Question

The value of the series \( \frac{2}{3!} + \frac{4}{5!} + \frac{6}{7!} + \dots \) is:

• A. \( 2e^{-2} \)
• B. \( e^{-2} \)
• C. \( c^{-1} \)
• D. \( 2e^{-1} \)

Hint:

For series involving factorials, try identifying if the series resembles a known expansion for functions like \( e^x \).

Answer 1

The Correct Option is C

Step 1: Identify the general term of the series. The series is of the form: \[ S = \frac{2}{3!} + \frac{4}{5!} + \frac{6}{7!} + \dots, \] where the general term can be written as: \[ T_r = \frac{2r}{(2r+1)!}. \] This is a series involving terms that are related to the exponential series, particularly the series for \( e^x \). 
Step 2: Recognize the sum. This series can be related to an exponential series and converges to a known value. By analyzing the sum and recognizing the pattern, we find that it evaluates to: \[ c^{-1}. \] Thus, the correct answer is: \[ \boxed{c^{-1}}. \]