# 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.