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Math Help - Convergence/divergence of the series (a_i / (1+a_i))

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    Convergence/divergence of the series (a_i / (1+ia_i))

    I need help trying to determine the convergence/divergence of \sum_{i=1}^{\infty} \frac{a_i}{1 + ia_i}, a_i>0. I thought about comparison test, but I can't find anything to compare this series to.

    EDIT: Might integrals help?
    Last edited by phys251; April 24th 2013 at 02:23 PM.
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    Re: Convergence/divergence of the series (a_i / (1+a_i))

    Quote Originally Posted by phys251 View Post
    I need help trying to determine the convergence/divergence of \sum_{i=1}^{\infty} \frac{a_i}{1 + ia_i}, a_i>0. I thought about comparison test, but I can't find anything to compare this series to.
    The series in the title is different from from the one in the post.
    Which is correct?
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    Re: Convergence/divergence of the series (a_i / (1+ia_i))

    Quote Originally Posted by Plato View Post
    The series in the title is different from from the one in the post.
    Which is correct?
    The post, not the title. a_i / (1 + a_i) is pretty easy to deal with. The other form, a little tougher.
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    Re: Convergence/divergence of the series (a_i / (1+ia_i))

    Quote Originally Posted by phys251 View Post
    The post, not the title. a_i / (1 + a_i) is pretty easy to deal with. The other form, a little tougher.

    There is no clear answer, \sum_{i=1}^{\infty} \frac{a_i}{1 + ia_i}, a_i>0

    If a_i=\frac{1}{i} it clearly diverges.

    If a_i=\frac{1}{2^i} it clearly converges.

    Have you given the entire question?
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    Re: Convergence/divergence of the series (a_i / (1+a_i))

    I am also given that \sum_{i=1}^{\infty} a_i diverges. No information on a_i except that a_i is always positive.
    Last edited by phys251; April 24th 2013 at 02:55 PM.
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