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Math Help - Random Series Question

  1. #1
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    Random Series Question

    Hello there,
    My friends and I were talking and we have been wondering the following:
    If the absolute value of a series converges, does the series converge?
    This came up when we were talking about alternating series'.

    Thanks,
    Rob
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  2. #2
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    Re: Random Series Question

    Quote Originally Posted by RobertXIV View Post
    Hello there,
    My friends and I were talking and we have been wondering the following:
    If the absolute value of a series converges, does the series converge?
    This came up when we were talking about alternating series'
    If this is what you mean: If \sum\limits_{n = K}^\infty  {\left| {{a_n}} \right|} converges then \sum\limits_{n = K}^\infty  { {{a_n}} } also converges.
    Yes that is a theorem.

    But you posted " the absolute value of a series converges". That makes no sense.

    Which is it?
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  3. #3
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    Re: Random Series Question

    Sorry, sorry, I meant the first one. Thanks a lot!
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  4. #4
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    Re: Random Series Question

    Quote Originally Posted by RobertXIV View Post
    Sorry, sorry, I meant the first one. Thanks a lot!

    The first one is the series of absolute values whereas what you posted is the absolute value of a series. You can see the difference.

    Here why it works. Series convergence is all about the convergence of a sequence of partial sums.

    See that 0\le \left| {\sum\limits_{n = 1}^K {{a_n}} } \right| \leqslant \sum\limits_{n = 1}^K {\left| {{a_n}} \right|} .

    Thus if the sequence  { \sum\limits_{n = 1}^K {\left| {{a_n}} \right|} converges so does  {\sum\limits_{n = 1}^K {{a_n}}
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