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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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What would it mean for mathematics if it was proven that $\pi$ is a normal number?

Whether or not $\pi$ is a normal number has yet to be determined. That is to say, we do not know for sure that its base expansion’s infinite sequence of digits is distributed uniformly. Speaking ...
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1answer
79 views

Find : $\int_0^{\pi}\frac{(1-\cos px)(1+\cos x)}{\sin x}dx$

How I can evaluate in closed form this trigonometry integral $$I_p=\int_0^{\pi}\frac{(1-\cos (px))(1+\cos x)}{\sin x}dx$$ , $p≥0$ positive integer Original question is find : $(p+2)I_{p+2}-2I_{p+1}...
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2answers
33 views

Product Pi Notation [on hold]

I wonder what is the properties of Product Pi Notation? I can't found anywhere about the properties. First of all, i have: $X=\beta\alpha\\ X^2=\beta^2\alpha(\alpha + 1) \\ X^3=\beta^3\alpha(\alpha +...
8
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1answer
74 views

What reason is there to conjecture that every finite string is really in the decimal expansion of $\pi$?

One of my students asked me this, and it occurred to me that I had never really questioned it. Apparently, it is only conjectured but widely believed that the decimal expansion in base $10$ of $\pi$ ...
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3answers
332 views

What is the value of $\frac11+\frac13-\frac15-\frac17+\frac19+\frac1{11}-\dots$?

The series $\sum_{k=1}^{\infty }\frac{(-1)^{k+1}}{2k-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$ converges to $\frac{\pi}{4}$. Here, the sign alternates every term. The series $\sum_{k=...
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0answers
26 views

Question about the Gamma function

My question is fairly simple: I was wondering if $\,\,\,\Gamma\left(\pi\right) = 2.2880377\ldots\,\,\,$ had any special meaning. Is it irrational ?. transcendental ? is it useless ? ...
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34 views

Would sampling the decimal digits of $\pi$ generate a white noise signal?

Discrete r.v. $X = \pi(d)$ (defined in another q of mine). Discrete r.v. $Y = X - 4.5$. q1: Would it be incorrect to deduce $Y\sim U(-4.5,4.5)$ from $X\sim U(0,9)$? q2: If you answered no to q1, ...
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39 views

Decimal digit extraction of $\pi$

I've seen many folks asking for this so I thought I'd take a shot @ answering it: This function $\pi[d]:\mathbb{N}\rightarrow\left\{\mathbb{W}<10\right\}$, based on the BBP closed form expression, ...
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1answer
48 views

How can i prove the following Equality? involving these infinite products

$$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^3}\right)= \frac{\cosh(\frac{\pi}{2}\sqrt3)}{3\pi} $$ This can be found here (http://mathworld.wolfram.com/InfiniteProduct.html) Line 22 It is known that $$...
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1answer
477 views
+50

An interesting formula for $\pi$

Looking through some old notebooks I found this monster of a formula: For any integer $r>1$, we have $$\pi=(-1)^{\left\lfloor\frac{r}{2}\right\rfloor-\left\lfloor\frac{2r-1}{4}\right\rfloor}\...
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1answer
85 views

Bernoulli numbers and $\pi^2$.

It is probably well-known that: $$ \lim_{n\to\infty}\frac{b_{2n}n^2}{b_{2n+2}}=-\pi^2, $$ where $b_n$ are the Bernoulli numbers. By a numerical experiment I have found that the quotient $$ \frac{b_{...
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176 views

The number $\pi$ in an unexpected context

[This is a follow-up question to this one: Figures and Numbers: Relating properties of geometric shapes and their Fourier series.] Drawing shapes by some predefined Fourier series I found this square ...
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0answers
10 views

Normal vs disjunctive vs lexicon

Apologies for lack of rigour but I'll attempt to phrase this in an answerable way. In this question, @Charles writes: [Being a normal number] (or even the weaker property of being disjunctive) ...
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4answers
61 views

For Fourier series, where does the $\pi$ and the $2\pi$ come from?

If a signal is given on an interval $[0, 2\pi]$, the Fourier series can be written as $$ f(t) = c_0 + \sum_{n = 1}^\infty a_n\cos nt + b_n\sin nt $$ with coefficients $$ c_0 = \frac1{2\pi}\int_0^...
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1answer
694 views

Are my results new? [closed]

I'm eighteen and sometimes I like doing math on my own when I'm inspired. I would like to know if some of my "discoveries" are new (I don't think so :) ). These are some of the results I found in the ...
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86 views

Digits of $\pi$ in other bases?

