$($a$)$ Using the power series representation

$e^y=1y\frac{1}{2!}{y}^2\frac{1}{3!}{y}^3$ dots$={\displaystyle \underset{n=}{\overset{}{}}}\frac{1}{n!}{y}^n,$

find a power series representation for the function:

$f(x)={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{-x^2}dx,f(0)=0$

$(6$ mar

$($b$)$ Find the interval and radius of convergence for the following power series:

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(100{)}^n}{n!}(x7{)}^n$