Show that the conditional convergence of $\sum_{k=1}^\infty a_k $ does not necessarily imply that $\sum_{k=1}^\infty a_k^3$ converges conditionally.

Question

Show that the conditional convergence of $\sum_{k=1}^\infty a_k $ does not necessarily imply that $\sum_{k=1}^\infty a_k^3$ converges conditionally.

Let $a_k = (-1)^k \frac{1}{\sqrt[3]{k}}$, then $$\sum_{k=1}^\infty a_k < \infty, \sum_{k=1}^\infty |a_k|= \infty.$$

That is $\sum_{k=1}^\infty a_k$ converges conditionally. However $$a_k^3 = (-1)^{3k} \frac{1}{k}$$ and $$\sum_{k=1}^\infty a_k^3 < \infty$$ by the alternating series test, but $$\sum_{k=1}^\infty |a_k^3| = \sum_{k=1}^\infty \frac{1}{k}$$ which is the harmonic series and thus divergent. Is what they were after with the question? That is to come up with a counterexample for the proposition?