I'm working through how Taylor's Remainder Theorem works using this guide, and on page 2, I saw something that made me puzzled.
The first step is to acknowledge that there is some value of K for which
$$\sin(1/2) = T_3(1/2)+K(1/2)^4$$
Where $T_3$ is the third-order Maclaurin polynomial for $\sin(x)$.
After applying Rolle's theorem up to the fourth derivative, we get:
$$K = \frac{\sin^{(4)}(c)}{24}$$
But then it says that:
$$|\sin(1/2) - T_3(1/2)| ≤ K(1/2)^4≤\frac{1}{24 * 2^4}$$
I don't get why the $|\sin(1/2) - T_3(1/2)|$ would ever be less than $K(1/2)^4$. Wasn't these defined to be exactly equal to each other? That there is some $c$ value which makes $\sin(1/2) = T_3(1/2)+K(1/2)^4$? I'm interested to see because I'm still shaky on the concept and maybe there's an something I'm missing.
The absolute value bars is what does it. You have to account for the fact that $\sin( 1/2) -T_3(1/2)$ could be negative.