a question about Taylor's theorem

Question

I know if I have $a < b$ and If a function an it's derivatives satisfy the conditions of Taylor's theorem then $f (b)=f (a)+f'(a)(b-a)+ \frac {f''(a)}{2}(b-a)^2$ $+......+$ $\frac {f{(a)}^n {(b-a)}^n}{(n!)}$. can I write the same formula to express $f (b)$ in terms of $f (a)$ if $b <a$ ?

Answer 1

The way you have written Taylor's formula it is only valid for arbitrary $a$ and $b$ if $f$is a polynomial of degree $\leq n$. In all other cases there is an error term $R_n$, which can be presented in various forms, all of them exhibiting that $|R_n|$ is very small if $n\gg1$ and $|b-a|\ll1$.

One more thing: Yes, $b$ may be $<a$ as well.