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.

  • $\begingroup$ Try to show that if one of the two series converges absolutely, then $\sum \lvert a_n b_n\rvert < +\infty$. $\endgroup$ – Daniel Fischer Sep 19 '15 at 11:54
  • $\begingroup$ No, that's not correct. In general, we have $\sum a_n b_n \neq \biggl(\sum a_n\biggr)\biggl(\sum b_n\biggr)$, and you need some other property of the sequence $(b_n)$. If $\sum b_n$ converges (not necessarily absolutely), what properties of the sequence $(b_n)$ does that imply? $\endgroup$ – Daniel Fischer Sep 19 '15 at 12:00
  • $\begingroup$ That $b_n \rightarrow 0$ as $n \rightarrow \infty$ $\endgroup$ – continental Sep 19 '15 at 12:01
  • $\begingroup$ Okay. And that implies something else. Any guess what might help? $\endgroup$ – Daniel Fischer Sep 19 '15 at 12:02
  • $\begingroup$ Maybe that the sequence ($b_n$) is bounded since if it converges it is Cauchy and if it is Cauchy it is bounded? $\endgroup$ – continental Sep 19 '15 at 12:03

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.

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    $\begingroup$ More generally if $(a_n)_n$ is a bounded sequence and $\sum_nb_n$ converges absolutely then $\sum_na_nb_n$ converges absolutely, regardless of whether $\sum_na_n$ converges.............+1 $\endgroup$ – DanielWainfleet Mar 16 '18 at 9:09

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