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

Question 1Select one:

a. 

∑∞n=0−3n(x+1)n,|x+1|<3

b. 

∑∞n=0(−1)n3n(x+1)n,|x+1|<3

c. 

∑∞n=0−3n(x+1)n,|x+1|<13

d. 

∑∞n=0(−1)n3n(x+1)n,|x+1|<13




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.

Learn how to find a power series for the function f(x) = 1/2 + 3x centered at x0 = -1. Understand the steps involved in expressing the function as a power series expansion. Suitable for advanced high school students and beyond.

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