Chapter 8, Problem 31SE

To determine

(a)

To show:

$G^1(x)=G(x)+e^x$ and $G(0)=1$, if G(x) be the generating function for the sequence $\left\{a_n\right\}$.

To determine

(b)

To show:

${\left(e^{-x}G(x)\right)}^1=1$ and hence conclude that $G(x)=x{e}^x+e^x$

To determine

(c)

$a_n$ by using $G(x)=x{e}^x+e^x$