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.