10
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Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student:

Find an approximation of $\pi$ formed of the digits $0$ through $9$, each used at most once, combined with (unlimited quantities of) such basic symbols as the elementary arithmetical operations, parentheses, root signs, superscripts indicating exponentiation, and factorial signs.

I don't recall anymore how specific they were about which symbols were allowed, but this is probably close enough. I came up with

$$ \pi = \left(\left( - \frac{1}{2} \right)! \right)^{6/3}, $$

which unfortunately was denied the prize. However, that's not the point of the present question.

I also considered expressions of the form $$ \pi \stackrel{?}{\approx} \sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}} $$ for $n > 2$ not violating the requirements of the puzzle. However, the question I couldn't answer was whether or not suitable combinations of factorial signs and square roots would provide arbitrarily close approximations of $\pi$.

Generalizing this, the question I couldn't answer was this one:

Let $n>2$ be an integer. Let $r_{k,m} = \sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$ with $k$ factorial signs and $m$ square roots. Do the $r_{k,m}$ lie dense in $\mathbb{R}_{>1}$? Does it make a difference which $n$ we start with?
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  • $\begingroup$ $\frac{31415}{10000}$ ? Or, simpler still, just 3. $\endgroup$ – wltrup Aug 10 '15 at 11:31
  • $\begingroup$ Wait, what question are you answering? My real question is not the original challenge, but the problem I mentioned at the end. $\endgroup$ – RP_ Aug 10 '15 at 11:40
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    $\begingroup$ IN my point of view, to see things clearly, I would not prefer writing nested squares, I would write:$$ r_{k,m} =(n!!\cdots !)^{\frac{1}{2^m}}$$. Now you may thing of using Stirling's approximation of the factorials ! $\endgroup$ – Elaqqad Aug 10 '15 at 12:31
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    $\begingroup$ @MarkHurd It does but back when I wrote it, the question didn't say anything about digits being used just once each. Nonetheless, $\pi \approx 3$ is still a valid answer, and still tongue in cheek. :) $\endgroup$ – wltrup Aug 11 '15 at 8:14
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    $\begingroup$ @MostafaAyaz: I mean: does the set $\{ r_{k,m} \}_{k,m \in \mathbb{N}}$ lie dense in $\mathbb{R}_{>1}$? Or weaker, can anything else be said about the set of its accumulation points? $\endgroup$ – RP_ Jan 3 '18 at 18:34

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