Answer by prAnjal for Suppose a series $a_n$ is greater than 0 for all positive integer n, and that $\sum \frac {a_n} n$ converges, then does the following also converge?

let $v_n = \frac{a_n}{m+n}$, and $u_n = \frac{a_n}{n}$. Now observe that $\frac{a_n}{m+n} \le \frac{a_n}{2n}$, so by comparison test, we can say that as each term of $v_n$ is smaller or equal to that of $\frac{u_n}{2}$, and as $u_n$ converges, so $v_n$ also converges as $m \to \infty$.

I think this is it.

:)