Which statement best describes the convergence behavior of a power series relative to…

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Which statement best describes the convergence behavior of a power series

\displaystyle\sum_{n=0}^{\infty} a_n\,(x-c)^n

relative to its radius of convergence

R

? (a) The series converges absolutely for

|x-c|<R

and

|x-c|=R

, and diverges for

|x-c|>R

. (b) The series converges conditionally for

|x-c|<R

, but diverges for

|x-c|\ge R

. (c) The series converges absolutely for

|x-c|<R

, may converge conditionally or diverge at

|x-c|=R

, and diverges for

|x-c|>R

. (d) The series converges absolutely for

|x-c|<R

, diverges at

|x-c|=R

, and may converge or diverge for

|x-c|>R