I have this problem here that I'm not sure how to solve. Consider the polynomial $1 - \large\frac {x^2}{2!}$ as an approximation to $\cos(x)$ on the closed interval $-1 \le x \le 1$. What is the best bound on the error that is given by Taylor's inequality?
I tried finding the $3^{rd}$ derivative of the function, which gives $\sin(x)$. Maximum value of this is $\sin(x)$, taking $x = 1$. Largest absolute value of $x^{n+1}$ is $x^4$, which equals to $1$. So I thought the answer, according to the error formula, would be $\large\frac{\sin(1)}{6}$. But apparently the correct answer is $\frac {1}{24}$.
Can someone please explain why this is so? Would really appreciate it!
Thanks!