Series problem

  • #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?
 

Answers and Replies

  • #2
mathman
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Let a0=1 and an=n for n>0. Both series are the same and diverge.
 
  • #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.
 
  • #4
HallsofIvy
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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?
 
  • #5
mathman
Science Advisor
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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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