If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges? I know if $\sum{a_n}$ converges absolutely then $\sum{\frac{a_n}{n}}$ converges since $0\le \frac{|a_n|}{n} \le |a_n| $ for all $n$ so it converges absolutely by the basic comparison test and therefore converges. However, I cannot prove the convergence of $\sum \frac{a_n}{n}$ if $\sum{a_n}$ converges but not absolutely even though I suspect it to be true. Can you give me a proof or a counterexample for this?

Obviously, this conclusion is correct.
In mathematical analysis, the Abel discriminant of the series of real terms is described as follows.

If $a_n$ is monotonous and $\sum b_n$ converges, $\sum b_na_n$ converges.

