A Question about the Alternating Series Test

  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. Stephen Tashi

    Stephen Tashi 4,411
    Science Advisor
    2014 Award

    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. 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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