Mathematics

Problem

Consider an arbitrary power series

S(x)=\sum_{n=0}^{\infty} a_{n}x^{n}

with radius of convergence

R>0

. (a) Explain why

S(x)

converges absolutely for all

x

satisfying

|x|<R

. (b) Provide a reasoning argument showing that the formal derivative series

S'(x)=\sum_{n=1}^{\infty} n\,a_{n}x^{n-1}

also converges to

S'(x)

for

|x|<R

. (c) Discuss how the radius of convergence of the differentiated series compares with that of the original series and explain what additional analysis is required to decide convergence at

x=\pm R

. Give an example illustrating one possible outcome at an endpoint.

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