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A Question about the Alternating Series Test

  1. Apr 7, 2011 #1
    The definition I am working from is "Let Z=(z(sub n)) be a decreasing sequence of strictly positive numbers with lim(Z)=0. Then the alternating series, Sum(((-1)^n)*Z) is convergent.

    My question is how to solve the following:
    If the hypothesis that Z is decreasing is dropped, show the Alternating Series Test may fail.

    I am aware of a proof utilizing some Z that is also alternating, but this breaks the condition that Z is strictly positive. I am unaware of an such sequence that has a limit of 0, all elements of the series are positive, yet is divergent.

    This question is due within 10 hours. Please help!
     
  2. jcsd
  3. Apr 8, 2011 #2

    Stephen Tashi

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    Look at a sequence whose odd terms are a convergent sequence of positive terms, like a geometric series and whose even terms are the terms of the divergent harmonic series {1/n}. The alternating sequence of negative odd terms and positive even terms should diverge. Intuitively this is because what is begin taken away reaches a limit while what is being added doesn't.

    It will be a slight nuisance to get the indexing written correctly. I think harmonic terms will be [tex] \frac{1}{ (n/2)+1} [/tex] , for [tex] n = 0,2,4,...[/tex]
     
  4. Apr 8, 2011 #3
    Thanks!

    I never really assessed the thought of a limit being reached while another series (running in parallel) continued decreasing. I appreciate the quick input!
     
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