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A general convergent series under different circumstances.

  1. Oct 1, 2012 #1
    1. The problem statement, all variables and given/known data

    Ʃan (sum from n=1 to ∞) converges.

    1) Determine whether the series Ʃln(1+an) (sum from n=1 to ∞) converges or diverges. Assume that an>0 for all n.

    2) Show each of the following statements or give a counter-example that establishes that it is false:
    a)Ʃai2 (sum i=1 to ∞) converges if the ai are alternating.
    b)Ʃai2 converges if the ai are non-negative.

    2. Relevant equations



    3. The attempt at a solution

    For the first question (1), if an converges, then the terms go to zero as n goes to ∞. This means that as n goes to ∞ for Ʃln(1+an), the series will diverge to -∞, because ln(0)=-∞. I'm not sure if that's right, and if that's a good enough proof. Any ideas?

    For the second question (2a), if Ʃai converges, then the square should converge, but faster. I'm not sure how it alternating would effect the convergence, since the absolute value of the terms are still getting smaller as i gets bigger.

    (2b) The same as the last one, I don't see how the terms being positive makes a difference, since the series is still converging. It just means that the terms don't dip below zero.

    Not sure if I'm on the right track or not. Any thoughts would be greatly appreciated.

    Thanks!!
     
  2. jcsd
  3. Oct 1, 2012 #2

    jbunniii

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    This isn't right. If [itex]a_n[/itex] goes to 0, then [itex]1 + a_n[/itex] goes to 1, so [itex]\ln(1 + a_n)[/itex] goes to zero. Thus you have not ruled out convergence.

    Consider an alternating series that converges, but not absolutely.

    Hint: How does [itex]a_n^2[/itex] compare with [itex]a_n[/itex] if [itex]0 \leq a_n \leq 1[/itex]?
     
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