Consider the function $f(x)=e^{-x^2}.$

a$)$ Use the Maclaurin series for $e^x,$ shown below, to determine the Maclaurin series for $f(x).$ Write your answer in summation notation. $(6$ points$)$

$e^x={\displaystyle \underset{n=0}{\overset{}{}}}\frac{x^n}{n!}$

b$)$ Use your answer to part a to determine the Maclaurin series for

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{-x^2}dx$

Write your answer in summation notation. $(6$ points$)$

c$)$ Determine the minimum number of terms required to evaluate

${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^2}dx$

accurate to within $0\mathrm{.}0001.$ Then use this number of terms to estimate the value of the integral. $(6$ points$)$