If $r_i=r_{i+t}$ for some $t$ and all $i$ you can split the sum into a finite sum and the rest $$\sum_{i=1}^t \frac{r_i}{i}+\sum_{i=t}^\infty \frac{r_i}{i}=\sum_{i=1}^t \frac{r_i}{i}+r_t\sum_{i=t}^\infty \frac{1}{i} $$ by your assumption. So the latter is just the harmonic series and your series diverges unless $r_t=0$.
This reduces your question to a finite question. Now the answer depends on what you call non-trivial. Eventually all $r_i$ have to be 0, but but of course solutions exist. For example $r_i=0$ for all $i \neq 1,2$ and set $r_1=1$ and $r_2=-2$.