Let

$a\_n=\left\{\begin{matrix}
 & \frac{1}{n^2}, n \text{ is not a product of 3} & \\ 
 & -\frac{1}{n}, n \text{ is a product of 3} & 
\end{matrix}\right.$

Does $\sum a\_n$ converge or diverge?

Obviously, I know that this series must diverge since the negative term will obviously dominate the positive terms eventually (which, we know, will converge to a number lesser than $\frac{\pi^2}{6}$). Is it possible to offer a rigorous proof, however?