Mathematics

Problem

Consider the general power series

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

with a positive radius of convergence

R

. (a) Define the radius of convergence

R

in terms of the sequence

\{a_n\}

. (b) Explain—in the context of convergence tests—why the open interval

(r-R,\,r+R)

remains the set of interior points of convergence for the series

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

. (c) Provide two distinct reasons why convergence at an endpoint of

S(x)

does not automatically imply convergence at the same endpoint for

S'(x)

. (d) Give an explicit example of a power series whose sum converges at

x=r+R

while its derivative fails to converge there, and justify your claim.