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\[ S = \frac{2}{3!} + \frac{4}{5!} + \frac{6}{7!} + \cdots = \sum_{n=1}^{\infty} \frac{2n}{(2n+1)!} \]
This is a known convergent series, and its sum is:
\[ \boxed{e^{-1}} \]
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