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Math Help - Unconditional Convergence

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Unconditional Convergence

    Is there a series that's unconditionally convergent but not absolutely convergent?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by chiph588@ View Post
    Is there a series that's unconditionally convergent but not absolutely convergent?
    Isn't that the defintion. A series is conditionally if it's convergent but it's modulus isn't. So, if it's unconditionally convergent either (I would assume) that it doesn't converge at all or it's absolutely convergent.
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    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Drexel28 View Post
    Isn't that the defintion. A series is conditionally if it's convergent but it's modulus isn't. So, if it's unconditionally convergent either (I would assume) that it doesn't converge at all or it's absolutely convergent.
    That's what I thought, but I've only seen that absolutely convergent implies unconditional convergence.
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Wait a minute: "Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general".

    This is from Wikipiedia.

    I'm really curious to see a series with this property.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by chiph588@ View Post
    Wait a minute: "Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general".

    This is from Wikipiedia.

    I'm really curious to see a series with this property.
    http://en.wikipedia.org/wiki/Unconditional_convergence
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    Quote Originally Posted by chiph588@ View Post
    Is there a series that's unconditionally convergent but not absolutely convergent?
    The two properties are equivalent in R and in finitely-dimensional spaces.

    Try X=\ell_2 and the series x_n=\frac 1{\sqrt{n}}e_n, where e_n denotes the sequence which has one on n'th place and all other terms are zeros.

    If I remember correctly, this is precisely the most basic example given in the book
    Kadets, Kadets: Series in Banach spaces: conditional and unconditional convergence
    which is devoted basically to this topic.

    However, many texts on functional analysis mention the relationship between various modes of convergence in Banach spaces.
    I remember I have seen this in Wojtaszczyk's Banach spaces for analysts (with proofs) and in Megginson's Intorduction to Banach spaces theory (as an exercise - if I remember correctly).
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