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

*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:

*a*and

*x*, and

*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.