Consider the power series e$^($x$)=1+$x$+($x$^(2))/(2)+($x$^(3))/(3!)+$cdots. Let f be the function given by f$($x$)=$e$^(-($x$^(2))/(3)).$

a$.$ Find P$\_(4)($x$),$ the fourth degree Maclaurin Polynomial for f$($x$).$

b$.$ Write f$($x$)$ as a power series using summation notation.

c$.$ Use the polynomial P$\_(4)($x$)$ in part $($a$)$ to approximate f$(0\mathrm{.}3)$ to $5$ decimal places.

d$.$ Find the interval of convergence of the power series for f$($x$)$ about x$=0.$ Show the analysis

that leads to your conclusion.

e$.$ For what value of n$,$ will $|$f$($x$)-$P$\_($n$)($x$)|<mn>0</mn><mn>.</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>1</mn>$ for x