Question

Suppose \( \sum_{n=1}^{\infty} a_n (x - 2)^n \) is convergent at \( x = -5 \), then it need not be convergent on which interval?

• A. \( |x - 2| \leq 5 \)
• B. \( |x - 2|<5 \)
• C. \( |x - 2| \leq 7 \)
• D. \( |x - 2|<7 \)

Hint:

The radius of convergence of a power series is determined by the distance from the center of the series to the nearest point where the series fails to converge. The series converges within the radius, but may not converge at the boundary.

Answer 1

The Correct Option is C

The given series is a power series \( \sum_{n=1}^{\infty} a_n (x - 2)^n \), which is convergent at \( x = -5 \). For a power series, the radius of convergence \( R \) defines the interval within which the series converges. 
If the series converges at \( x = -5 \), the distance from \( x = -5 \) to the center \( x = 2 \) gives the radius of convergence: \[ R = | -5 - 2 | = 7 \] Thus, the series is convergent within the interval \( |x - 2| \leq 7 \), i.e., on the interval \( [-5, 9] \). 
However, the series need not be convergent outside this interval. Specifically, the series may fail to converge on the interval \( |x - 2| \leq 7 \) since convergence at \( x = -5 \) does not guarantee convergence at the endpoints of the interval. 
Therefore, the correct answer is that the series need not be convergent on the interval \( |x - 2| \leq 7 \).