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Questions tagged [pi]

The number $\pi$ is the ratio of a circle's circumference to its diameter. Understanding its various properties and computing its numerical value drove the study of much mathematics throughout history. Questions regarding this special number and its properties fit in here.

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How to find the roots of sin(x) using series theory

If we define $$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n \ x^{2n+1}}{(2n+1)!}$$ How to find the roots of $\sin(x)$, i.e. $$\pi =4 \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$ satisfies $\sin (\pi)=0$
Phy-zr's user avatar
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57 votes
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Is the new series for 𝜋 a Big (or even Medium) Deal?

There's been some oohing and ahhing in the science press recently over the discovery of a new formula for $\pi$ obtained as a side effect of computing amplitudes in string theory: $$\pi=4+\sum_{n=1}^\...
Steven Stadnicki's user avatar
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If I have x and y coordinates of a point along the arc, how do I convert that to a percentage of PI?

I am using Javascript to create shapes in canvas. I am creating an arc where you specify the start and end angle of the arc to show where along a circle the arc begins and ends. They are initially set ...
Jared H's user avatar
1 vote
0 answers
122 views

The connection between $\pi$, $e$ and $20$ [closed]

It's well documented that $e^{\pi} \approx 20+\pi$. This can be explained using the following series: $$\sum\limits_{k=1}^{\infty}\frac{8\pi k^{2}-2}{e^{\pi k^{2}}} = 1$$ The series is quickly ...
Darmani V's user avatar
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Proving $ \sum_{k=0}^{\infty} \frac{k! \, (2k)! \, (25k - 3)}{(3k)! \, 2^k} = \frac{\pi}{2} $ [duplicate]

Wikipedia claims that $$ \sum_{k=0}^{\infty} \frac{k! \, (2k)! \, (25k - 3)}{(3k)! \, 2^k} = \frac{\pi}{2} $$ but does not give a citation. How can this be proved? Edit: This was correctly closed as a ...
Potato's user avatar
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What are some relations between the golden number or ratio $\phi$, and $\pi$?

What are some relations between the golden number or ratio $\phi$, and $\pi$? For example, by considering this answer https://math.stackexchange.com/a/744196/ ; by Steve Lewis. Now taking the point at ...
g.a.l.l.e.t.a's user avatar
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1 answer
69 views

Finding Number Sequences in the Digits of Pi?

I understand that π (pi) has an infinite number of digits, and this means any finite sequence of numbers can theoretically be found within its digits. For example, if you're looking for the sequence &...
asmgx's user avatar
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12 votes
2 answers
281 views

Bauer's series for $\frac{1}{\pi}$

Recently, someone asked a question involving the expression $$ \sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$ At first glance, I knew that the expression was the value of ...
seoneo's user avatar
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10 votes
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How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$

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 ...
Nomas2's user avatar
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2 votes
0 answers
135 views

PI 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 ...
jaredjbarnes's user avatar
1 vote
2 answers
161 views

Why 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 ...
asmaier's user avatar
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Proof 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} = \frac{4(1^2)}{4(1^2)-1}*\frac{4(2^2)}{4(2^2)-1}*\...
Craw Craw's user avatar
2 votes
2 answers
177 views

Why 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 = 3....
TOM's user avatar
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1 vote
1 answer
98 views

Efficient division by 99 by hand to compute digits of $\pi$ [closed]

I want to compute some digits of $\pi$ by hand. There are several formula which have been used in pre-computer time, among them: \begin{align} \frac{\pi}{4}&=4\arctan(\frac{1}{5})-\arctan(\frac{1}{...
L. Milla's user avatar
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2 votes
1 answer
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Approximation of $\pi$ by integral and rational number

Via WolframAlpha, I observed that $$\int_0^1\frac{x^{4n}(1-x^2)}{1+x^2}dx=\frac\pi2-\frac pq \to 0$$ when the integer $n\to\infty$. This gives an approximation of $\pi$ by a rational number. It is not ...
Bob Dobbs's user avatar
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3 answers
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Why 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 1}\...
pie's user avatar
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3 votes
2 answers
122 views

Might there be an $n^{\text{th}}$ digit of $\pi$ where the sequence becomes palindromic?

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 ...
Pickelhaube808's user avatar
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How to calculate the ratio of convergence for Euler's, Gauss' and Viète's approximation of $\pi$?

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 $2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\...
Marina's user avatar
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1 vote
0 answers
64 views

Parametric 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 function: \begin{equation}\...
Aninda Sinha's user avatar
1 vote
1 answer
113 views

Does there exist an irrational number $x$ such that both $x^π$ and $π^x$ are also irrational? [closed]

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?
Mods And Staff Are Not Fair's user avatar
2 votes
1 answer
198 views

Let : $\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$ then the minimum over $(0,\infty)$ verify a particular power series

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 : $$2+A-A^4-30A^6-...
Ranger-of-trente-deux-glands's user avatar
3 votes
2 answers
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Why does Im($(-1/10^n)^{-1/10^n}$) turn into the digits of pi as integer n gets larger?

