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Abs. Convergence Proof

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    if [itex]a_k \le b_k[/itex] for all [itex]k \in \mathbb{N}[/itex] and [itex]\sum_{k=1}^{\infty} b_k[/itex] is absolutely convergent, then [itex]\sum_{k=1}^{\infty} a_k[/itex] converges.

    2. Relevant equations
    It's either true or false.

    3. The attempt at a solution
    I think a counterexample to prove it's false is if we let [itex]a_k=-1, b_k = 0[/itex] which satisfies [itex]a_k \le b_k[/itex] and [itex]b_k[/itex] is abs. convergent but [itex]\sum_{k=1}^{\infty} a_k[/itex] diverges.
    Is this a correct counterexample?
  2. jcsd
  3. Apr 27, 2010 #2


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    Homework Helper

    that looks reasonable, if the ak & bk were positive numbers or it was written for maginutudes it would be true
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