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: \(\boxed{e^{-1}}\)