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Math Help - conceptual summation problem

  1. #1
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    conceptual summation problem

    Let \sum_{k=1}^{\infty}a_n be a convergent series.

    Let |\frac{a_{n+1}}{a_n}| < r < 1 , n \in N.

    Let (b_n) be a sequence such that |b_n|< M.

    Prove that \sum_{k=1}^{\infty}a_n b_n is convergent.

    ---------

    I see here that

    |\frac{a_{n+1}}{a_n}| < r < 1 looks like the ratio test.

    (b_n) has upperbound M.


    How do I go about this proof?
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  2. #2
    MHF Contributor
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    There is already a proof about this situation:

    Abel's test - Wikipedia, the free encyclopedia

    The only thing lacking is b_n being monotonic, but somehow I think that's assumed.
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  3. #3
    Super Member girdav's Avatar
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    We have \left|c_{n+1}\right| =\left|a_{n+1}b_{n+1}\right| \leq rM\left|a_n\right| \leq \cdots \leq r^{n+1}.M
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