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Inverting an Asymptotic Equality

  1. Oct 24, 2011 #1

    jgens

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    Gold Member

    1. The problem statement, all variables and given/known data

    Show that the following two statements are equivalent:
    1. [itex]a_n^2\log{(a_n)} \sim n[/itex]
    2. [itex]a_n \sim \sqrt{\frac{2n}{\log{(n)}}}[/itex]

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I can show [itex](2) \implies (1)[/itex] pretty easily. However, I am having some difficulties with the proof in the other direction. I can get the proof down to the point where [itex]a_n^2\log{(a_n)} = b_n^2\log{(b_n)}[/itex] where [itex]b_n = \sqrt{\frac{2n}{\log{(n)}}}[/itex] but I don't know how to make the final step concluding [itex]a_n \sim b_n[/itex]. Could anyone give me a little help with this?

    Thanks
     
  2. jcsd
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