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Math Help - Does a series converge if it's absolutely convergent?

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    MHF Contributor Also sprach Zarathustra's Avatar
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    Does a series converge if it's absolutely convergent?

    A stupid one... (just to be sure)

    If I have a series that is converge absolutely so the series is converges?

    (it is implies from Cauchy criterion?)

    Thanks!
    Last edited by mr fantastic; June 28th 2010 at 01:02 AM. Reason: Re-titled.
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  2. #2
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    Quote Originally Posted by Also sprach Zarathustra View Post
    A stupid one... (just to be sure)

    If I have a series that is converge absolutely so the series is converges?

    (it is implies from Cauchy criterion?)

    Thanks!
    If a series is absolutely convergent, then it is convergent.

    This is easily proven using the comparison test.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Prove It View Post
    If a series is absolutely convergent, then it is convergent.

    This is easily proven using the comparison test.

    I think you wrong by saying "This is easily proven using the comparison test."

    I know what you mean, but it's not so true....


    ehmmm...
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    Actually it is.

    Note that for any infinite series \sum{a}, it can never be any greater than the sum of its absolute values.

    So \sum{a} \leq \sum{|a|}.


    Now simply apply the comparison test.


    Note though that this only half proves the theorem, however you can also bound the series below...

    \sum{(-|a|)} \leq \sum{a}

    -\sum{|a|} \leq \sum{a}.



    So we can say

    -\sum{|a|} \leq \sum{a} \leq \sum{|a|}

    and since the series is bounded by the sum of the absolute values, that means if the sum of the absolute values converges, then the series is also convergent.
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