}.$I tried from arcsin(x) series and got $\frac{1-z^4}{(1+z^4)^{\frac{2}{3}}}=...
View ArticleMotivation for Ramanujan's mysterious $\pi$ formula
The following formula for $\pi$ was discovered by Ramanujan:$$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$Does anyone know how it works, or what...
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