Find a power series for the function f(x)=1/2+3x centered at x0=−1.

To find the power series representation of the function \( f(x) = \frac{1}{2} + 3x \) centered at \( x_0 = -1 \), we can express the function in the form:

\[
f(x) = \sum_{n=0}^{\infty} c_n (x - x_0)^n
\]

Here, \( x_0 = -1 \), so we want to write \( f(x) \) as a power series in \( (x + 1) \):

\[
f(x) = \sum_{n=0}^{\infty} c_n (x + 1)^n
\]

Given \( f(x) = \frac{1}{2} + 3x \), we can write it as:

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

Simplifying, we get:

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

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

\[
f(x) = -\frac{5}{2} + 3(x + 1)
\]

Now, express this as a power series centered at \( x_0 = -1 \):

\[
f(x) = -\frac{5}{2} + 3(x + 1)
\]

So the power series is:

\[
f(x) = -\frac{5}{2} + 3(x + 1)
\]

Since this is a linear function, the series truncates after the first degree:

\[
f(x) = -\frac{5}{2} + 3(x + 1)
\]

Thus, the power series for \( f(x) = \frac{1}{2} + 3x \) centered at \( x_0 = -1 \) is:

\[
f(x) = -\frac{5}{2} + 3(x + 1)
\]

This is the final answer, where the series has only two non-zero terms.

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Would you like more details, or do you have any questions? Here are some related questions:

1. How do you find the power series for higher-degree polynomials?
2. What are the general steps to find the power series of any function centered at \( x_0 \)?
3. Can the power series be used to approximate functions near \( x_0 \)?
4. How does the radius of convergence affect the power series?
5. What happens if you center the series at a different \( x_0 \)?

**Tip:** When working with power series, always consider the function's behavior around the center to ensure accuracy in approximation.

Learn how to find the power series representation for the function f(x) = 1/2 + 3x centered at x0 = -1. Detailed step-by-step solution covering power series representation and Taylor series expansion. Suitable for advanced undergraduate students and anyone interested in function approximation.

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