Series counterexample

The canonical example is $$a_n=b_n=\frac{(-1)^n}{\sqrt n}$$

ADD Given two sequences $a_n,b_n$, the sequence $c_k=\sum_{i=1}^k a_ib_{k-i}$ is usually called the Cauchy product or convolution of $a_n$ with $b_n$. It is a good exercise (and not an easy one) to prove that if $a_n$ is absolutely summable - that is $$\sum |a_n| $$ exists - and $b_n$ is summable, then the Cauchy product is summable and it converges to the product of the sums. This is known as Merten's theorem.

ADD There is a theorem (found in Spivak's calculus) that says that if both $a_n$ and $b_n$ are absolutely summable, then any sum of the form $$\sum_{i,j} c_{i,j}$$ where each product $a_\ell b_k$ appears exactly once will converge to $$\sum a_n\cdot \sum b_n$$