# Series converge?

• November 4th 2012, 03:22 AM
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.
• November 4th 2012, 04:43 AM
girdav
Re: Series converge?
Use Abel transform.
• November 4th 2012, 04:59 AM
Re: Series converge?
Would you please be in detail?
• November 4th 2012, 05:47 AM
girdav
Re: Series converge?
Sorry, it was a misleading hint. You can use Hölder's inequality to see it works if $\sum_{k\geq 0}|a_k|^3<\infty$.
• November 4th 2012, 07:13 AM
hollywood
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