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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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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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79 views

Show that $\sum\limits_{k=1}^\infty \frac{1}{2^k k^2}=\frac{\pi^2}{12}-\frac{1}{2}\log^2 2$ [duplicate]

I have tried to split the sum into two different parts where the first reduces to the well known Basel problem, but I'm not able to show that the remaining series converge to $\log^22$: $\sum\limits_{...
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1answer
42 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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48 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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65 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
51 views

Why do we consider $\pi$ as a irrational number?

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
72 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
62 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
48 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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38 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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114 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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231 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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62 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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156 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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56 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
74 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
254 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
200 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
85 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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52 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
81 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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170 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
40 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
42 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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112 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
44 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
44 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 ...
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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, ...
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1answer
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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\...
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4answers
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A nice relationship between $\zeta$, $\pi$ and $e$

I just happened to see this equation today, any suggestions on how to prove it? $$\sum_{n=1}^\infty{\frac{\zeta(2n)}{n(2n+1)4^n}}=\log{\frac{\pi}{e}}$$
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2answers
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Does this limit $lim_{n\to\infty}\sum_{i=0}^n 1/n \sqrt{1 - i^2/n^2}$ converge to $\pi/4$?

While trying to find an approximate area of a quarter of a circle by splicing it into small rectangles and summing their areas I've reached a point where I have this formula: $$\sum_{i=0}^n 1/n \sqrt{...
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3answers
77 views

Silly Question about $π$ [closed]

In our junior classes, we learnt that $π$ is an irrational number. Now, also we know about rational numbers. So, if I say that I have a thread of 44cm long and we may convert it into a circle. Then ...
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1answer
80 views

Help understanding the cause of this pattern when writing π as an infinite series with double factorials

I made a post about a year and a half ago: $\pi$ as an Infinite Series using Taylor Expansion on Equation of a Circle where essentially I used the Taylor series expansion on $\ y = \sqrt{r^2-x^2}$ (...
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10answers
4k views

How are the known digits of $\pi$ guaranteed?

When discussing with my son a few of the many methods to calculate the digits of $\pi$ (15 yo school level), I realized that the methods I know more or less (geometric approximation, Monte Carlo and ...
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1answer
106 views

How is Euler's Formula Wrong?

I figured out that if $x^{y} = z$, then $z^y = x^{y^2}$. Then we know Euler's Formula: $$e^{πi} = -1,\quad (e^π)^i = -1,\quad (e^{2π})^i = 1$$ Now, using the formula above, let $e^{2π}$ act as x, ...
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Has the arithmetic mean of the Leibniz series been used for computing $\pi$?

I'm sure this was asked before, but I wasn't able to find the answer. If we have an alternating series with slow convergence, can't we just use the arithmetic mean to get a much better approximation ...
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2answers
60 views

Is $2\pi\sum_{n=0}^\infty\frac1{n!(n+1)!}$ equal to $5/\pi$?

I was working through some contour integration questions, and when finding the residues of the integral $\int e^{z+1/z}\,dx$, I found that it was equal to the infinite sum $$2\pi\sum_{n=0}^\infty\...
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2answers
39 views

Adding $2\pi$ inches to a string about the Earth's equator makes a new circle reaching how far above the ground? [closed]

Sofia wraps a long string around the equator of the earth, pulling it snug. If the earth were a perfect sphere, the string would be touching the ground all along its length. Suddenly, Sofia stretches ...
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1answer
198 views

Can The Existence Of $\pi$ Be Proved Without Formal Analysis?

I hope this question is not too long, but I have included some extra information to clarify the context of the question and hopefully avoid the 'circular' arguments which inevitably occur on this ...
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2answers
108 views

Is there an intuitive explanation for the occurrence of e and pi in Stirling's approximation? - $n!\approx \sqrt{2\pi n} (n/e)^n$

Is there an intuitive explanation for the occurrence of e and pi in Stirling's approximation? $$n!\approx \frac{n^n}{e^n}\sqrt{2\pi n}$$ Any help would be much appreciated.
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1answer
117 views

Is there a closed form for $\zeta(\pi)$?

What is $\zeta(\pi)$? I know that $\operatorname{Re}(\pi)>1$, thus $$\zeta(\pi)=\sum_{n\geq1}\frac{1}{n^\pi}$$ Yet I have no idea how to even begin evaluating this series. It's probably ...
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33 views

EXACT measurements of a circle. Possible? [closed]

I understand simple circle equations like c=pi×d and a=pi×(rr) (sorry keyboard doesn't have pi or exponents) if pi is irrational with an infinitely long decimal, doesn't that make it impossible to ...
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37 views

How many terms it takes for the Leibniz series to converge to three decimal places of accuracy?

I need to find out how many terms it takes for the this series to converge to three decimal places of accuracy of Pi. e.i how many it terms it takes to obtain the value 3.141 from, the series: Leibniz ...
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1answer
116 views

Proof of Bellard's formula

I'm reading Bellard's proof for his eponymous formula computing pi digits, and I can't get past the first line. Given that: $\displaystyle-\ln(1-x) = \sum_{n=1}^\infty \frac{x^n}{n}$ for $|x| < 1$...
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1answer
26 views

What are some statistical distributions with the irrational numbers e and pi in their functions? (apart from the most common - Normal, Poisson)

I've been researching on the application and origin of irrational numbers in probability theory and statistical distributions, so far having derived a unique proof of Stirling's approximation, and ...
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2answers
33 views

If two real numbers are immesurable can an integer sum between the two get as close to any real number as we like?

Say for example we have $\pi$ and $1$. Can the sum $m\pi + n$ for $ m, n\in \mathbb{Z}^+$ get as close to a real number as we like? At first I tried using the fact that you could have $n = -floor(m \...
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1answer
46 views

Construction of the quadratrix by one motion only

Ancient geometers geometrically constructed the number $\pi$ using a special curve, called the quadratrix of Hippias (or Dinostratus). One way to construct the quadratrix is by tracing the path of the ...
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3answers
126 views

Which is greater, $\left(\frac{e}{2}\right)^\sqrt{3}$ or $(\sqrt{2})^{\pi/2}$? (no calculators)

From a math contest in 1985: Determine which of the following is greater: (no calculators) $$\left(\frac{e}{2}\right)^\sqrt{3} \, \hspace{3mm} \text{or} \hspace{3mm} \, (\sqrt{2})^{\pi/2}$$ Hints ...
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1answer
80 views

How can I proof that a circle's circumference is well defined? And How do I find it? [closed]

I searched here and I got the definition that the circumference of a curve would be the smallest upper bound of the sequence of the sum of lenghts of polygonal paths along the curve, but how can I ...