Pi day got me interested in thinking about whether $\pi$ is a normal number - one whose digits are uniformly distributed no matter what base the number is written in. The OEIS sequence for $\pi$ in ...
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0answers
97 views

Two definitions of $\pi$

I have the feeling that ancient mathematicians (like Greek or Chinese), trying to find good approximations of $\pi$ used two definitions: If $A$ is the area of a disk and $r$ is its radius, $\pi=A/r^...
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1answer
305 views

Proof of identity for $\pi$: $\frac{\pi}{3} = \frac{2}{\sqrt{2+\sqrt{3}}}\frac{2}{\sqrt{2+\sqrt{2+\sqrt{3}}}}\cdots$

While browsing the internet today, I came across the following picture: (full image can be found here - credit to Цогтгэрэл Гантөмөр) Now, it would naturally seem we can extend this to an infinite ...
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1answer
91 views

Can the value of the Riemann zeta function at $n=2$ be derived from the Wallis formula for $\pi$?

It is well known that the Riemann zeta function, defined for all positive integers $n>1$ by $$ \zeta(n) = \sum_{m=1}^{\infty} m^{-n} $$ takes the value $\displaystyle \frac{\pi^2}{6}$ at $n=2$. On ...
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2answers
470 views

$-1$ to the power of a irrational number

According to Wolfram Alpha, $(-1)^\pi \approx -0.90 + 0.43i$. But $\pi$ has proven to be irrational (we can't write $\pi$ in terms of a fraction $a/b$ with $a$ and $b$ integers) and, in the real ...
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1answer
36 views

Transition from $n$ Polygon to $2n$ Polygon which encloses the unit circle.

I want to calculate the area of the $2n$ polygon outside I already know the side length of the $2n$-polygon which is within the unit circle I have called the lengt $l_{2n}$, the length for the $n$ ...
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1answer
51 views

Evaluating $S_n$ where $n=1,2,3,\dots$ and $S_n=\sum_{k=1}^{\infty }\frac{1}{(4k^2-1)^n}$

If $n$ is a natural number and $S_n=\sum_{k=1}^{\infty }\frac{1}{(4k^2-1)^n}$, then $S_1=\frac{1}{2}, S_2=\frac{\pi^2-8}{16}, S_3=\frac{-3\pi^2+32}{64}, S_4=\frac{\pi^4+30\pi^2-384}{768},\dots$. How ...
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0answers
58 views

Proof that 2*2/sqrt(2)*2/sqrt(2+sqrt(2))*2/sqrt(2+sqrt(2+sqrt(2)))*… equals PI?

I found this formula that $\pi=2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}\cdot\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdot...$ I tested it out and it seems to be true, but I don't get why ...
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1answer
81 views

What is the numerical value of $(-3)^{\pi}$

As the title suggest what is the numerical value of $(-3)^{\pi}$? could we derive an answer using numerical analysis something along the lines of well if its basically $(-3) \cdot(-3) \cdot (-3) \...
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1answer
59 views

Why do we consider $\pi$ as a irrational number? [on hold]

Why do we consider $\pi$ as a irrational number? Why is that? We all know that $\pi$ is the solution of circumference / diameter of a circle and there could be infinite amount of circles which can ...
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0answers
76 views

Proving that $π^e$ is irrational [closed]

I tried for a few hours to come with a proof that $π^e$ is irrational. I mainly tried with the method "proof by contradiction" and didn't use calculus at all, but couldn't come up with a proof. Can ...
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1answer
65 views

How far would a person have to search through Pi to get a 50% of getting a million consecutive ones? [closed]

We know that Pi is a pseudo random sequence that continues indefinitely, so we know that there is a million consecutive ones(or any other combination) contained within Pi somewhere. So then, if we ...
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3answers
58 views

Why is $\left\lfloor\frac{\lfloor a\pi\rfloor}{a}\right\rfloor=3$ for $a>0$?

Why is this true? $$\left\lfloor\frac{\lfloor a\pi\rfloor}{a}\right\rfloor=3 \text{, for } a>0$$ I need this to solve the Ukraine Math Olymipiad 1999. "$\lfloor\cdot\rfloor$" indicates the floor ...
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2answers
41 views

What is the travelled distance of the red mark on the upper surface of the rotating cube?

Each side of a cube is 2 unit in length. This cube is kept on a table such a way that one surface (i.e., 4 vertices) of it completely touches the table. At this position, a red point is drawn on ...
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122 views

Proof of this formula for $\sqrt{e\pi/2}$ and similar formulas.

\begin{align} \sqrt{\frac{e\pi}{2}}=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}} \end{align} as seen here. ...
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1answer
246 views

Using a gamblers race to approximate $\pi$

Imagine two wealthy gamblers start tossing their own separate fair coins, winning 1\$ on heads and losing 1\$ on tails. Both start at 0\$ and have infinite bank balances. Both of them want to get to k\...
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0answers
68 views

Looking for a proof that $\pi$ is irrational using a series representation.

I'm searching for a proof that $\pi$ is irrational using a series representation for $\pi$. I've seen that Apery proved that $\zeta(2)$ is irrational by using the series \begin{align} \zeta(2) = \...
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2answers
159 views

A Series for $\pi$

Question: Can we show that $$\sum_{n=0}^\infty(-1)^{n+1}\frac{(2n-3)!!}{(2n+3)!!}=\frac{\pi}{8} $$ ? According to wolfram alpha this result is true. Just amateur curiosity, not sure of a starting ...
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2answers
58 views

Why do $4\cdot 2^n\sin\frac{45}{2^n}$, $2\cdot 2^n\sin\frac{90}{2^n}$, and $1\cdot 2^n\sin\frac{180}{2^n}$ all tend to $\pi$?

