Problem $2$: Understanding of Errors $($using an example of the cosine series$)$

The algorithm for the cosine Taylor's series expansion is given as follows:

$cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+$cdots$={\displaystyle \underset{k=1}{\overset{}{}}}\frac{(-1{)}^{k+1}{x}^{(2k-2)}}{(2k-2)!}$

a$)$ Create a user defined function to approximate the value of cosine $($the algorithm above$)$ using a while$-$loop to ensure that the algorithm runs until the relative error is below a given error tolerance $($maxtol$).$ A partially completed code has been provided $($remove the "part" from the function handle before running$).$

b$)$ Test the function created in part $($a$)$ by evaluating at $x=5$ and maxtol $=1e-3.$