$(-.001)^{-.001} \approx 1.007 - .0031634i$, $(-.000001)^{-.000001} \approx 1.000014 - .00000314164$, and $(-.000000001)^{-.000000001} \approx 1.0000000207 - .00000000314159271i$. Notice that as we ...
Alexandra's user avatar
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0 answers
134 views

How practical is this formula to calculate $\pi$?

So I recently stumbled upon a maths post which caught my eye. Here is the link It talked about calculating pi by perceiving the circle as a $n$-sided regular polygon. Eventually ending up with the ...
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BBP algorithm for calculating $\pi$ digits, sequentially

The wiki article https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula says $$ \pi = \sum_{k=0}^{\infty} \left[ \frac{1}{16^k} \left( \frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 +...
user9137's user avatar
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12 votes
2 answers
324 views

Prove that Wallis' product and Euler's formula directly imply that $(-1/2)!=\sqrt{\pi}$

(This occured to me recently, and I was pretty sure that it was true, so I was pleased that it really was. This has almost certainly been published many times before, but I didn't see it in either of ...
marty cohen's user avatar
0 votes
1 answer
71 views

Is my proof of this fact about pi correct?

I recently have thought of a proof but I can’t tell if it is correct or not. The proof is of $n \pi$ being irrational if n is an integer and non zero. The proof is below: We assume that $n \pi$ is not ...
Evon Z's user avatar
  • 3
4 votes
2 answers
145 views

Which numbers can be expressed as $a+b\pi$, with $a$ and $b$ being integers?

Since $a$ and $b$ are both integers, there is only a countable amount of numbers $a+b\pi$. Thus not every real number can be expressed as such. But is there a way to determine if $x$ can be expressed ...
Quintium's user avatar
  • 162
3 votes
1 answer
155 views

Series $\sum_{n=0}^{\infty}(-1)^n\binom{2n}{n}^5\left(\frac{1+4n}{2^{10n}}\right)=\frac{\Gamma^4(1/4)}{2\pi^4}$

$$\sum_{n=0}^{\infty}(-1)^n\binom{2n}{n}^5\left(\frac{1+4n}{2^{10n}}\right)=\frac{\Gamma^4(1/4)}{2\pi^4}$$ Is there any way to prove this? I don't even know where to start with this one. The following ...
Miracle Invoker's user avatar
15 votes
1 answer
499 views

Proof $\pi$ 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 ...
Alex Pawelko's user avatar
13 votes
1 answer
468 views

Covering a circle using rectangles

What is the maximum area that can be covered with $3$ rectangles inside a radius $1$ circle?(i.e. maximum area $=\pi$) The rectangles can be any length and height you want, and can rotate and reflect. ...
A Math guy's user avatar
4 votes
1 answer
132 views

Perpendicular distance from stright line to point on arc

I have a line with a circular arc cutting through it at 2 points, A and B. Given that I know the length of line CD (which is the distance between line AB and the highest point on the arc - ...
Jralford's user avatar
  • 143
2 votes
0 answers
99 views

How to show sequence converges to $\pi$ [duplicate]

The question is as follows: Let $a_{0} = 2\sqrt{3}$ and $b_{0} = 3$ and define two sequences recursivly as $$a_{n} = \frac{2a_{n - 1}\cdot b_{n - 1}}{a_{n - 1} + b_{n - 1}} \text{ and }b_{n} = ...
Eric's user avatar
  • 33
2 votes
0 answers
103 views

Power Series with digits of $\pi$

Sorry if this has already been asked, but I haven't found a post. Can anything be said about the function $$f(z)=\sum_{n=0}^\infty a_n z^n$$ where the $a_n$'s are the digits of $\pi= 3.14159...$, so $$...
Diger's user avatar
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3 votes
0 answers
368 views

A nice formula for pi [closed]

I would like to share a simple derivation of an iterative formula for Pi. The formula I have derived: $\begin{align} A_1 &= 4 \\ A_{n+1} &=2*4^{n} \left(1 - \sqrt{1 - \frac{A_n}{4^{n}}}\right)....
Rotem Tsafrir's user avatar
0 votes
0 answers
51 views

ln n and pi breakup sequence

To compute the logarithm of an integer efficiently using pen and paper, one can first write the number as a product of numbers close to $1$. For example, $$10 = \left(\frac{64}{63}\right)^{63}\left(\...
Axel's user avatar
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0 votes
0 answers
44 views

Doubt in Niven's proof given in wikipedia.

