Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
f. Diverges for x = 4.9

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Identify the center of the power series, which is at \(x = 2\), since the series is given as \(\sum a_n (x - 2)^n\).

Use the information about convergence and divergence at specific points to find the radius of convergence \(R\). The series converges at \(x = -1\) and diverges at \(x = 6\).

Calculate the distance from the center to these points: \(|-1 - 2| = 3\) and \(|6 - 2| = 4\). Since the series converges at \(x = -1\) (distance 3) and diverges at \(x = 6\) (distance 4), the radius of convergence \(R\) satisfies \(3 \leq R < 4\).

Determine whether the series converges or diverges at \(x = 4.9\) by calculating the distance from the center: \(|4.9 - 2| = 2.9\).

Since \(2.9 < 3 \leq R\), the point \(x = 4.9\) lies within the radius of convergence, so the series converges at \(x = 4.9\). Therefore, the statement that it diverges at \(x = 4.9\) is false.

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주요 개념

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Radius of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges absolutely. For a series centered at x = 2, it converges for all x such that |x - 2| < R, where R is the radius. Knowing convergence and divergence at specific points helps estimate R.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It is centered at the series' center (here, x = 2) and extends R units in both directions. Convergence at boundary points must be checked separately, as behavior can differ.

Testing Convergence at Specific Points

To determine if a series converges or diverges at a particular x, compare the distance |x - center| to the radius of convergence. If the distance is less than R, the series converges; if greater, it diverges. If equal, convergence must be tested individually.

Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.f. Diverges for x = 4.9