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Math Help - Absolute convergence of log function

  1. #1
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    Absolute convergence of log function

    How should I show the absolute convergence of \sum \log(1 + a_n)?
    I can't think of a test to use for a log function
    Thanks!
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by cubs3205 View Post
    How should I show the absolute convergence of \sum \log(1 + a_n)?
    I can't think of a test to use for a log function
    Thanks!
    What do you mean by absolute convergence? Do you mean if |a_n| converges? If so just replace a_n by |a_n| in the following, it still holds. First we must have the convergence of \sum{a_n}. If that is true, then by definition \lim_{n\to\infty}a_n=0. So using the limit comparison test we have \lim_{n\to\infty}\frac{\log(1+a_n)}{a_n}. Now letting \varphi=a_n it is clear that since a_n is a null sequence (it covnerges to zero) that As n\to\infty\implies\varphi\to{0} and we are left with \lim_{\varphi\to{0}}\frac{\ln(1+\varphi)}{\varphi}  =1
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