Mathematics

Problem

A power series

\sum_{n=0}^{\infty}a_n\,(x-3)^n

has radius of convergence

2

and converges to a function

p(x)

. It is known that the series converges at

x=1

but diverges at

x=5

. (a) In complete sentences, describe what information the radius of convergence provides about the behavior of

p(x)

near

x=3

. (b) Explain why term-by-term differentiation of the series yields a new series that converges to

p'(x)

on a specific interval. Identify that interval and justify your answer. (c) Based on the given convergence at

x=1

and divergence at

x=5

, state the set of real numbers

x

for which the original series may be substituted directly into

p(x)

to obtain numerical values of

p(x)

. Provide reasoning for your conclusion.