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Product of convergent infinite series converges?

  1. Jun 15, 2010 #1
    1. The problem statement, all variables and given/known data

    Given two convergent infinite series such that \sum a_n -> L and \sum b_n -> M, determine if the product a_n*b_n converges to L*M.

    2. Relevant equations



    3. The attempt at a solution

    If know that if a_n -> L this means that the sequence of partial sums of a_n = s_n converges to L. Similarly for the sequence of partial sums of b_n = t_n converges to M. I am not sure how to multiply these two sequences of partial sums.
     
  2. jcsd
  3. Jun 16, 2010 #2
    A_n = a_0 + a_1 + ... + a_n
    B_n = b_0 + b_1 + ... + b_n

    (A_n)(B_n) = (a_0 + a_1 + ... + a_n) (b_0 + b_1 + ... + b_n)
    = sum of i=0 to n (inner sum of j = 0 to n) a_i b_j

    Sorry I don't know how to use latex on this forum.
    Does that help with multiplying the sequences?

    so for example A_1 * B_1 = (a_0 + a_1) ( b_0 + b_1) = a_0b_0 + a_0b1 + a_1b_0 + a_1b_1
     
  4. Jun 16, 2010 #3
    OK, that makes sense. Unfortunately, I still have no idea how to start this proof. I know that I have to do an epsilon proof that the limit is L*M, but it seems like I am going to need something more than that.
     
  5. Jun 16, 2010 #4
    Maybe I'm wrong but I'll throw my idea at you=)

    [tex]\sum_{k=0}^\infty a_k \mbox{\Rightarrow\ \forall\ \epsilon>0\ \exists\ N_1>0\ so\ \forall\ m>n>N_1, } |\sum_{k=n+1}^{m} a_k|< \epsilon \mbox{ and } |\sum_{k=0}^\infty a_k-L|<\epsilon[/tex]

    [tex]\sum_{k=0}^\infty b_k \mbox{\Rightarrow\ \forall\ \epsilon>0\ \exists\ N_2>0\ so\ \forall\ m>n>N_2, } |\sum_{k=n+1}^{m} b_k|< \epsilon \mbox{ and } |\sum_{k=0}^\infty b_k-M|<\epsilon[/tex]

    [tex]\mbox{So if I'm right, after } N> \max \{N_1, N_2\} \mbox{ good things should happen. =) } [/tex]
     
  6. Jun 16, 2010 #5
    But if I have a geometric series, say \sum (1/2^n) - > 2 and another geometric series, say \sum (3/5)^n -> 5/2. Now say I multiply these two series giving \sum ((3/10)^n). The product of the series converges, but it converges to 10/7 which is different from 2(5/2), so the conjecture cannot be proved. Thanks for your help!
     
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