Compute Taylor error of $10^{-5}$ for $\cos(\pi/20)$

Question

I need to calculate $\cos(\pi/20)$ with an error less than $10^{−5}$ of Taylor/Maclaurin series, What I did is like that:

$$|R_n(X)|=\left|\frac{(-1)^nC}{(n+1)!}\right|=\left|\frac{C}{(n+1)!}\right|_{0<C<\frac{\pi}{20}}<\frac{\frac{\pi}{20}}{(n+1)!}=\frac{\pi}{20(n+1)!}<10^{-5}$$

The statement is true when n=7, is my answer correct?

Answer 1

No, it is not correct. You wrote $R_2(X)$, but I suppose that you meant $R_n(X)$. Furthermore,$$\left|R_n\left(\frac\pi{20}\right)\right|\leqslant\frac{\left(\frac\pi{20}\right)^{n+1}}{(n+1)!}.$$Using the fact that $\pi<4$ (and that therefore $\frac\pi{20}<\frac15$), you can see that taking $n=4$ is enough.