# Math Help - Series converge?

1. ## Series converge?

Suppose the series $\sum_{n=1}^\infty a_n^3$ converges, whether or not the series $\sum_{n=1}^\infty a_n/n$ converge?

I think it does not necessarily converge, but I could not find a counterexample. Would you help me? Thank you.

2. ## Re: Series converge?

Use Abel transform.

3. ## Re: Series converge?

Would you please be in detail?

4. ## Re: Series converge?

Sorry, it was a misleading hint. You can use Hölder's inequality to see it works if $\sum_{k\geq 0}|a_k|^3<\infty$.

5. ## Re: Series converge?

Yes, so a counterexample would have to have to converge conditionally.

I think you might be able to prove that it converges, though. If the first series converges, then $\lim_{n\rightarrow\infty}a_n=0$. Can you use this to show that the second series converges?

- Hollywood