\mathop {\lim }\limits_{n \to \infty } S_n = \mathop {\lim }\limits_{n \to \infty } \left( {q\frac{{q^n - 1}}{{q - 1}}} \right) = q\mathop {\lim }\limits_{n \to \infty } \frac{{q^n - 1}}{{q - 1}} = \frac{q}{{1 - q}}