1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.