Taylor's Ineqaulity
If $|f^{(n+1)}(x)|\leq M$ for $|x-a|\leq d$, then the remainder $R_n(x)$ of the Taylor series satisfies the inequality
$$|R_n(x)|\leq \dfrac{M}{(n+1)!}|x-a|^{n+1}\text{ for } |x-a|\leq d$$
I'm trying to understand the theorem. Can someone explain it to me intuitively what does it mean?
Best Answer
It's a way of saying how close the Taylor approximation is. It also means that Taylor polynomials are generally very good approximations very near the center of the expansion, but might not be good very far away. In particular, if $|x - a| < 1$, then taking high powers of $|x - a|$ is very very small.
As an aside, you may like a note I wrote for my students about Taylor Series. If you think of Taylor polynomials primarily as a tool to get approximations of functions, then it makes sense to understand how good of an approximation you get. [Error estimates are in section 3 of my note]