Suppose the power series ∑ₖ₌₀∞ cₖ(x−a)ᵏ has an interval of convergence of (−3,7]. Find the center a and the radius of convergence R.

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Recall that the interval of convergence of a power series \( \sum_{k=0}^\infty c_k (x - a)^k \) is centered at \( a \), and the radius of convergence \( R \) is the distance from \( a \) to either endpoint of the interval.

Given the interval of convergence \( (-3, 7] \), identify the center \( a \) as the midpoint of the interval endpoints. Calculate \( a = \frac{-3 + 7}{2} \).

Calculate the radius of convergence \( R \) as the distance from the center \( a \) to either endpoint. Use \( R = 7 - a \) or \( R = a - (-3) \).

Verify that the radius \( R \) is the same when calculated from both endpoints to confirm consistency.

Summarize the results: the center \( a \) and the radius of convergence \( R \) define the power series' interval of convergence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Power Series and Center

A power series is an infinite sum of terms in the form cₖ(x−a)ᵏ, where a is the center of the series. The center a is the point around which the series is expanded, and it determines the horizontal shift of the interval of convergence.

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges. It is typically an interval centered at a, possibly including or excluding endpoints, and is crucial for understanding where the series represents a valid function.

Radius of Convergence

The radius of convergence R is the distance from the center a to either endpoint of the interval of convergence. It measures how far from a the series converges and is calculated as half the length of the interval of convergence.

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