Disprove that if $\sum_n a_n$ converges, then $\sum_n a_n^2$ converges

Question

I am attempting to answer a question from some Real Analysis exercises. The question asks if the series $\sum_n a_n$ converges, then does $\sum_n a_n^2$ converge, diverge, or is it impossible to tell. Then there is a part b that asks what changes if $\sum_n a_n$ converges absolutely. Intuition tells me that $ \sum_n a_n^2$ won't converge unless an converges absolutely (this proof I know how to do and is quite simple). However, I am struggling to come up with examples so that $\sum_n a_n$ converges and $\sum_n a_n^2$ diverges. Could anyone help?

Answer 1

Try $a_n = \displaystyle\frac{(-1)^n}{\sqrt{n}}$

Answer 2

Let $a_n = \frac{\left(-1\right)^n}{ln\left(n\right)}$ and here's your answer.