Stumped on \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}

[Nexus[Free-DC]]

This thing has me tearing my hair out:

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

Does \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}} 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?

 

[mathman]

Let a₀=1 and a_n=n for n>0. Both series are the same and diverge.

 

[Nexus[Free-DC]]

Sorry, I guess I wasn't clear enough. Do ALL such series diverge? I already know all series of the form a_n=kn+c do since a_an = k(kn+c)+c=k^2n+kc+c, but that doesn't cover all divergent series.

 

[HallsofIvy]

Doesn't your requirement that a_an make sense require that a_n be an increasing, unbounded, sequence of positive integers- and so any subsequence will diverge?

 

[mathman]

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