# Series problem

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?

## Answers and Replies

mathman
Science Advisor
Let a0=1 and an=n for n>0. Both series are the same and diverge.

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.

HallsofIvy
Science Advisor
Homework Helper
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?

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