Tagged Questions

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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3
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58 views

Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
2
votes
2answers
57 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
92 views

Prove this formula for $\pi$

I have to use a certain approximation for $\pi$ for my computer science class, but I don't really understand what's going on, other than that this is related to the Taylor polynomial for arctangent. ...
6
votes
4answers
162 views

How prove $\pi^2>2^\pi$

show that $$\pi^2>2^\pi$$ I use computer found $$\pi^2-2^\pi\approx 1.044\cdots,$$ can see this I know $$\Longleftrightarrow \dfrac{\ln{\pi}}{\pi}>\dfrac{\ln{2}}{2}$$ so let ...
0
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1answer
24 views

Approximating Pirrational Numbers

A while back I wrote this question on PPCG.SE about the numbers I termed Pirrational numbers. They are defined as follows: Let $P_i$ be the $i$th Pirrational number for some $i \in \mathbb{N}_0$ ...
0
votes
1answer
28 views

A question about infinitie series and pi

This is the sequence that can be used to find an exact value of pi 4/1−4/3+4/5−4/7+4/9−4/11…..(to infinity) = 𝜋 Or (1/1−1/3+1/5−1/7+1/9−1/11….. (to infinity) )= 𝜋/4 Given that we have this ...
12
votes
4answers
238 views

$\lim_{n\to\infty}\sqrt{6}^{\ n}\underbrace{\sqrt{3-\sqrt{6+\sqrt{6+\dotsb+\sqrt{6}}}}}_{n\text{ square root signs}}$

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 ...
2
votes
0answers
33 views

How do we know that the first few digits of an approximation for $\pi$ are correct?

For Gregory–Leibniz series, wikipedia has - "after 500,000 terms, it produces only five correct decimal digits of π.". But how do you know that those five decimal values are correct when you reach ...
0
votes
2answers
57 views

Different ways of approximation of $\pi$

i am studying trig n knows that pi can be approxated using gregory series ruthrford sries etc . Aso its strange n mysterious that pi is just ratio of cirumference n diameter . this profoundly shows ...
3
votes
2answers
49 views

Correcting Error in the Leibniz $\pi$ formula… why does it work?

You are probably familiar with the Leibniz $\pi$ formula: $$ 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4} $$ For a CS homework assignment I had to write a ...
0
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1answer
34 views

evaluating Pi with imaginairy unit i leads to contradiction!

I was reading about evaluating $i^i$ so I tried that with Mathematica and got a real (R) result and Mathematica suggested an alternative form being $e^{-pi/2}$ so I solved for $\pi$: $i^i = ...
7
votes
2answers
121 views

Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...
0
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0answers
55 views

Using $1/\pi$ to approximate $\pi$

Using Ramanujan's formula for approximating $\pi$ gives $1/\pi.$ How can $1/\pi$ be used to approximate $\pi?$
7
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5answers
191 views

Which number its greater $\pi^3$ or $3^\pi$?

$ \pi^3$ or $3^\pi$ using algebra please, I arrive the solution whith $a^x > 1 + x$ but I am interested in more solutions.
0
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2answers
24 views

What is the meaning of the PI function.

I am solving for a configuration problem and i have seen a function π This is a function not 3.14 which is the value of pi. While accessing some lectures i found out that they also call this symbol ...
5
votes
1answer
147 views

Why does summation of infinite series end up in powers of pi?

For instance, we have $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$$ $$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...=\frac{\pi^2}{6}$$ ...
2
votes
3answers
61 views

Why does this loop yield $\pi/8$?

Was doing some benchmarks for the mpmath python library on my pc with randomly generated tests. I found that one of those tests was returning a multiple of $\pi$ consistently. I report it here: ...
0
votes
1answer
46 views

Bailey–Borwein–Plouffe formula

$\displaystyle \pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right]$ How does the BBP formula for ...
1
vote
1answer
68 views

What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? [duplicate]

"homework" What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? I know that both $\pi$ and $e$ are irrational. What can be said about $\pi+e$, and $\pi e$?
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votes
0answers
77 views

Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
6
votes
2answers
87 views

Calculating value of $\pi$ independently using integrals.

Recently I noticed this integral: $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}7-\pi\approx0$$ Which is a very interesting result which gives us the value of ...
23
votes
6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
-1
votes
2answers
90 views

Why Doesn't this equation work?

$$\frac{n2^nn!^2}{(2n)!}$$ Is supposed to output the nth decimal of pi, this works fine with $$n = 1$$, but why not with $$n = 2$$? (I ommited the sigma, equation was found here)
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votes
3answers
42 views

How Does This Evaluate?

I just relized, I have no idea how to evaluate: $$n2^2n!^2$$ Google didn't help me find a breakdown could anybody just space it out?
6
votes
8answers
212 views

Calculate $\pi$ By Hand?

