Question

The sum of the following infinite series is: \[ 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \dots \]

• A. \pi
• B. 1 + e
• C. e - 1
• D. e

Hint:

Taylor series expansions are useful for understanding the properties and behaviors of exponential functions like \( e^x \), particularly in mathematical and engineering applications.

Answer 1

The Correct Option is C

This series is similar to the Taylor series expansion for \( e^x \), but it starts at 0, not at 1 as the typical \( e \) expansion would. The series actually represents \( e - 1 \) since: \[ e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots \] Removing the first term (which is 1) from the equation, we are left with: \[ e - 1 = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots \]