A$1($a$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x{e}^{-x^4}dx$ as an infinite series.

$($b$)$ Use the series you found in part $($a$)$ to evaluate ${\displaystyle \underset{0}{\overset{1}{}}}x{e}^{-x^4}dx$ correct to within an error of $0\mathrm{.}001.$

A$2$ Write down the Mclaurin series $($Taylor series centred at $x=0)$ for the function $y(x)=e^x,$ with $$ a real number.

$($a$)$ Differentiate the series, and compare $y^{\text{'}}(x)$ to $y(x).$ What relationship between and $y$ and $y^{\text{'}}$ do you get?

$($b$)$ Use the series to find $\underset{x0}{lim}\frac{e^x-1}{x}.$