$4.($a$)$ Consider the power series

X$\backslash$infty

n$=0$

x

$2$n

n$!$

Show that the series converges for every x in R$.$ Include your explanation in the handwritten answers.

$($b$)$ Use Matlab to evaluate the sum of the above series. Again, include a screenshot of your command

window showing $(1)$ your command, and $(2)$ Matlab$$s answer.

$($c$)$ Use Matlab to calculate the Taylor polynomial of order $4$ of the function f$($x$)=$ e

x

$2$

at the point

a $=0.$ Include a screenshot of your command window showing $(1)$ your command, and $(2)$ Matlab$$s

answer.

$($d$)$ Explain how the series from Point $4$a$)$ is related to the Taylor polynomial from Point $4$c$).$ Include

your explanation in the handwritten answers