Finding example of a convergent series $\sum a_n $ such that $\sum a_n^2 $ does not converge I'm feeling that examples for this would arise from some alternating series, in particular of the form $\sum (-1)^nx_n$. But I'm having trouble finding a concrete example. Thanks!

Maybe it is worth noting that in your example the series $\sum_{n=1}^{\infty}a_n$ cannot be absolutely convergent because:
$$\sum_{n=1}^{\infty}|a_n| < \infty \Rightarrow \lim_{n\to\infty}|a_n|= 0 \Rightarrow \exists N \in \mathbb{N}: |a_n| < 1 \forall n\geq N $$ $$\Rightarrow \sum_{n=\color{blue}{N}}^{\infty}a_n^2 \leq \sum_{n=\color{blue}{N}}^{\infty}|a_n|$$
So, for example, any sequence $a_n = (-1)^n\frac{1}{n^{\alpha}}$ with $0 < 2\alpha < 1$ would give a series satisfying your conditions.