I am not sure what question or inquiries to ask actually, but I just think this is really awesome Can someone explain to me why the graphs of $$4\cdot 2^n\sin\frac{45}{2^n}, \qquad 2\cdot 2^n\sin\...
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3answers
79 views

Sum of Infinity of Trigo to Pi

I am currently working on a proof with a good friend of mine that involves adding more and more triangles to the sides of a regular polygon but keeping the longest diagonal constant until eventually, ...
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5answers
283 views

Relationship between Catalan's constant and $\pi$

How related are $G$ (Catalan's constant) and $\pi$? I seem to encounter $G$ a lot when computing definite integrals involving logarithms and trig functions. Example: It is well known that $$G=\...
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1answer
50 views

How to use infinite series to bound $\pi$.

Given that: $\pi = \sum_{k=0}^{\infty} \frac{1}{16^k}\left(\frac{4}{8k+1} - \frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)$ and $0 \le\left(\frac{4}{8k+1} - \frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}...
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0answers
216 views

Relation between $\pi$ and $e$ - Coincidence or not coincidence? [closed]

Is any explanation known of why $$ \pi^4+\pi^5\approx e^6 $$ holds at ridiculously high precision?
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2answers
87 views

Sum to Infinity of Trigonometry to $\pi$

For $$y=\sum_{n=0}^a2\cdot2^n\cdot\tan\left(\frac{45}{2^n}\right)\cdot\sin\left(\frac{90}{2^n}\right)^2$$ I am currently working on a proof with a good friend of mine that involves adding more and ...
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0answers
54 views

Determination of $\pi$ [duplicate]

How to prove $$ \frac{\sqrt{8}}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^4 396^{4n}}=\frac{1}{\pi}, $$ which actually looks like coincidence?
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1answer
89 views

Who was Dalzell? $\pi$ < 22/7

The Dalzell-Integral reads: $$0<\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ It proves that $\pi<\frac{22}{7}$. See also Wikipedia. It was introduced by D.P.Dalzell in 1944 (see ...
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2answers
273 views

Proof that $ \pi > 2 \cdot \sqrt{2}$ and $ \pi > 3 $

I need to prove that $\pi > 3$ and $ \pi > 2 \cdot \sqrt{2}$ only in use of definition of cosine (by series) or $\cos(x) = \frac{e^{iz}+ e^{-iz}}{2}$and definition of $\pi$ as $\pi = 2\cdot x_0$...
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2answers
41 views

Definition of the approx. symbol

Take an unending number, say e.g $π$. If we want to show $π$'s value, should we use the approximately notation or equal sign when writing: $π = 3.14...$ or $π ≈ 3.14...$ This might be a really ...
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3answers
44 views

Does decimal point count as a digit of pi?

My friend includes the decimal point as a digit of pi. Is this right? He says the first 5 digits of pi are 3.141 because he counts the decimal point as a digit. I told him that decimal point does not ...
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2answers
118 views

Help finding the formula for this sequence {$23$, $114$, $187$, $473$, $2792$, $5624$, $19640$, $75884$, $187211$, $479798$, $1452835$, $5102237$…}

Excuse the large title (The 'good title' page said not to be afraid to make it too long) $\{23,114,187,473,2792,5624,19640,75884,187211,479797,1452795,5102858,14872865,72392867,146262888\}$ I'm ...
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1answer
45 views

Why does this 2Pi-Periodic function looks like this?

I am given this function 1 on an interval $[0,\pi]$ and am asked to make a sketch of it on an interval $[-\pi,\pi]$. When plugging it into Maple, I am given a sketch graph like this 2, however, that ...
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1answer
45 views

Confusion over the word “ratio” in the definition of $\pi$

According to Wikipedia, pi is "a mathematical constant that was originally defined as the ratio of a circle's circumference to its diameter." However, when I think of the word "ratio", something like ...
52
votes
3answers
4k views

What is the formula for pi used in the Python decimal library?

(Don't be alarmed by the title; this is a question about mathematics, not programming.) In the documentation for the decimal module in the Python Standard Library, ...
3
votes
1answer
106 views

Nice result that I can't prove: $\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x²\text{erf}(x)) \bigg) \;dx=\pi$

I'm always trying to find the integral representation of $\pi$ using some interesting special function, at this time I have got the below representation $$I=\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x^2\...
5
votes
4answers
121 views

A nice relationship between $\zeta$, $\pi$ and $e$

I just happened to see this equation today, any suggestions on how to prove it? $$\sum_{n=1}^\infty{\frac{\zeta(2n)}{n(2n+1)4^n}}=\log{\frac{\pi}{e}}$$