Taylor’s Theorem, Version 1

If all the derivatives of the function f up through f^{(N+1)} exist in an interval I containing the number a, then for all x in I, f(x) is well approximated by its Nth-degree Taylor polynomial,

T_{N}(x) = ∑_{j=0}^{N} \frac{f^{(j)}(a)}{j!}(x − a)^{j} ,

in the following sense:

f(x) = T_{N}(x) + R_{N}(x),

where

|R_{N}(x)| ≤ \frac{M|x − a|^{N+1}}{(N + 1)!}

with

M = max_{z∈I} |f^{(N+1)}(z)|.