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Let $\sum a_n$ be a convergent series. Does $\sum \frac{(-1)^na_n}{n}$ always converge? Prove or show an example to contradict.

I tried both finding an example/proving it, however – I couldn't manage to prove it as it is not known whether $a_n$ is a positive series, thus the usual convergence tests cannot be applied.

I tried finding an example but couldn't find such. Any hint?

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    $\begingroup$ Counter-example: $a_n=\frac {(-1)^{n}} {\ln (n+1)}$. $\endgroup$
    – Kavi Rama Murthy
    Apr 12, 2021 at 11:26

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Hint: It is much easier for an alternating series to converge than for a series were all the terms have the same sign. And the $(-1)^n$ factor can transform one into the other.

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