Converse of Abel's test I am considering the converse of Abel's test. Suppose $\sum_{n}a_{n}b_{n}$ is convergent, $b_{n}$ is monotonic, bounded and positive, is $\sum_{n}a_{n}$ converge?

Let, $a_n=(-1)^n$ and $b_n=\frac{1}{n}$. So $b_n$ is monotonic and bounded as $0<b_n \leq1$. Also $\sum_n a_n b_n$ is convergent but $\sum_n a_n$ is not convergent..
