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

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?

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

## 1 Answer

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