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Suppose $\sum a_n$ converges conditionally and $\sum b_n$ converges absolutely, then will $\sum a_nb_n$ converge absolutely?

I know that $\sum|a_n|$ does not converge while $\sum a_n$ does and that $\sum|b_n|$ does converge, and also that $\lim_{n \to \infty} |a_nb_n|=0$ but I'm not sure how to proceed from here.


marked as duplicate by Micah, Robert Israel sequences-and-series Mar 8 '18 at 1:41

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$|a_{n}|<1$ eventually and so $|a_{n}b_{n}|\leq|b_{n}|$ eventually, but $\displaystyle\sum|b_{n}|<\infty$, then so is $\displaystyle\sum|a_{n}b_{n}|<\infty$.

  • $\begingroup$ how when |$a_n$| diverges $\endgroup$ – user524644 Mar 8 '18 at 1:27
  • $\begingroup$ I don't use that. Rather, if $\displaystyle\sum a_{n}$ exists, then its tail goes to zero. $\endgroup$ – user284331 Mar 8 '18 at 1:27
  • $\begingroup$ ok but what so what if $\sum |a_nb_n| < \infty$ $\endgroup$ – user524644 Mar 8 '18 at 1:29
  • $\begingroup$ I don't understand your question? $\endgroup$ – user284331 Mar 8 '18 at 1:30
  • $\begingroup$ how does that fact show absolute convergence $\endgroup$ – user524644 Mar 8 '18 at 1:30

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