The problem was as follows: Let $a_n$ be a non-negative series. Assuming $\sum_{n=1}^{\infty}a_n$ converges does
a) $\sum_{n=1}^{\infty}\left(a_n\right)^{\frac{3}{2}}$ converge?
b) $\sum_{n=1}^{\infty}\left(a_n\right)^{\frac{3}{2}}\cdot b_n$ converge, assuming that $b_n$ is some bounded series?
I answered the first one as true since $a_n < \left(a_n\right)^{\frac{3}{2}}$ and $a_n$ converges so $\left(a_n\right)^{\frac{3}{2}}$ should, too. As for b), I answered false since it is missing the $\lim_{x\to \infty }b_n=0$ condition. I have more doubts about the second, though.
Yes they both converge. For the first one use $a_n^3 \le a_n^2 \implies a_n^{\frac{3}{2}} \le a_n$ for some $n \ge N$ because $a_n \le 1$ for some $n \ge N$. The other one converges absolutely since the series $\displaystyle \sum_{n=1}^\infty |a_n^{\frac{3}{2}}||b_n|$ converges.