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.