If $\sum_{n=1}^{\infty}a_n$ converges, $\sum_{n=1}^{\infty}b_n$ diverges, but $b_n \not=0 \forall n \in \mathbb{N}$, then $\sum_{n=1}^{\infty} \frac{a_n}{b_n}$ converges? No, I think it's not true. Let $a_n=\frac{1}{n^2}$, $b_n = \frac{1}{n}$. Then it diverges.
If $\lim_{n \to \infty}a_n=0$ and $\sum_{n=1}^{\infty}b_n$ converges, then $\sum_{n=1}^{\infty}a_nb_n$ converges? No, I think it's not true. $a_n=n$, $b_n = \frac{1}{n^2}$, so $\sum a_nb_n$ diverges.