Taylors theorem

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Taylor's theorem, a theorem in analysis 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 continously differentiable on the closed interval [a, x] and n+1 times differentiable on the open intervall (a, x), then we have

                  f'(a)        f(2)(a)               f(n)(a)
  f(x)  =  f(a) + ---- (x-a) + ----- (x-a)2 + ... + ----- (x-a)n  +  R
                   1!            2!                   n!

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:

        f(n+1)(ξ)
  R  =  -------  (x-a)n+1
         (n+1)!      

where ξ is a number between a and x, and

         x  f(n+1)(t) 
  R  =  ∫  -------- (x-t)n dt  
        a     n!

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.