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
We first prove Taylor's theorem with the Lagrange remainder term by induction.
For n = 0, Taylor's theorem states that
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The mean value theorem asserts that this holds.
Now, suppose that Taylor's theorem holds for a particular n, that is, suppose that
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We can rewrite the integral using integration by parts. An antiderivative of (x − t)n as a function of t is given by −(x−t)n+1 / (n + 1), so
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Substituting this in (*) proves Taylor's theorem for n + 1, and hence for all nonnegative integers n.
Taylor's theorem with the Cauchy remainder term now follows by applying the intermediate value theorem.