All over the internet the only hand equation i found was $$\frac\pi4 = 1 - \frac13 + \frac15 - \frac17+\cdots.$$ But this takes something like a thousand iterations to get to four digits, is there a ...
2
votes
1answer
86 views

Plouffe's formula for $\pi$

Plouffe established the following formula for $\pi$ in $2006$ $$\pi = 72\sum_{n = 1}^{\infty}\frac{1}{n(e^{n\pi} - 1)} - 96\sum_{n = 1}^{\infty}\frac{1}{n(e^{2n\pi} - 1)} + 24\sum_{n = ...
3
votes
1answer
64 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
5
votes
1answer
91 views

Show that $\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$

MathWorld states that (see equation $(130)$) $$\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$$ and attributes it to Gosper. I believe an approach to establish the formula ...
1
vote
1answer
64 views

Can $\pi$ and the $\pi$ in radians simplify?

I saw in a proof for the limit $$\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1$$ that, in one of the steps, you had to take the area of a section of a circle, in which you had to do $\frac{\pi r^2 ...
2
votes
3answers
58 views

Why does $y = x\sin(\frac{180}{x})$ approach $\pi$?

A few days ago I was playing on my scientific calculator and I ran over an interesting little equation: $180\sin(1)$ is extremely close to $\pi$. At first I thought it was a coincidence, but then I ...
0
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0answers
36 views

Calculate digits of pi without needing to reuse them

I am looking for some algorithms that can calculate digits of pi. without needing to reuse previous digits. I would like to find the most simple and fast algorithms possible. Thanks!
2
votes
1answer
50 views

Find Pi using integral

I am just started learning calculus and wonder why: $$\int_0^1 \frac{4}{1+x^2}$$ Allows to find $\pi$? It would be great if someone could provide very detailed explanation.
1
vote
2answers
46 views

Finding the exact area of a circle?

Background: I recently began taking calculus and it has come to alter the way I look at circles, and curves. The equation of a circle is $\pi r^2$, traditionally in school we have always left the ...
7
votes
1answer
108 views

Showing that $ \int_{0}^{\pi / 4} \arctan \! \left( \sqrt{\frac{\cos 2x}{2 \cos^{2} x}} \right) \mathrm{d}{x} = \frac{\pi^{2}}{24} $.

I was wondering if an expert in integration could kindly solve the following problem, which was posed in a mathematics competition (I can’t remember which one) and was unsolved by any participant. ...
2
votes
2answers
63 views

How can I visually imagine the area of a circle divided by $\pi$?

If I have a circle with an area of 100 units^2, and I divide it by $\pi$, how can I imagine that visually in my mind? Since 100 / $\pi$ =~ 31.83, and the square of that is =~ 5.64, I currently ...
0
votes
2answers
67 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
13
votes
2answers
369 views

Ramanujan's approximation for $\pi$

In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of $\pi$, such as $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ...
2
votes
1answer
59 views

Continued fraction expansion of Pi (oeis A001203). [duplicate]

I would like to understand how you get the numbers $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+...}}}}$$ i.e. $\{3,7,15,1,292,...\}$ (A001203). In the comments of A046965 is explained a method ...
1
vote
0answers
40 views

What is the limit of $k^2|\pi-n(k)/k |$, where $k$ minimizes $|k\pi -n|$?

Let $k\in \mathbb N$ and for any such n, let $k=k(n)$ minimizes the distance $|k\pi-n|\leq 2 \pi$. It is clear that, by fixing the value of $n$, it is possible to choose $k$ (and vice versa). ...
4
votes
1answer
83 views

Intuitive reason for why the Gaussian integral converges to the square root of pi?

This is a very famous problem, which is commonly taught when students begin learning about multivariable integration in polar coordinates. However, it has always bothered me that we recieved such an ...
0
votes
2answers
95 views

Lambert's Original Proof that $\pi$ is irrational.

I am trying to find Lambert's original proof that $\pi$ is irrational. Wikipedia has a little description but it is quite lacking. Can someone direct me to Lambert's original proof or post his proof ...
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votes
2answers
98 views

If pi (or tau) is a constant for a circle, what is the equivalent of a sphere called? [closed]

I never learned this in math class, but was just thinking that a sphere should have some equivalent to pi, and that it too should be a constant... what would that be? Thinking more about it, if the ...
4
votes
1answer
115 views

How do you use the BBP Formula to calculate the nth digit of π?

I know what the Bailey-Borweim-Plouffe Formula (BBP Formula) is—it's $\pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} ...
4
votes
2answers
141 views

What is the importance of $\pi$ in mathematics? [closed]

In what specific fields is $\pi$ relevant in mathematics and how is its accuracy important? Is there any field in which its precision leads to some results despite others?
12
votes
3answers
416 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
0
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3answers
59 views

Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ (corrected inequation)

Prove that Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ algebraically or geometrically. $n\sin\frac{2\pi}{n}-n\sin\frac{\pi}{n}$ means the area of a regular n-gon + the area ...
5
votes
1answer
143 views

Finding a specific sequence of digits in pi

Looking at the pifs project on GitHub and this question on SO has made me curious as to how feasible it is to find a specific sequence of digits within Pi. Essentially, on average, how many digits of ...
1
vote
1answer
80 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 ...
9
votes
2answers
229 views

Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?

I have heard that it is unknown whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer. How can this be? $\pi$ is known to many digits and it seems like only a matter of time on a computer to find ...
26
votes
4answers
522 views

Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$?

Following reading this great post : What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis ...