Construct a convergent series $\sum_{n=1}^{\infty} a_n$ an such that $\sum_{n=1}^{\infty} a_n^3$ is divergent.

Question

My class was given a set of practice problems and I can't figure this one out.

Construct a convergent series $\sum_{n=1}^{\infty} a_n$ an such that $\sum_{n=1}^{\infty} a_n^3$ is divergent.

I noticed that $a_n$ cannot be a positive series.According to the Cauchy convergence criterion, there exists an $\exists N \in \mathbb{Z}$ such that for all n>N ,$a_n<1$. $\forall m,n\in\mathbb{Z},m>n>N st.\sum_{i=n}^m a_i^3<\sum_{i=n}^m a_i$ .Therefore, a positive series does not satisfy this property.After that, I have no idea how to construct such a series.