Problem

Part I Multiple Choice

(A) If both $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ are absolutely convergent, then $\sum_{n=1}^\infty a_nb_n=\left(\sum_{n=1}^\infty a_n\right)\cdot\left(\sum_{n=1}^\infty b_n\right)$.

(B) If $\sum_{n=1}^\infty a_n$ is convergent and $\sum_{n=1}^\infty b_n$ is divergent, then $\sum_{n=1}^\infty (a_n+b_n)$ must be divergent.

(C) Suppose that $f(x)$ is a positive and continuous function on $[1,\infty)$, and the improper integral $\int_1^\infty f(x)\,\mathrm{d}x$ is convergent. Let $a_n=f(n)$, then $\sum_{n=1}^\infty a_n$ is convergent.

(D) If $a_n\le b_n$ for all $n\in\mathbb{N}$ and $\sum_{n=1}^\infty b_n$ is convergent, then $\sum_{n=1}^\infty a_n$ must be convergent.