After getting stuck with the previous positivity (it probably sounds too complex), I would like to give a version of the problem which is of most interest to me.

Consider a sequence of real numbers
$a_1,a_2,\dots,a_n,\dots$ with absolute values bounded above by the first term $a_1=a>0$,
which satisfies, for all $n=1,2,\dots$,
$$
|A_n|\le A \qquad\text{where}\quad A_n=a_1+a_2+\dots+a_n.
$$
In addition, assume that *infinitely many terms of the sequence are nonzero*.
These settings and Dirichlet's convergence test
guarantee that the series
$$
\sum_{n=1}^\infty\frac{a_n}n
$$
converges.

Assume, in addition, that $$ \max_{1\le k\le n}A_k+\min_{1\le k\le n}A_k\ge0 \qquad\text{for all}\quad n=1,2,\dots. $$

The problem is to show that $$ \sum_{n=1}^\infty\frac{a_n}n>0 $$ and to provide, in terms of $a$ and $A$, a lower (strictly positive) bound for the series. (The latter is optional, as I am not sure that such a bound exists.)