The function is defined by for all real

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The function

h

is defined by

h(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n+2}}{(n+2)!}

for all real

x

. Which option gives a power‐series expression for

h''(x)

centered at

x=-1

along with the radius of convergence of

h''

? (a)

h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n}}{(n+2)!}

; radius of convergence

\infty

(b)

h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(n+1)(x+1)^{n}}{n!}

; radius of convergence

1

(c)

h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n}}{n!}

; radius of convergence

\infty

(d)

h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n}}{n!}

; radius of convergence

1