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Thread: Does the converse hold?

  1. #1
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    Does the converse hold?

    Suppose that both series $\displaystyle \Sigma a_k $ and $\displaystyle \Sigma b_k $ are absolutely convergent. Show that then so too is the series $\displaystyle \Sigma a_k b_k $. Does the converse hold?
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    Consider $\displaystyle a_k= b_k= \frac{(-1)^k}{k}$.
    Last edited by Jhevon; Sep 25th 2010 at 10:45 PM. Reason: fixed LaTeX
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by rondo09 View Post
    Suppose that both series $\displaystyle \Sigma a_k $ and $\displaystyle \Sigma b_k $ are absolutely convergent. Show that then so too is the series $\displaystyle \Sigma a_k b_k $. Does the converse hold?
    Yust for clarity's sake: are You asking if, given a series $\displaystyle \Sigma a_{k} b_{k}$ that absolutely converges, the same holds for $\displaystyle \Sigma a_{k}$ and $\displaystyle \Sigma b_{k}$ ?...

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  4. #4
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    Yes, the "converse" of "if A then B" is "if B then A".
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    MHF Contributor chisigma's Avatar
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    ... very well!... what about $\displaystyle \displaystyle a_{k} = \frac{1}{k^{2}}$ and $\displaystyle \displaystyle b_{k} = \sin k$?... the series $\displaystyle \displaystyle \sum_{k=1}^{\infty} \frac{\sin k}{k^{2}}$ converges absolutely... is the same also for $\displaystyle \displaystyle \sum_{k=1}^{\infty} \sin k$ ?...

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    $\displaystyle \chi$ $\displaystyle \sigma$
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  6. #6
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    Quote Originally Posted by chisigma View Post
    Yust for clarity's sake: are You asking if, given a series $\displaystyle \Sigma a_{k} b_{k}$ that absolutely converges, the same holds for $\displaystyle \Sigma a_{k}$ and $\displaystyle \Sigma b_{k}$ ?...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    What am I asking if there are series $\displaystyle \Sigma a_{k}$ and $\displaystyle \Sigma b_{k}$ which are absolutely convergent, then show that the series $\displaystyle \Sigma a_{k}b_{k}$ is also absolutely convergent. Then, P.S, does the converse hold?
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  7. #7
    MHF Contributor chisigma's Avatar
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    If $\displaystyle \Sigma a_{k}$ and $\displaystyle \Sigma b_{k}$ absolutely converge then is $\displaystyle \displaystyle \lim_{k \rightarrow \infty} |a_{k}| = \lim_{k \rightarrow \infty} |b_{k}| = 0$ , so that $\displaystyle \forall k>K$ is $\displaystyle \displaystyle |a_{k}\ b_{k}| < |a_{k}|$ or also $\displaystyle \displaystyle |a_{k}\ b_{k}| < |b_{k}|$ and the series $\displaystyle \Sigma a_{k}\ b_{k}$ converges...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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