
Series converge?
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.

Re: Series converge?

Re: Series converge?
Would you please be in detail?

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

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 $\displaystyle \lim_{n\rightarrow\infty}a_n=0$. Can you use this to show that the second series converges?
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