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Math Help - Series theorem proof

  1. #1
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    Series theorem proof

    This is a proof from Principles of mathematical analysis. A series of non negative decreasing terms a_k converges iff the series 2^k*a_2^k converges. I need to show their sequence of partial sums are either both bounded or both unbounded. What I don't understand is he lets n<2^k but does he mean this is for k=n or something else?

    Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Series theorem proof

    Quote Originally Posted by Duke View Post
    This is a proof from Principles of mathematical analysis. A series of non negative decreasing terms a_k converges iff the series 2^k*a_2^k converges. I need to show their sequence of partial sums are either both bounded or both unbounded. What I don't understand is he lets n<2^k but does he mean this is for k=n or something else?

    Thanks
    Oh yeah, Rudin gives a bad proof for that, it's a little sloppy. This theorem is known as Cauchy's Condensation test, a proof can be found here.
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  3. #3
    Super Member girdav's Avatar
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    Re: Series theorem proof

    I don't know what the n you talk about is.
    To show the result, notice that if \sum_{k=0}^{+\infty} a_k is convergent then \sum_{n=0}^{+\infty}\sum_{k=2^n}^{2^{n+1}-1}a_k is convergent (it works because the a_k are non-negative).
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  4. #4
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    Re: Series theorem proof

    Thanks for that proof drexel
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  5. #5
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    Re: Series theorem proof

    I would still like to know how rudin did it. He does very slick proofs.
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