Consider the function

$f(x)=cosx$

$($a$)$ Determine the Taylor polynomial of degree $n$ for $f$ at the point $x=0.$ Write the answer in sigma notation. $[5$ marks$]$

$($b$)$ Show that for all $x,$ we have

$\underset{n}{lim}{R}_{n,0}f(x)=0$

$[6$ marks$]$