Let $(a_n)$ be some sequence of real (or maybe even complex) numbers. For which sequences does
$$S=\sum_{n=1}^\infty \frac{a_n}n$$
converge to a finite value?
Let $p$ denote the period of $a_n$, i.e. $a_n = a_{n+p}$ for all $n$.
$\large{p=1}$
The series $S$ will only converge if $a_1=0$ because otherwise, $S$ is the Harmonic series.
$\large{p=2}$
The series will converge if $a_2=-a_1$. Then the series converges to $S= a_1\ln 2$. If $a_2=-a_1+\Delta$ then $S$ can be split into two series, one converging to $a_1\ln 2$, and one diverging like $\Delta$ times the harmonic series.
$\large{p\geqslant3}$
For this case I have only the conjecture that $$S \text{ converges } \quad\iff\quad \sum_{n=1}^p a_n = 0$$ but I have no idea how to proceed?
I have worked this out before. The sum converges if and only if $A=\sum_{k=1}^p a_k = 0 $.
This can be proved by looking at each group of $p$ consecutive terms starting at $n$ and subtracting $\frac{A}{n}$.
I'll work out the details if you want.