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Infinitie Series: Converge/Diverge question

  1. Mar 28, 2005 #1
    this question doesn't look hard when i read it, but then when i tried doing it, i can't :frown:

    Suppose [tex]\sum[/tex]Ak and [tex]\sum[/tex]Bk are both convergent series with positive terms. Does it follow that [tex]\sum[/tex]AkBk converges? Show your answer is correct with an appropriate proof/ counter example.

    Thanks for help.
     
  2. jcsd
  3. Mar 28, 2005 #2

    matt grime

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    Yes, it converges. Here's the hint:

    since sum A_n converges, A_n tends to zero, hence for all k greater than some N, 0<=A_n< 1

    now apply some comparison test to reach the conlcusion.


    to make it more interesting how about:

    Secondly, use the Cauchy schwartz inequality to prove the result (hint, you'll need to show sum (A_n)^2 exists.

    Thirdly, show that if we drop the positivity conditions on the A_n and B_m then the result is false (hint alternating series test - if C_m is a sequence of positive terms tending to zero then sum (-1)^mC_m converges)
     
  4. Mar 28, 2005 #3

    mathman

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    Since the tails of both the A series and the B series must ->0, the tail of the product series goes to 0 faster. To put it rigorously, there exists a K such that for all k>K, AkBk<Ak, therefore the series of products must converge.
     
  5. Mar 28, 2005 #4

    matt grime

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    You're not using the positivity condition, which is required. Though that is a small point.
     
  6. Mar 29, 2005 #5

    mathman

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    (matt grime) I presume you remark was addressed to me. I didn't use the words. However, I know the positive condition was necessary, and I didn't bother using | | for the terms involved.
     
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