If they are two general geometric progression series, then you will get a mess as each term has to be distributed out across the sum. Each term that is distributed across the sum will be itself a geometric progression. (a+ar+ar^2+..ar^n.)(b+bk+bk^2+...bk^n) a(b+bk+bk^2+...)+ar(b+bk+bk^2+...+bk^n)+ar^2+(b+bk+bk^2+...+bk^n)+...+ar^n((b+bk+bk^2+...+bk^n) The first term in the sum will be itself a geometric series, first term ab and common ratio k, the second another geometric series, first term abr and common ratio again k, and so on.
So, I suppose the answer is that you will get a series of geometric series unfortunately not something simple, as the summation operator is linear: the rules exist for summing a variable times a constant and summing over a sum, but sadly there is no simple rule for when you multiply two sums together.