Write down the Maclaurin series of $$ e^{x} $$

The Maclaurin series for $ e^x $ is a Taylor series expansion of $ e^x $ around $ x = 0 $. The Maclaurin series is given by:

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

This can be written out explicitly as:

\[
e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots
\]

### Do you want more details or have any questions?

Here are some related questions:

1. What is the general form of a Maclaurin series for any function?
2. How can the Maclaurin series be used to approximate $ e^x $ for small values of $ x $?
3. What is the radius of convergence of the Maclaurin series for $ e^x $?
4. How does the Maclaurin series for $ e^x $ relate to its derivatives?
5. Can you find the Maclaurin series for other exponential functions like $ e^{-x} $?

**Tip:** When using the Maclaurin series for practical calculations, truncating the series after a few terms often provides a good approximation, especially for small values of $ x $.

Explore the Maclaurin series for e^x with a detailed explanation and examples. Learn how the series expansion e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots relates to calculus and Taylor series. Suitable for undergraduate students and anyone interested in mathematical series expansions.

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