Can the product of absolutely converging sequences diverge?

Question

Let $a_n, b_n$ be $2$ sequences with the property that $\sum |a_n|$ and $\sum |b_n|$ are something, that is, the sums converge. Is it possible that $\sum a_n b_n$ diverges?

If I didn't put absolute values around the sums then you have this obvious example: $a_n = b_n = \frac{(-1)^n}{\sqrt{n}}$ with the product being the Harmonic Series.

Answer 1

For large $n$, $|a_n|\geq a_n^2,|b_n|\geq b_n^2$ therefore $ \sum_{n}a_n^2<\infty$ and $\sum_{n}b_n^2<\infty$.

Now, using Cauchy Schwarz, the result follows namely

$$ \sum_{n}a_nb_n \leq \sqrt{ \sum_{n}a_n^2 } \sqrt{ \sum_{n}b_n^2 } <\infty $$

Answer 2

No, $\sum a_n b_n$ will be absolutely convergent: since $\sum b_n$ converges, then $b_n \to 0$, so $b_n$ is bounded.