## Example of Convergent Series [closed]

I apologize if this question does not belong here, but I have worked on it for hours and cannot find an answer anywhere. Maybe it has a very obvious answer that I cannot think of; if so, sorry for wasting anyone's time.

Can anyone think of sequences ${a_n}$, ${b_n}$ such that $\sum a_n$ diverges, $b_n\to\infty$, but $\sum a_nb_n$ converges?

Thank you.

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This would fit better on some of the other sites mentioned in the FAQ. – Douglas Zare Mar 9 2011 at 7:31
by the way, you should try $a_n=b_n=\frac 1{n+1}$. – Henri Mar 9 2011 at 7:51
Thank you for your help, Henri, but I need a sequence $\{b_n\}$ such that $b_n\to\infty$. This is the hard part, I think. – Daniel Mar 9 2011 at 7:53
(I will make an edit to make it less ambiguous) – Daniel Mar 9 2011 at 7:54
You should probably ask this question elsewhere, as suggested in Douglas Zare's comment. But just to help you clarify the question: do you actually want $(b_n)$ to tend to $+\infty$, do you just want it to be unbounded, or do you want $(|b_n|)$ to tend to $+\infty$? I think this affects whether or not you can find such sequences. (Incidentally, where does the question come from? you don't make it very clear if you sequences are supposed to be real-valued, positive, etc.) – Yemon Choi Mar 10 2011 at 5:45