# 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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### Does a set of all decimal expansions of $\pi$ contains $\pi?$ [duplicate]

Let's say there is a set containing all finite decimal expansions of $\pi$: $$A = \{3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... \}$$ Does this set contains $\pi$? I see that it is probably not true ...
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### PI calculation and error with approximations

$\Pi$ can be calculated with infinite product, series or polygonal methods. It's my understanding approximation using these methods give a bound on what the value of $\Pi$ could be. For instance, ...
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### Is $\frac{0}{\pi}$ rational?

In math lesson, our teacher showed this formula. $Q = \{\frac{a}b\}\land (a\land b\in Q)\land (b\neq 0)$ According to this formula, $\frac{0}\pi$ is... strange. You know, $\frac{0}\pi$ is 0 and 0 is ...
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### Why Stirling formula at n = -1 and n= -2 is so close to 2*pi*i [closed]

We know asymptotic formula for gamma function as Stirling formula $$\Gamma(z+1) \approx F(z+1) = \sqrt{2\pi z}\left({\cfrac{z}{e}}\right)^z\cdot \left({1+\cfrac{1}{12z} +\cfrac{1}{288z^2}+...}\right)$$...
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### An uncommon continued fraction of $\frac{\pi}{2}$

I'm currently stuck with the following infinite continued fraction: $$\frac{\pi}{2}=1+\dfrac{1}{1+\dfrac{1\cdot2}{1+\dfrac{2\cdot3}{1+\dfrac{3\cdot 4}{1+\cdots}}}}$$ There is an obscure clue on this: ...
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### Why this is equal to $\pi$?

I just recently came across: $$\lim_{N\to \infty}\bigg[2\prod_{k=1}^{N}\biggl(\frac{(2k)^2}{(2k)^2-1}\biggr)\biggr]$$ which appears to be approaching $\pi$... Can anyone explain to me why this is, or ...
### Are $\pi$ and $\tan^{-1}\left(2\right)$ rational multiples of each other? [duplicate]
For a proof of quantum universality, I need to show that $\tan^{-1}\left(2\right)$ is not a rational multiple of $\pi$. How do I show this? I feel like showing algebraic independence over the ...
The Wallis sieve is a variation on Sierpinski's carpet, where you start with a square, and in the $i$th step you divide each square into $(2i+1)^2$ smaller squares and remove the middle one. The total ...