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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Is there any similar equation for π that someone has figured out?
I'm first off sorry if my title is a little vague, but I couldn't find any other way to put it. I'm also very sorry because I am extremely bad at math, but I thought that this was interesting none-the-...
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Formula involving exponents of Mersenne primes
Recently I formulated the following claim:
$$\frac{2}{\pi}=\left(\frac{7}{8}+\frac{3}{4h_2(2)}\right) \cdot \left(\displaystyle\prod_{\substack{p \equiv 1 \pmod{4}\\ p \in \mathbb{M}} } \frac{p+1}{p-...
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Conjecture: The largest root of $\sum_{k=0}^n\cos\left(\frac{k\pi}{2}\right)\binom{n}{k}x^k$ is approximately $\frac{n}{\pi}$ or $\frac{2n}{\pi}$.
Is the following conjecture true:
For large $n,ドル the largest root of $\displaystyle\sum\limits_{k=0}^n\cos\left(\frac{k\pi}{2}\right)\binom{n}{k}x^k$ is approximately $\dfrac{n}{\pi}$ for odd $n,ドル or ...
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Problem with proof in Spivak's Calculus
This is question (b) from problem 11 in Chapter 8 and it has to do with approximating the area of the circle by inscribing appropriate polygons. If $P$ is a regular polygon inscribed inside a circle ...
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Why does this discrete product built from floor and ceiling of squares converge to pi or 1?
I have been experimenting with a structure I call the Discrete Square Residual Structure (DSRS).
For a fixed integer $\mu > 0,ドル define
$U(n) = \lceil \tfrac{n^2}{\mu} \rceil, \quad L(n) = \lfloor \...
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When does the overlap of two circles equal $\pi$?
Suppose two equal circles of radius $R$ are centered at $(-a,R)$ and $(a,R)$.
The green area $A$ is given by
$$A(a,R) = 2Ra - a\sqrt{R^2 - a^2} - R^2 \arcsin\!\Bigl(\tfrac{a}{R}\Bigr).$$
For example, ...
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Showing $\sum_{n=1}^\infty\arctan\frac1{F_{2n + 1}}=\sum_{n=0}^\infty\arctan\frac1{\sqrt5F_{2n + 1}},ドル without showing each sum is $\pi/4$
This reference contains two similar series, which are
\begin{align}
\arctan \frac{1}{F_{2n + 1}} = \arctan \frac{1}{F_{2n}} &- \arctan \frac{1}{F_{2n + 2}} \\[6pt]
\Longrightarrow \qquad \sum_{n = ...
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Why does 355/113 show up in mediant approximation of $\pi$?
I recently came across a method used by ancient Chinese mathematicians to come up with the rational approximation $\pi \approx \frac{355}{113}$. They has already calculated $\pi \approx 3.1416$ and ...