Mathematics

Problem

A student finds that for a power series centered at

3

\displaystyle\lim_{n\to\infty}\left|\frac{b_{n+1}}{b_n}\right|\,|x-3| = \frac{|x-3|}{4}

. Which conclusion about the convergence of the series is most accurate? a) The series converges when

|x-3| \le 4

. b) The series converges when

|x-3| < \dfrac14

. c) The series converges when

|x-3| < 4

and diverges when

|x-3| > 4

, while the endpoints must be analyzed separately. d) The series converges when

|x-3| < 4

and diverges when

|x-3| \ge 4