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Thread: An inverse question on Cauchy product of series

  1. #1
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    An inverse question on Cauchy product of series

    Given two series \sum\limits_{n = 0}^\infty a_n, \sum\limits_{n = 0}^\infty b_n, and their Cauchy product \sum\limits_{n = 0}^\infty c_n where c_n=\sum\limits_{k = 0}^n a_kb_{n-k}, if series \sum c_n converges, dose series \sum a_n or \sum b_n converge too? Thanks!
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  2. #2
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    No, you don't know that both \sum a_n and \sum b_n converge. Take a_n=1 and b_n=0.
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by putnam120 View Post
    No, you don't know that both \sum a_n and \sum b_n converge.
    That does not answer the question :
    Quote Originally Posted by zzzhhh
    does series \sum a_n or \sum b_n converge too?
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  4. #4
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    then is it possible that both \sum a_n and \sum b_n diverge?
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  5. #5
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    I don't think it is possible for both \sum a_n,\sum b_n to diverge. This is clear if one of the sequences has only positive terms (or similarly only negative terms). For the case with mixed terms try violating a condition for the alternating test. I haven't worked out all the details but think that might work. I'll let you know I have figured out more.
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  6. #6
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    From the wonderful book Counterexamples in analysis, an example of two (very!) divergent series whose Cauchy product is absolutely convergent. Define a_0=-1, a_n=1 for n≥1. Define b_0=2, b_n=2^n for n≥1. Then it's easy to check that c_0=-2 but c_n=0 for n≥1.
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  7. #7
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    what an ingenious construction, must by Cauchy himself :-)
    Thank u Opalg, as well as the great book you mentioned. thank you all!
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