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Math Help - Big Oh question

  1. #1
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    Big Oh question

    I am trying to prove an idientity I have come accross in a book:

    \log\left(\log t-\log\log t+O\left(\frac{\log\log t}{\log t}\right)\right)=\log\log t-\frac{\log\log t}{\log t}-\frac{1}{2}\left(\frac{\log\log t}{\log t}\right)^{2}+O\left(\frac{\log\log t}{\left(\log t\right)^{2}}\right)

    I have managed to prove,

    \log\left(\log t+O\left(\log\log t\right)\right)=\log\log t+O\left(\frac{\log\log t}{\log t}\right)

    using various facts such as

    • O\left(O\left(f\left(z\right)\right)\right)=O\left  (f\left(z\right)\right),
    • g\left(z\right)O\left(f\left(z\right)\right)=O(g(z  )f(z))
    • and \log z=O\left(z\right).

    But I can't prove the above more difficult identity. Can anyone help me with this?
    Thanks for reading.
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  2. #2
    Grand Panjandrum
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    Re: Big Oh question

    Quote Originally Posted by aukie View Post
    I am trying to prove an idientity I have come accross in a book:

    \log\left(\log t-\log\log t+O\left(\frac{\log\log t}{\log t}\right)\right)=\log\log t-\frac{\log\log t}{\log t}-\frac{1}{2}\left(\frac{\log\log t}{\log t}\right)^{2}+O\left(\frac{\log\log t}{\left(\log t\right)^{2}}\right)

    I have managed to prove,

    \log\left(\log t+O\left(\log\log t\right)\right)=\log\log t+O\left(\frac{\log\log t}{\log t}\right)

    using various facts such as

    • O\left(O\left(f\left(z\right)\right)\right)=O\left  (f\left(z\right)\right),
    • g\left(z\right)O\left(f\left(z\right)\right)=O(g(z  )f(z))
    • and \log z=O\left(z\right).

    But I can't prove the above more difficult identity. Can anyone help me with this?
    Thanks for reading.
    I think you need to observe that for large enough t then \log(t)\gg \log(\log(t)) +O\left(\frac{\log\log t}{\log t}\right) so:

    \log\left(\log (t)-\log\log (t)+O\left(\frac{\log\log (t)}{\log (t)}\right)\right)=(\log \log (t)) + \log\left(1+\frac{-\log\log (t)+O\left(\frac{\log\log (t)}{\log (t)}\right)}{\log(t)}\right)

    and now expand the big log on the right as a power series.

    CB
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  3. #3
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    Lightbulb Re: Big Oh question

    This is precisely the part I was stuck on, was not sure how to deal with all terms that arise from the 3rd order expansion of the "big" log term on the right. But then I realized the O(\frac{\log \log t}{(\log t)^2}) term absorbs most of them.
    Last edited by aukie; November 23rd 2011 at 01:03 PM.
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