Use the established MacLaurin series to find the first three non-zero terms of
f (x) = (1 − e−x)/x

We are tasked with finding the first three non-zero terms of the Maclaurin series for the function:

\[
f(x) = \frac{1 - e^{-x}}{x}
\]

### Step 1: Find the Maclaurin series for \( e^{-x} \)

The Maclaurin series for \( e^{-x} \) is:

\[
e^{-x} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots
\]

### Step 2: Substitute the series into \( f(x) \)

Now, substitute the series for \( e^{-x} \) into the expression for \( f(x) \):

\[
f(x) = \frac{1 - e^{-x}}{x} = \frac{1 - \left( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots \right)}{x}
\]

Simplifying the numerator:

\[
f(x) = \frac{1 - 1 + x - \frac{x^2}{2} + \frac{x^3}{6} - \dots}{x} = \frac{x - \frac{x^2}{2} + \frac{x^3}{6} - \dots}{x}
\]

### Step 3: Simplify the expression

Now, divide each term in the series by \( x \):

\[
f(x) = 1 - \frac{x}{2} + \frac{x^2}{6} - \dots
\]

### Step 4: Conclusion

The first three non-zero terms of the Maclaurin series for \( f(x) = \frac{1 - e^{-x}}{x} \) are:

\[
f(x) = 1 - \frac{x}{2} + \frac{x^2}{6}
\]

Would you like further details on how to expand this series or how to apply it? Here are some follow-up questions to explore:

1. How is the general form of a Maclaurin series derived for other functions?
2. What is the radius of convergence for the Maclaurin series of \( e^{-x} \)?
3. How can you use Maclaurin series to approximate functions near \( x = 0 \)?
4. What is the difference between a Maclaurin and a Taylor series?
5. How can the error term in a truncated Maclaurin series be estimated?

**Tip:** The Maclaurin series is just a special case of the Taylor series, where the expansion occurs around \( x = 0 \).

Use the established Maclaurin series to find the first three non-zero terms of f(x) = (1 − e^(-x))/x

Find the first three non-zero terms of the Maclaurin series for the function f(x) = (1 − e^(-x))/x. This solution involves expanding the exponential function and simplifying the result using series techniques. Suitable for undergraduate calculus students.

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