Problem 6. (12 pts) Determine whether the following statement is true or false. If the statement is true, prove it, and, if it is false, provide a counterexample. \text {

If } \sum_{n \in \mathbb{N}} a_{n} \text { is conditionally convergent, then } \sum_{n \in \mathbb{N}} n^{k} a_{n} \text { is divergent, for each } k \in(1,+\infty)