Only evens allowed

Let $S_n = \frac{1}{1} + \frac {1}{2} + \frac {1}{2 \times 4} + \ldots + \frac { 1 } { 2 \times 4 \times \cdots \times 2n }$ . Evaluate

$\lim_{n\rightarrow \infty} S_n .$

Details and assumptions

Note: The initial term $\frac {1}{1}$ occurs because the empty product (product of no terms) is defined to be 1.