Chapter 6.3, Problem 133E

In the following exercises, find the smallest value of n such that the remainder estimate $\left|Rn\right|\le \frac{M}{(n+1)!}{(x-a)}^{n+1}$ where M is the maximum value of $|f^{(n+1)}(z)|$ on the interval between a and the indicated point, yields $\left|R_n\right|\le \frac{1}{1000}$ on the indicated interval.

133. f(x) = cos x on $\left[\frac{\pi }{2},\frac{\pi }{2}\right]$ , a = 0