If (i) $\sum a_n$ converges absolutely (ii)$\sum a_n = A$ (iii)$\sum b_n = B$ (iv)$c_n = \sum_{k=0}^n a_k b_{n-k}$, Then $\sum c_n = AB$.
Here, what if (iii) is changed to ' $\sum b_n$ diverges'? If it is inconclusive, please give me an example.
Plus, if $\sum a_n$ and $\sum b_n$ both diverge, does $\sum c_n$ diverge?