Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
The interval of convergence of the power series ∑ cₖ(x−3)ᵏ could be (−2,8).

Verified step by step guidance

1. Recall that the interval of convergence of a power series \( \sum c_k (x - a)^k \) is always centered at \( a \). In this problem, the series is centered at \( x = 3 \).
2. The interval of convergence is an interval \( (a - R, a + R) \), where \( R \) is the radius of convergence. Since the center is \( 3 \), the interval must be symmetric around 3.
3. Check if the given interval \( (-2, 8) \) is symmetric about 3. Calculate the distance from 3 to each endpoint: \( 3 - (-2) = 5 \) and \( 8 - 3 = 5 \). Both distances are equal, so the interval is symmetric about 3.
4. Since the interval \( (-2, 8) \) is symmetric about 3, it could represent the interval of convergence of the power series \( \sum c_k (x - 3)^k \) with radius of convergence \( R = 5 \).
5. Therefore, the statement is true because the interval \( (-2, 8) \) is a valid interval of convergence centered at 3 with radius 5.

Key Concepts

Power Series and Interval of Convergence

A power series is an infinite sum of the form ∑ cₖ(x−a)ᵏ centered at a point a. The interval of convergence is the set of x-values for which the series converges. This interval is always centered at a and extends symmetrically or asymmetrically around a, possibly including or excluding endpoints.

Radius of Convergence

The radius of convergence (R) is the distance from the center a to the endpoints of the interval of convergence. It determines how far from a the power series converges. The interval of convergence is (a−R, a+R), possibly including endpoints, so the length of the interval is always 2R.

Testing Interval Endpoints and Counterexamples

To verify if a given interval can be the interval of convergence, check if it is centered at a and if its length matches twice the radius of convergence. If the interval is not symmetric about the center or does not match the radius, it cannot be the interval of convergence. Counterexamples help illustrate when intervals fail these conditions.