Taylor's theorem
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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.

Proof:

Assume that : is a function that can be expressed in terms of a polynomial (it does not have to appear to be one). The n-th derivative of that function will have a constant term as well as other terms. The "zeroth derivative" of the function (plain old :) has what we will call the "zeroth term" (term with the zeroth power of x) as its constant term. The first derivative will have as a constant the coefficient of the first term times the power of the first term, namely, 1. The second derivative will have as a constant the coefficient of the second term times the power of the second term times the power of the first term: coefficient * 2 * 1. The next will be: coefficient * 3 * 2 * 1. The general pattern is that the n-th derivative's constant term is equal to the n-th term's coefficient times n factorial. Since a polynomial, and by extension, one of its derivatives, equals its constant term at x=0, we can say:

So we now have a formula for determining the coefficient of any term for the polynomial version of :. If you put these together, you get a polynomial approximation for the function.

Calculus Series
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