Example of two convergent series whose product is not convergent.
Could someone give me an example of two convergent series $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ such that $\sum_{n=0}^\infty a_nb_n$ diverges?
$$a_n = b_n = \dfrac{(-1)^n}{\sqrt{n+1}}$$ where $n \in \{0,1,2,\ldots\}$