Mathematics

Problem

Let

f(x)=\displaystyle \sum_{n=0}^{\infty} \dfrac{(x-2)^{n}}{(n+1)!}

. (a) Determine the radius and interval of convergence for

f

(include endpoint analysis). (b) Write a power series for

f'(x)

centered at

x=2

, listing the first three nonzero terms and the general term. (c) Determine the radius and interval of convergence for

f'(x)

and compare your result with part (a). (d) Derive a closed‐form expression for

f^{(k)}(2)

for any positive integer

k

. Show all reasoning.