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Math Help - Help classifying a series

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    8

    Help classifying a series

    I need to classify the following series:

    \sum_{n=n_0}^\infty\sqrt{a_n}

    with (a_n)\ge0 and \sum_{n=n_0}^\infty a_n \mathbb{C}

    I guess the question on the whole is whether i can classify any of the following series:

    \sum_{n=n_0}^\infty (a_n)^r , \forall r\in\mathbb{Z}

    based on \sum_{n=n_0}^\infty a_n being convergent

    As far as I can tell (using the Comparison Test), \sum_{n=n_0}^\infty (a_n)^r will converge if r > 1, but what if r < 1?

    Thanks in advance.
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  2. #2
    Newbie
    Joined
    Aug 2010
    Posts
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    Hi,
    if r<0, it never converges
    if 1>r>0, sometimes it converges, sometimes not...
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