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Series problem

  1. Apr 29, 2004 #1
    This thing has me tearing my hair out:

    Let {a0, a1,...} be a sequence such that
    [tex]\sum_{n=0}^{\infty}{\frac{1}{a_{n}}}[/tex] diverges.

    Does [tex]\sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}[/tex] diverge?

    My first instinct was to say no, but then I couldn't find any counterexamples. Now I am thinking it might actually be true but it has defied all the tests I've tried. Any ideas?
  2. jcsd
  3. Apr 29, 2004 #2


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    Let a0=1 and an=n for n>0. Both series are the same and diverge.
  4. Apr 29, 2004 #3
    Sorry, I guess I wasn't clear enough. Do ALL such series diverge? I already know all series of the form an=kn+c do since aan = k(kn+c)+c=k^2n+kc+c, but that doesn't cover all divergent series.
  5. Apr 30, 2004 #4


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    Doesn't your requirement that aan make sense require that an be an increasing, unbounded, sequence of positive integers- and so any subsequence will diverge?
  6. Apr 30, 2004 #5


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    If an=n2, both series converge. It looks like it would be hard to construct an example where the first diverges and second converges.
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