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For example, one can derive an approximation of $\pi$ from Stirling's approximation with one additional term as $$ \lim_{n \to \infty} \frac{72n(n!)^2}{n^{2n} e^{-2n} (12n+1)^2} $$

but is it correct to write

$$ \lim_{n \to \infty} \frac{72n(n!^2)}{n^{2n} e^{-2n} (12n+1)^2}? $$

Is the ! sufficient to separate the factoriand (if that's the word) from the exponent? I've seen both used in various places.

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I changed \frac{72n(n!)^2}{{n^{2n}} {e^{-2n}}{(12n+1)^2}} to \frac{72n(n!)^2}{n^{2n} e^{-2n} (12n+1)^2}. The former is hard to read and edit. – Michael Hardy May 1 '13 at 21:57
up vote 3 down vote accepted

This is really a matter of preference. I think either one is fine since $!^2$ has no meaning on its own, so the only way to interpret $n!^2$ is as $(n!)^2$. That being said, I think $(n!)^2$ looks a bit nicer.

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Following from that, are the brackets even necessary after 72n in my formula? – Lee Sleek May 1 '13 at 22:37
I would say yes, since $n n!$ could be interpreted as $n(n!)$ or $(nn)!$. Whenever there's a chance for ambiguity, you should use parentheses. – Alistair Savage May 1 '13 at 23:37
This makes me think that the old notation for factorial (a long division sign rotated 180 degrees, with a line under all the affected math and a horizontal line at the right) should be brought back now that we have modern typesetting. Also, that's precisely why I used all the brackets in the two formulas above, which Michael Hardy edited. – Lee Sleek May 1 '13 at 23:49

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