Series converge?

Jun 2011
73
2
Suppose the series \(\displaystyle \sum_{n=1}^\infty a_n^3\) converges, whether or not the series \(\displaystyle \sum_{n=1}^\infty a_n/n\) converge?

I think it does not necessarily converge, but I could not find a counterexample. Would you help me? Thank you.
 
Jul 2009
678
241
Rouen, France
Use Abel transform.
 
Jul 2009
678
241
Rouen, France
Sorry, it was a misleading hint. You can use Hölder's inequality to see it works if \(\displaystyle \sum_{k\geq 0}|a_k|^3<\infty\).
 
Mar 2010
1,055
290
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 \(\displaystyle \lim_{n\rightarrow\infty}a_n=0\). Can you use this to show that the second series converges?

- Hollywood