Given the infinite series:

$$S = \frac{2}{3!} + \frac{4}{5!} + \frac{6}{7!} + \cdots = \sum_{n=1}^{\infty} \frac{2n}{(2n+1)!}$$

This is a known convergent series, and its sum is:

$$\boxed{e^{-1}}$$

✅ Final Answer: $\boxed{e^{-1}}$