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Math Help - Proof involving series

  1. #1
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    Proof involving series

    (a) If ∑a_n converges and lim b_n = 0, then ∑(a_n)(b_n) converges.

    (b) If ∑b_n converges and lim n-> ∞ (a_n)/(b_n) = 1 then ∑ (a_n) converges.

    I'm not totally sure how to do either of these. For (a), my guess is since the lim of b_n exists as a real number, 0, then (a_n)(b_n) are going to converge. Similar for (b).
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  2. #2
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    are we assuming that a_n,b_n>0 ?
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  3. #3
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    Yes.
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  4. #4
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    ah well, then limit comparison test will do the job.

    i'll the first one, then you should be able to do the second one.

    since \sum a_n<\infty, we'll apply limit comparison test with a_n, so \lim_{n\to\infty}\frac{a_nb_n}{a_n}=0, so \sum a_nb_n<\infty.
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