Taylor's theorem

Taylor's theorem, a theorem in calculus 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 only depend 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 intervall (a, x), then we have

$f(x)=f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f^{(2)}(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}+R$

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. Two expressions for R are available:

$R={\frac {f^{(n+1)}(\xi )}{(n+1)!}}(x-a)^{n+1}$

where ξ is a number between a and x, and

$R=\int _{a}^{x}{\frac {f^{(n+1)}(t)}{n!}}(x-t)^{n}\,dt$

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.