The Taylor's theorem reference article from the English Wikipedia on 24-Jul-2004
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Taylor's theorem

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Topics in Calculus
Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials
Differentiation
Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem
Integration
Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Lists of integrals
Vector Calculus
Vector | Vector field | Matrix | Partial Derivative | Gradient | Flux | Divergence | Divergence Theorem | Del | Curl | Stokes' 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

The exponential function <i>y</i> = <i>e<sup>x</sup></i> (red) and its corresponding Taylor's polynomial of degree 4 (blue)Enlarge

The exponential function y = ex (red) and its corresponding Taylor's polynomial of degree 4 (blue)

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.

Proof

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

The fundamental theorem of calculus states that

This proves the theorem for n = 0.

Integration by parts yields the case n = 1

By repeating this process, we may derive Taylor's theorem for higher values of 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

We can again rewrite the integral using integration by parts. An antiderivative of (x − t)n as a function of t is given by −(xt)n+1 / (n + 1), so
Substituting this in (*) proves Taylor's theorem for 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:

The last integral can be solved immediately, which leads to