# 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

*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

The exponential function *y* = *e ^{x}* (red) and its corresponding Taylor's polynomial of degree 4 (blue)

*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.

## Proof

We first prove Taylor's theorem with the integral remainder term.

The fundamental theorem of calculus states that

*n*= 0.

Integration by parts yields the case *n* = 1

*n*.

This can be formalized by applying to the technique of induction. So, suppose that Taylor's theorem holds for a particular *n*, that is, suppose that

*x*−

*t*)

^{n}as a function of

*t*is given by −(

*x*−

*t*)

^{n+1}/ (

*n*+ 1), so

*n*+ 1, and hence for all nonnegative integers

*n*.

The remainder term in the Lagrangian form can be derived by the mean value theorem in the following way: