Suppose a function has a power-series representation centered at

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

Problem

c

, given by

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

with radius of convergence

R>0

. The derivative also has a power-series representation

f'(x)=\sum_{n=0}^{\infty} b_n (x-c)^n

. Which statement best describes the relationship between the radii of convergence of these two series? (a) The radius of convergence of

f'(x)

is at least

R

and may be larger than

R

. (b) Both series have the same radius of convergence

R

. (c) The radius of convergence of

f'(x)

R-1

whenever

R>1

. (d) The radius of convergence of

f'(x)

\dfrac{R}{2}