To find the power series for the function $f(x) = \frac{1}{2} + 3x$ centered at $x_0 = -1$, let's follow these steps:

Step 1: Rewrite the function in terms of $(x + 1)$

Given $f(x) = \frac{1}{2} + 3x$, we can express $x$ as $x = (x + 1) - 1$. Substituting this into the function:

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

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

Step 2: Express $f(x)$ as a power series

Now we need to express $3(x + 1)$ as a power series centered at $x_0 = -1$.

The function $f(x) = -\frac{5}{2} + 3(x + 1)$ is already linear, meaning it can be directly represented as:

$f(x) = -\frac{5}{2} + 3(x + 1) = \sum_{n=0}^{\infty} a_n (x + 1)^n$

To match this with a power series form:

• The coefficient of the constant term $a_0$ should be $-\frac{5}{2}$.
• The coefficient of the first-degree term $a_1$ should be $3$.
• Higher-order coefficients $a_n$ for $n \geq 2$ will be 0.

Thus:

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

But the given multiple-choice options involve summing a series, so let's match it with the standard form of a geometric series.

Step 3: Compare with the given options

The options are focused on geometric series. But since our function is linear, the power series expansion only has the first two terms, making higher-order terms unnecessary. The correct match is with the series where the first coefficient is non-zero for $n = 1$ only.

Analysis of Options:

Given our form of the function and comparing it with the geometric series forms:

• Option a: $\sum_{n=0}^{\infty} -3^n (x+1)^n$ does not match since the series here would expand for higher powers.
• Option b: $\sum_{n=0}^{\infty} (-1)^n 3^n (x+1)^n$ does not match as it suggests an alternating series which is not the case here.
• Option c: $\sum_{n=0}^{\infty} -3^n (x+1)^n$ also involves powers not present in our linear function.
• Option d: $\sum_{n=0}^{\infty} (-1)^n 3^n (x+1)^n$ contains higher powers and alternating signs which do not apply here.

Upon further analysis, none of the options seem to match the exact function $f(x)$ as a simple linear function cannot be matched with higher-degree terms unless we consider the linear term explicitly.

Conclusion:

The correct approach suggests there's no accurate match among the given options for the specific form of a power series centered at $x_0 = -1$ as requested. The function does not naturally align with a geometric series but is rather a linear expansion.

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Do you want more details on expanding power series, or have any specific questions?

5 Related Questions:

1. How can we express a non-linear function as a power series centered at a specific point?
2. What are the conditions under which a function can be expressed as a geometric series?
3. How does the radius of convergence affect the representation of a power series?
4. What is the significance of the center point $x_0$ in a power series expansion?
5. Can any function be expressed as a power series, or are there restrictions?

Tip:

Remember that linear functions can often be directly expanded into a power series with very few terms, focusing primarily on the first-degree and constant terms.