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Math Help - Convergence tests for series

  1. #1
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    Convergence tests for series

    Hi I was wondering if anyone could help me with a quick question:
    If both \sum_{n=1}^{\infty }a_{n} and \sum_{n=1}^{\infty }|a_{n}| converge and both sums are equal what can you conclude about the two series?
    If one of \sum_{n=1}^{\infty }a_{n} and \sum_{n=1}^{\infty }|a_{n}| converges and the other one diverges, which converges and which diverges?
    Thanks
    Last edited by Dragonkiller; August 24th 2012 at 04:12 AM.
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  2. #2
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    Re: Convergence tests for series

    Quote Originally Posted by Dragonkiller View Post
    Hi I was wondering if anyone could help me with a quick question:
    If both \sum_{n=1}^{\infty }a_{n} and \sum_{n=1}^{\infty }|a_{n}| converge and both sums are equal what can you conclude about the two series?
    If one of \sum_{n=1}^{\infty }a_{n} and \sum_{n=1}^{\infty }|a_{n}| converges and the other one diverges, which converges and which diverges?
    There is a standard theorem that says: If a series converges absolutely then it converges conditionally.
    i.e. If \sum_{n=1}^{\infty }|a_{n}| converges then \sum_{n=1}^{\infty }a_{n} converges.

    Now that answers both questions, but how?
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