How can I show that if 2 power series are conditionally convergent then it can happen that $\sum_{k=1}^{\infty}c_k$ diverges?

Question

Assume that $\sum_{k=1}^{\infty}a_k =A \in R$ and $\sum_{k=1}^{\infty}b_k =B \in R$ with partial sums $r_n =\sum_{k=1}^{n}a_k$ and $s_n=\sum_{k=1}^{n}b_k.$ If one wants to sum the double series $\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}a_kb_l$ the sum may depend on the order of the summation. If we sum over square we obtain
$\lim_{n \to \infty}\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}a_kb_l = \lim_{n \to \infty} r_ns_n =AB$
If the series $\sum_{k=1}^{\infty}a_k$ and $\sum_{k=1}^{\infty}b_k$ are absolutely convergent, then all orders of summation five the same result, i.e. AB. If we sum over the triangles we obtain
$\lim_{n \to \infty}\sum_{k=1}^{\infty}c_k \quad$ where $c_k =\sum_{l=1}^{k-1}a_lb_{k-l}.$
How can I show that if the series $\sum_{k=1}^{\infty}a_k$ and $\sum_{k=1}^{\infty}b_k$ are conditionally convergent then it can happen that $\sum_{k=1}^{\infty}c_k$ diverges? Allegedly I can use the following hint: consider $a_k=b_k=(-1)^{k+1}/\sqrt{k}.$

Answer 1

Hint: $\sqrt{l}\sqrt{k-l} = \sqrt{kl - l^2} \leqslant k$ and $|c_k| \not\to 0$ as $k \to \infty$.