If two real series $\sum a_n$ and $\sum b_n$ converge but $\sum a_n b_n$ does not can $\sum a_n$ or $\sum b_n$ converge absolutely?

Question

If two real series $\sum a_n$ and $\sum b_n$ converge but $\sum a_n b_n$ does not can one of $\sum a_n$ or $\sum b_n$ converge absolutely?

I can show that if $\sum a_n$ converges absolutely then the partial sums of the series $a_n b_n$ are bounded, but this is equivalent to saying that the series converges only when all the terms are positive, but that is not the case if $b_n<0$.

Obviously both cannot converge absolutely because then $a_n b_n$ converges absolutely and so the series itself converges.

Answer 1

No. This is impossible. If $\sum a_{n}$ is convergent, and $\sum b_{n}$ is absolutely convergent, then $\sum a_{n}b_{n}$ converges absolutely.

Since $\sum a_{n}$ converges, then $\lim a_{n}=0$. So, the sequence $(a_{n})$ is bounded and there exists $c\geq 0$ such that $|a_{n}|\leq c$ for all $n\geq 1$. Therefore $$\sum |a_{n}b_{n}|\leq c \sum |b_{n}|$$ and we are done by the comparison test. No need to worry about the constant $c$. We could start off looking at the series $\sum \frac{1}{c} a_{n}b_{n}$ and get $$\sum \frac{1}{c}|a_{n}b_{n}|\leq \sum |b_{n}|$$ and notice that multiplication by constant does not affect the convergence.