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Can anyone help me demonstrate these two statements?

$$ n! = o(n/2)^n $$

$$ n! = o(n/3)^n $$

I am sure about the first one but I don't know how to demonstrate it. As for the second one I am not sure whether it is true or not.

David Richerby
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dan369
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  • Well, since factorial is only defined for integers, you could do something along the lines of nn-1n-2...21 always being less than nnn...n*n, and then take a log base n of both sides, then use the change of base formula to get rid of that pesky division that's in there. – childofsoong Mar 19 '15 at 19:49
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    They're not statements about time complexity: they're statements about the growth rate of functions. Those functions could be used to measure anything at all. – David Richerby Mar 19 '15 at 21:43
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    What have you tried? Where did you get stuck? We want to help you with your specific problems, not just do your exercise for you. However, as it is we don't know what this problem is and thus how to help. See here for a relevant discussion. – D.W. Mar 19 '15 at 21:45
  • Yes, you're right, it's just growth rate of functions. So far I tried using induction, but I got to a point that I had to demonstrate this: $$ \left(\frac{n}{2}\right)^n(n+1) \leq \left(\frac{n+1}{2}\right)^{(n+1)} $$ And I did not know how to demonstrate it. I tried using Stirling's formula and apply the ln function to both sides of the disequation but I could not solve that either. But now I just tried with limits: $$ \lim_{n\to\infty} \frac{n!}{\left(\frac{n}{2}\right)^n} $$ And using Stirling's formula is much easier!! Thank you for linking that page!! – dan369 Mar 20 '15 at 10:43

1 Answers1

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These statements have nothing to do with time complexity. They have everything to do with asymptotics.

Hint: Use Stirling's approximation: $$ n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n. $$ Use the fact that $2 < e < 3$.

Yuval Filmus
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