Aspire Faculty ID #11671 · Topic: NIMCET 2024 · Just now
NIMCET 2024

The value of the series $\frac{2}{3!}+\frac{4}{5!}+\frac{6}{7!}+\cdots$ is

Solution

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

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