I am writing to seek clarification on a specific aspect of Niven's proof, as presented in the Wikipedia article. I have attached an image for your reference. My inquiry pertains to the value of and ...
Akash's user avatar
  • 1
8 votes
2 answers
349 views

(1/2)! from the infinite product definition of gamma

I wanted to derive $$\left(\frac{1}{2}\right)! = \frac{\sqrt\pi}{2}$$ from the infinite product definition of the gamma function $$\Gamma(z+1)=\prod_{n=1}^{\infty}\frac{(1+\frac{1}{n})^z}{1+\frac{z}{n}...
serpens's user avatar
  • 344
4 votes
2 answers
159 views

Can we find the exact value of a double sum with cosine without differentiation?

After finding an interesting double sum $$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{(-1)^{m+n}}{(m+n)^2} = \frac{\pi^2}{12}-\ln 2 ,$$ I started to investigate a harder one $$\displaystyle \sum_{m=...
Lai's user avatar
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1 vote
0 answers
38 views

Expected value of index of an n-digit number found in pi

Let $E(n)$ be the expected value of the index of finding an n-digit number in the digits of pi e.g. Number 0 is found at index 32 Number 1 is found at index 1 Number 2 is found at index 6 Number 3 is ...
Winter's user avatar
  • 936
3 votes
0 answers
57 views

Find the coincidence series with value $π-3.1$.

I discovered an interesting series that seems to yield rational numbers related to π: $$ \sum_{k=0}^∞\frac{m}{\Pi_{i∈I}(4k+i)}=\left|\frac{p}{q}-π\right| $$ I don't know if it's a coincidence or if ...
Aster's user avatar
  • 1,230
2 votes
1 answer
89 views

Approximating $\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$ with stable decimal places

Consider the Leibniz formula for $\pi$ $$ \pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}. $$ What is the minimum number of terms needed to calculate $\pi$ accurate to $k$ decimal places, in the sense ...
sam wolfe's user avatar
  • 3,425
6 votes
1 answer
136 views

How to prove this general form of the BBP formula?

Question How to prove the general form of the BBP(Bailey–Borwein–Plouffe) formula? Prove this formula is always true, or find a counterexample $k$ that makes the formula invalid: $$ π≟\sum_{n=0}^∞ \...
Aster's user avatar
  • 1,230
2 votes
1 answer
129 views

Understanding a geometric proof that $\pi \neq 4$

There are a few steps in a constructive proof that $\pi \neq 4$ that I cannot fully justify. Here is a sketch of the proof, with the step I don't fully understand bolded. Inscribe a circle of radius $...
Mathematical Endeavors's user avatar
1 vote
2 answers
231 views

Can you prove that $\frac{f_n}{f_{n-1}}$ converges to $2\pi$ and $\frac{1}{2\pi}$ if $f_n=\frac{1}{f_{n-1}}+f_{n-2}$ where $f_0=0$ and $f_1=2$?

I imagine this is already found but I cannot find the proof. The formula also works for $f_1=\sqrt{2}^{1/x}$ to get many multiples of $\pi$ Can you also prove that when $f_1=\sqrt{\frac{2}{\pi}}$ then ...
Joe's user avatar
  • 534
8 votes
2 answers
256 views

Minimum number of terms to approximate $\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$

Consider the Leibniz formula for $\pi$ $$ \pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}. $$ What is the minimum number of terms needed to calculate $\pi$ accurate to $k$ decimal places? My attempt: ...
sam wolfe's user avatar
  • 3,425
3 votes
1 answer
163 views

Does a closed-form expression exist for $ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $?

I am trying to find a closed-form expression for the following integral $$ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $$ There are specific values that I would like to generate (Table of ...
Gabriel Demirdag's user avatar
29 votes
2 answers
6k views

Find a simple proof that π is irrational

I know there are many questions on the site about finding a proof that π is irrational, but I'm posting the question separately to discuss a particular proof further We know that the Wallis Product is ...
زكريا حسناوي's user avatar
11 votes
1 answer
239 views

What is the limit $\lim_{n\to \infty} \cos(n)^{n^2} $

Two years ago I found this question and wasn't able to make the slightest advancement. $\lim_{n\to \infty} \cos(n)^{(n²)} $ The question is does the limit exist, where n is natural only (of course it ...
Oren Dubin's user avatar
0 votes
0 answers
57 views

Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers.

This is one of the exercises in my abstract algebra book (Nicholson) and it's just the title: Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers. All I know what to do ...
iwjueph94rgytbhr's user avatar
1 vote
0 answers
40 views

How do I calculate the digit number n of pi?

I want to create a pi calculator with no limit to precision, but I cannot find a formula for finding only the digit number n of pi. Here is what I mean: Let's say I want to find the 4th digit of pi. ...
Rchat42's user avatar
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