Taylor's theorem

In calculus, Taylor's theorem, 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 depend only 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 continuously differentiable on the closed interval [a, x] and n+1 times differentiable on the open interval (a, x), then we have

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. Three expressions for R are available. Two are shown below:

where ξ is a number between a and x, and

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.