Use a Maclaurin series in this table to obtain the Maclaurin series for the given function.
f(x) = 4ex + e8x

To find the Maclaurin series for the function \( f(x) = 4e^x + e^{8x} \), we can start by recalling the Maclaurin series for the exponential function \( e^x \).

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

\[
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
\]

Using this, let's express the Maclaurin series for each part of the function:

### 1. \( 4e^x \)
Multiply the Maclaurin series for \( e^x \) by 4:

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

This simplifies to:

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

### 2. \( e^{8x} \)
Substitute \( 8x \) for \( x \) in the Maclaurin series for \( e^x \):

\[
e^{8x} = \sum_{n=0}^{\infty} \frac{(8x)^n}{n!} = 1 + \frac{8x}{1!} + \frac{(8x)^2}{2!} + \frac{(8x)^3}{3!} + \cdots
\]

This simplifies to:

\[
e^{8x} = 1 + 8x + \frac{64x^2}{2!} + \frac{512x^3}{3!} + \cdots
\]

### Combine the series
Now, add the two series to obtain the Maclaurin series for \( f(x) = 4e^x + e^{8x} \):

\[
f(x) = \left(4 + 4x + 2x^2 + \frac{4x^3}{3!} + \cdots \right) + \left(1 + 8x + 32x^2 + \frac{512x^3}{3!} + \cdots \right)
\]

Combining like terms, we get:

\[
f(x) = (4 + 1) + (4x + 8x) + \left(2x^2 + 32x^2\right) + \left(\frac{4x^3}{6} + \frac{512x^3}{6}\right) + \cdots
\]

Simplifying further:

\[
f(x) = 5 + 12x + 34x^2 + \frac{516x^3}{6} + \cdots
\]

Thus, the Maclaurin series for \( f(x) = 4e^x + e^{8x} \) is:

\[
f(x) = 5 + 12x + 34x^2 + 86x^3 + \cdots
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you derive the Maclaurin series for \( \sin(x) \)?
2. What is the radius of convergence for the Maclaurin series of \( e^x \)?
3. How does the Maclaurin series differ from the Taylor series?
4. How can the Maclaurin series be used to approximate functions?
5. What is the Maclaurin series expansion for \( \ln(1 + x) \)?
6. How do higher-order terms in a Maclaurin series affect the accuracy of the approximation?
7. Can you derive the Maclaurin series for \( \cos(x) \)?
8. What is the significance of the constant term in a Maclaurin series?

**Tip:** Always check the radius of convergence of a series when using it to approximate a function over a range.

Explore the Maclaurin series expansion for the function f(x) = 4e^x + e^{8x}. This detailed solution includes step-by-step calculations using the Maclaurin series formula and exponential series expansion. Suitable for advanced undergraduate students and anyone interested in mathematical series expansions.

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