Mathematics

Problem

A power series

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

converges for

|x|<R

where

R>0

. Let

T(x)

denote the derivative of

S

, so that

T(x)=S'(x)=\sum_{n=1}^{\infty} n a_n x^{n-1}.\;

Answer the following questions about the relationship between

S

and

T

. (a) Explain why

T(x)

must also converge for every

|x|<R

. In your explanation, cite any relevant convergence theorems for power series. (b) Describe explicitly how the coefficients of

T

are related to the coefficients of

S

, and interpret the effect that differentiation has on the general term of the series.