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Math Help - Problems proving a summation

  1. #1
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    Problems proving a summation

    I have been trying for a week to prove a result from a textbook and have gotten nowhere. Please help.

    I have the facts q^{-\delta} logq $\rightarrow$ $\infty$ for each $\delta$>0 and I'm trying to prove for each \epsilon>0
     \sum_{q=1}^{\infty}((logq)/(q^{1+\epsilon}))<\infty
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  2. #2
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    Re: Problems proving a summation

    Quote Originally Posted by klw289 View Post
    I have been trying for a week to prove a result from a textbook and have gotten nowhere. Please help.

    I have the facts q^{-\delta} logq $\rightarrow$ $\infty$ for each $\delta$>0 and I'm trying to prove for each \epsilon>0
     \sum_{q=1}^{\infty}((logq)/(q^{1+\epsilon}))<\infty
    This is series of tricks. I am going to use easier notation.
    Suppose that p>1 and let r=\frac{p-1}{2}.

    Set some things up first. \log(n)=r^{-1}\log(n^r)\le r^{-1}n^r

    So \dfrac{\log(n)}{n^p}\le\dfrac{n^r}{r\cdot n^p}=\dfrac{1}{r\cdot n^{p-r}}

    If we note that p-r=\frac{p+1}{2}>1.

    So you have a p-series which converges.
    Last edited by Plato; May 5th 2012 at 04:06 PM.
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