Taylor's theorem, a theorem in analysis named after the mathematician Brook Taylor who stated it in 1712, allows the approximation of a differentiable function near a point by a polynomial whose coefficients only depend on the derivatives of the function at that point. The precise statement is as follows: If n≥0 is an integer and f is a function which is n times continously differentiable on the closed interval [a, x] and n+1 times differentiable on the open intervall (a, x), then we have
f'(a) f(2)(a) f(n)(a) f(x) = f(a) + ---- (x-a) + ----- (x-a)2 + ... + ----- (x-a)n + R 1! 2! n!
Here, n! denotes the factorial of n, and R is a remainder term which depends on x and is small if x is close enough to a. Two expressions for R are available:
f(n+1)(ξ) R = ------- (x-a)n+1 (n+1)!
where ξ is a number between a and x, and
x f(n+1)(t) R = ∫ -------- (x-t)n dt a n!
If R is expressed in the first form, the so-called Lagrange form, Taylor's theorem is exposed as a generalization of the mean value theorem (which is also used to prove this version), while the second expression for R shows the theorem to be a generalization of the fundamental theorem of calculus (which is used in the proof of that version).
For some functions f(x), one can show that the remainder term R approaches zero as n approaches ∞; those functions can be expressed as a Taylor series in a neighborhood of the point a and are called analytic.
Taylor's theorem (with the integral formulation of the remainder term) is also valid if the function f has complex values or vector values. Furthermore, there is a version of Taylor's theorem for functions in several variables.