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Math Help - Infinite Series Problem

  1. #1
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    [SOLVED]Infinite Series Problem

    show that if the seriesΣuk and Σvk both diverge, then seiresΣ(uk + vk) and Σ(uk - vk) may either converge or diverge

    Thank you !
    Last edited by soleilion; April 5th 2008 at 02:43 PM.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    that is completely ambiguous

    I am sorry but if youd like me to answer you must specify do you mean \sum_{u,k=0}^{\infty}v\cdot{k} and \sum_{u,k=0}^{\infty}u\cdot{k} or do you mean \sum_{k=0}^{\infty}u(k) and \sum_{k=0}^{\infty}v(k)? or what?
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  3. #3
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    and

    this one
    thank you
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  4. #4
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    Quote Originally Posted by soleilion View Post
    show that if the seriesΣuk and Σvk both diverge, then seiresΣ(uk + vk) and Σ(uk - vk) may either converge or diverge
    Consider: u_k  = \frac{1}{k}\,\& \,v_k  = \frac{{ - 1}}{k}
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  5. #5
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    Quote Originally Posted by Plato View Post
    Consider: u_k = \frac{1}{k}\,\& \,v_k = \frac{{ - 1}}{k}

    your examples are both diverges by p-series

    then, if they plus, Σ(uk + vk)=0 converges

    if they minus Σ(uk + vk)=2/k, still diverges

    is my proof right???
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  6. #6
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    Then how about

    v(k)=\frac{1}{\sqrt{k}} and u(k)=\frac{-1}{\sqrt{k}} since both are divergent P-series but their addition is 0 which is convergent...
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  7. #7
    Moo
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    Hello,

    Take the general example of 1/x^a and -1/x^a with a =< 1 (Riemann series)

    You can also take a look at alternated series
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