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Conv. Conditionally Proff

  1. Apr 28, 2010 #1
    1. The problem statement, all variables and given/known data

    If [tex]a_k[/tex] is decreasing and it's limit is 0 as [tex]k \to \infty[/tex] and [tex]\sum_{k+1}^{\infty} b_k[/tex] converges conditionally, then [tex]\sum_{k=1}^{\infty} a_k b_k[/tex] converges


    2. Relevant equations
    This is true or false.


    3. The attempt at a solution
    I think it is false because if we let [tex]a_k = \frac{1}{\sqrt{k}}, b_k= \frac{(-1)^k}{\sqrt{k}}[/tex] we satisfy our initial conditions but [tex]a_k \cdot b_k = \frac{1}{k}[/tex] so [tex]\sum_{k=1}^{\infty} a_k b_k[/tex] diverges.
    Is this correct?
     
  2. jcsd
  3. Apr 28, 2010 #2

    lanedance

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    i think you missed a (-1)^k in the a_k term?
     
  4. Apr 28, 2010 #3
    that would make [tex]a_k b_k = \frac{(-1)^{2k}}{k}[/tex] which is convergent, I think. I meant what I put but apparently it does not work?
     
  5. Apr 28, 2010 #4

    lanedance

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    isn't it
    [tex]a_k b_k = \frac{(-1)^{2k}}{k} = \frac{((-1)^2)^k}{k} = \frac{1}{k}[/tex]

    i wasn't sure where the alternating negative went in your 1st post...
     
  6. Apr 28, 2010 #5
    You are right, I owned my self by basic algebra :tongue2:
     
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