If the series $\sum a_n$ and the sequence $(b_n)$ converge, does series $\sum a_n b_n$ also converge?

I think there should be some extra conditions to make this true, like $(b_n)$ is monotone or $\sum a_n$ converges absolutely.

So is there a counter example to show it's not necessarily true?

  • $\begingroup$ $b_n = \frac{1}{(n - a)^2}$ for $a > 0$ will counter it if $b_n$ is not required to be monotone. $\endgroup$ – Axoren Oct 26 '14 at 19:54

Hint: $a_n=b_n=(-1)^n/\sqrt{n}$, $a_nb_n=1/n$...


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