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Math Help - Series converge?

  1. #1
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    Series converge?

    Suppose the series \sum_{n=1}^\infty a_n^3 converges, whether or not the series \sum_{n=1}^\infty a_n/n converge?

    I think it does not necessarily converge, but I could not find a counterexample. Would you help me? Thank you.
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  2. #2
    Super Member girdav's Avatar
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    Re: Series converge?

    Use Abel transform.
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  3. #3
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    Re: Series converge?

    Would you please be in detail?
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  4. #4
    Super Member girdav's Avatar
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    Re: Series converge?

    Sorry, it was a misleading hint. You can use Hölder's inequality to see it works if \sum_{k\geq 0}|a_k|^3<\infty.
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  5. #5
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    Re: Series converge?

    Yes, so a counterexample would have to have to converge conditionally.

    I think you might be able to prove that it converges, though. If the first series converges, then \lim_{n\rightarrow\infty}a_n=0. Can you use this to show that the second series converges?

    - Hollywood
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