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Can somebody look over my proof? (Asymptotic notation and algorithms)

  1. Jul 16, 2012 #1
    Hey. I'm self studying my way through an algorithms book, and one of the questions at the end of a section is to prove that [itex] log(n!) \in \Theta (n log(n))[/itex]. I wrote up what I believe is a valid proof. However, I don't have much experience writing formal proofs (and even less experience with having my proofs corrected by professionals), so it could very well be invalid/lingered with errors. Can somebody take a quick look at it and comment on any mistakes that may exist? Can you comment on my style? Any advice would be appreciated.

    Here is my write-up. I apologize for not using the built-in equation editor.

    http://i.imgur.com/iyAto.png
     
  2. jcsd
  3. Jul 17, 2012 #2
    I think Stirling's formula is unnecessary here. An asymptotic expression for [itex]\log(n!)[/itex] can be obtained as follows:
    [tex]\log(n!)=\sum_{k=1}^{n}\log(k) \sim \int_{1}^{n}\log(x)dx[/tex]
    The integral on the right side can be evaluated using integration by parts:
    [tex]\int \log(x)dx = x\log(x) - \int dx = x\log(x)-x+C[/tex]
    and this yields
    [tex]\log(n!) \sim n\log(n)-n \sim n\log(n)[/tex]
    Can you take it from there?
     
  4. Jul 17, 2012 #3
    That wouldn't be rigorous, would it? In my proof, I kept the error terms and treated them rigorously.

    EDIT: The point of the problem is to proof the asymptotic behavior of the function, so it would be self defeating to use its asymptotic form. Please correct me if I am wrong, though.
     
    Last edited: Jul 17, 2012
  5. Jul 17, 2012 #4

    AlephZero

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    You can get the error bounds from post #2.
    $$\log(n!) = \sum_{k=2}^n \log k$$
    I changed the lower limit since ##\log 1 = 0##.

    Because ##\log## is a monotonic function you can bracket the sum by two integrals
    $$\int_2^n \log (x-1)\,dx < \sum_{k=2}^n \log k < \int_2^n \log x\,dx$$

    Draw a picture of the integrals and the sum as a set of rectangles, if you can't see where that came from.
     
  6. Jul 17, 2012 #5
    I see! That definitely looks like a valid proof, too. The book that I'm using mentioned bounding functions using integrals, but for some reason told me to use Sterling's formula to prove the theorem.
     
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