So for this question, I first had to find the Maclaurin polynomial of $\displaystyle f(x)=5\cos(0.2x) $ which I found to be $\displaystyle 5[1-\frac{(0.2x^2)}{2}+\frac{(0.2x^4)}{4!}]$, and the second part asks:

"Find the largest integerksuch that for all x for which |x|<1 the Taylor polynomial $\displaystyle T_{5}(x)$ approximates f(x) with error less than $\displaystyle 10^{-k}$.

I really don't know how to go about doing this. Do you have any suggestions?