Let $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ be two absolutely convergent series such that their values are $s_{1}$ and $s_{2}$ respectively.
If $\sum_{n=1}^{\infty} a_{n}b_{n} = s_{0}$, does a relation exist between $s_{0}, s_{1}$, and $s_{2}$?
No. For any converging series $\sum_{n=1}^\infty a_n$ with not all summands zero and any given $s_2,s_0$, you can find $b_n$ such that $\sum_{n=1}^\infty b_n=s_2$ and $\sum_{n=1}^\infty a_nb_n=s_0$. Just start with any convergent series $\sum b_n$, find $i,j$ such that $a_i\ne a_j$. Then by modifying $b_i,b_j$ you can influence $s_2$ and $s_0$ arbirarily.