Taylor’s Theorem, Version 2 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) = T_{N}(x) + R_{N}(x),

where T_{N} is the Nth-degree Taylor polynomial,

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

and there is some number z strictly between* a and x such that

R_{N}(x) = \frac{f^{(N+1)}(z)}{(N + 1)!}(x − a)^{N+1} .

* “Strictly between” means that either a < z < x or x < z < a, whichever is appropriate. Here is a fine point: If x = a, then there is no number strictly between them, so the theorem as stated is false. In that case, however, f(x) is exactly equal to T_{N}(x) (all of whose terms are zero except (possibly) the first (j = 0)).