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Math Help - Are the sum and/or product of two divergent sequences divergent?

  1. #1
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    Are the sum and/or product of two divergent sequences divergent?

    As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Are the sum and/or product of two divergent sequences divergent?

    Quote Originally Posted by CountingPenguins View Post
    As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?

    S_1=1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2n+1}

    S_2=-(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{  2n})

    \lim_{n\to\infty}(S_1+S_2)=\ln2
    Last edited by Also sprach Zarathustra; June 29th 2011 at 02:13 AM.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Are the sum and/or product of two divergent sequences divergent?

    Quote Originally Posted by CountingPenguins View Post
    As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?
    EDIT: Oooops... you asked for sequences and I give you a series...(But it can work as well)

    \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}} is converges.

    But, \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}}\cdot\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}} isn't.
    Last edited by Also sprach Zarathustra; June 28th 2011 at 10:28 PM.
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    Re: Are the sum and/or product of two divergent sequences divergent?

    Edit: disregard this. The left forum sidebar protruded over the formula for S_2 and obscured the minus.

    Quote Originally Posted by Also sprach Zarathustra View Post
    S_1=\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2n+1}

    S_2=-(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{  2n})

    \lim_{n\to\infty}(S_1+S_2)=\ln2
    Isn't S_1+S_2 the harmonic series? Then it does not converge.
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Are the sum and/or product of two divergent sequences divergent?

    Quote Originally Posted by emakarov View Post
    Edit: disregard this. The left forum sidebar protruded over the formula for S_2 and obscured the minus.

    Isn't S_1+S_2 the harmonic series? Then it does not converge.
    Actually I forgot '1' in the sum S_1... (I'll fix it in my post)
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