Chapter 8.2, Problem 15E

To determine

(a)

To show:

The series ${\displaystyle {\sum }_{n=0}^{\infty }{x}^n/2^n}$ is the Cauchy product of $q\left(x\right)$ and ${\displaystyle {\sum }_{n=0}^{\infty }{x}^n/n!}$.

To determine

(b)

To find:

The equations using the Cauchy-product

$\frac{1}{2^0}=a_0$, $\frac{1}{2}=a_0+a_1$, $\frac{1}{2^2}=\frac{a_0}{2}+a_1+a_2$, $\frac{1}{2^3}=\frac{a_0}{6}+\frac{a_1}{2}+a_2+a_3$, ….

To determine

(c)

To find:

The values of $a_0,a_1,a_2,a_3$ from part (b).