are we assuming that ?
(a) If ∑a_n converges and lim b_n = 0, then ∑(a_n)(b_n) converges.
(b) If ∑b_n converges and lim n-> ∞ (a_n)/(b_n) = 1 then ∑ (a_n) converges.
I'm not totally sure how to do either of these. For (a), my guess is since the lim of b_n exists as a real number, 0, then (a_n)(b_n) are going to converge. Similar for (b).