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Converges Proof

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

    If [tex]\sum_{k=1}^{\infty} a_k[/tex] converges and [tex]a_k/b_k \to 0[/tex] as [tex]k\to \infty[/tex], then [tex]\sum_{k=1}^{\infty} b_k[/tex] converges.

    2. Relevant equations
    It is true or false.

    3. The attempt at a solution
    I think it is false and here is my counterexample. Let [tex]a_k = 0,b_k=\frac{1}{k}[/tex]. This satisfies our initial conditions of [tex]\sum_{k=1}^{\infty} a_k[/tex] converges and [tex]a_k/b_k \to 0[/tex] as [tex]k\to \infty[/tex] but [tex]\sum_{k=1}^{\infty} b_k[/tex] diverges.
    Is this correct?
    Last edited: Apr 28, 2010
  2. jcsd
  3. Apr 28, 2010 #2


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    Science Advisor
    Homework Helper

    Looks okay.

    Your counterexample also looks correct, if you want to make it slightly less trivial you could use
    [tex]a_k = \frac{1}{k^2}[/tex]
    instead :)
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