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Consider the power series given by P an(x − 3)n and suppose that it converges at x = −1 and diverges at
x = 10. Determine what can be said about the convergence of the power series for the following x-values. You
should clearly state for each x-value whether the power series will converge, diverge, or that its behavior is
unknown.

To analyze the convergence of the power series \( \sum a_n (x - 3)^n \), we’ll use the information provided: it converges at \( x = -1 \) and diverges at \( x = 10 \). This helps establish the interval of convergence of the series around the center \( x = 3 \).

The convergence of a power series depends on its radius of convergence \( R \). The radius \( R \) is determined by the distance between \( x = 3 \) (the center) and the points where convergence and divergence behaviors are known:
- **Convergence at \( x = -1 \)** suggests that \( |x - 3| \leq R \), so \( R \geq |(-1) - 3| = 4 \).
- **Divergence at \( x = 10 \)** implies that \( |x - 3| > R \), so \( R < |10 - 3| = 7 \).

Thus, we conclude that the radius of convergence \( R \) lies between 4 and 7:
\[ 4 \leq R < 7 \]

Now, let’s examine the specific points given:

### (a) \( x = 3 \)
For \( x = 3 \):
- \( |x - 3| = 0 \), which is within any radius \( R > 0 \).
- Therefore, the series **converges** at \( x = 3 \).

### (b) \( x = 6 \)
For \( x = 6 \):
- \( |x - 3| = 3 \), which is less than both 4 and 7.
- This distance is within the radius of convergence, so the series **converges** at \( x = 6 \).

### (c) \( x = -5 \)
For \( x = -5 \):
- \( |x - 3| = 8 \), which is greater than both 4 and 7.
- Since this distance exceeds the radius of convergence, the series **diverges** at \( x = -5 \).

### Summary of Results:
1. **(a) \( x = 3 \)**: Converges.
2. **(b) \( x = 6 \)**: Converges.
3. **(c) \( x = -5 \)**: Diverges.

Would you like further details on power series convergence, or do you have any questions?

### Related Questions
1. How is the radius of convergence determined for a general power series?
2. What does it mean for a series to converge absolutely versus conditionally?
3. How can the Ratio Test be used to find the radius of convergence?
4. What is the interval of convergence for a series centered at a point other than zero?
5. How does the behavior at the endpoints of an interval of convergence vary?

### Tip:
For power series, knowing one point of convergence and one point of divergence can significantly narrow down the possible radius of convergence.

Consider the power series given by P an(x − 3)^n and suppose that it converges at x = −1 and diverges at x = 10. Determine what can be said about the convergence of the power series for the following x-values. You should clearly state for each x-value whether the power series will converge, diverge, or that its behavior is unknown.

Explore the convergence of a power series with center x = 3, given its convergence at x = -1 and divergence at x = 10. Determine convergence behavior for x-values 3, 6, and -5 using the radius and interval of convergence.

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