The sum of the series `1/(2!)-1/(3!)+1/(4!)-...` upto infinity is (1) `e^(-2)` (2) `e^(-1)` (3) `e^(-1//2)` (4) `e^(1//2)`

Explanation

To find the sum of the series \( S = \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \cdots \) up to infinity, we can use the Taylor series expansion of the exponential function \( e^x \).

### Step 1: Write the Taylor series for \( e^x \)

The Taylor series expansion for \( e^x \) is given by:

\[
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
...