proof of Taylor’s Theorem

Let f(x),a<x<b be a real-valued, n-times differentiable function, and let a<x0<b be a fixed base-point. We will show that for all xx0 in the domain of the function, there exists a ξ, strictly between x0 and x such that


Fix xx0 and let R be the remainder defined by


Next, define


We then have

F(ξ) =f(ξ)+k=1n-1(f(k+1)(ξ)(x-ξ)kk!-f(k)(ξ)(x-ξ)k-1(k-1)!)-R(x-ξ)n-1(n-1)!

because the sum telescopes. Since, F(ξ) is a differentiable function, and since F(x0)=F(x)=f(x), Rolle’s Theorem imples that there exists a ξ lying strictly between x0 and x such that F(ξ)=0. It follows that R=f(n)(ξ), as was to be shown.

Title proof of Taylor’s Theorem
Canonical name ProofOfTaylorsTheorem
Date of creation 2013-03-22 12:33:59
Last modified on 2013-03-22 12:33:59
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 8
Author rmilson (146)
Entry type Proof
Classification msc 26A06