The Maclaurin series expansion of $e^{-x}$ is given by:

$e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!}$

Expanding the first few terms:

$e^{-x} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!} - \frac{x^5}{5!} + \dots$

This series is valid for all real values of $x$.

Would you like a deeper explanation or any specific application?

Here are five related questions to expand on this topic:

1. How is the Maclaurin series derived from the Taylor series?
2. What is the radius of convergence of this series?
3. How can this expansion be used to approximate $e^{-x}$ for small values of $x$?
4. How do we use the Maclaurin series to estimate error in approximations?
5. Can the Maclaurin series be used to evaluate limits involving $e^{-x}$?

Tip: The Maclaurin series for $e^x$ and $e^{-x}$ are closely related and can be used to derive hyperbolic functions like $\cosh x$ and $\sinh x$.