Taylor Series/ Polynomials

• Taylor Theorem Proofs: The second proof uses the Cauchy mean value theorem which makes things very neat. An interesting point brought up by the article is that Taylor's theorem with a first degree remainder term is simply Lagrange's mean value theorem. The author suggests this might be inspiration for using Cauchy's generalized mean value theorem in proving the more general Taylor theorem.
• Single- and Multi-variable: Extends the Taylor series to multivariable functions.
• Asymptotic Nature of Taylor Polynomials: Defines the kth order Taylor polynomial centered at x=c as the unique k degree polynomial of the class o((x-c)^k) as x -->c. With this definition, some nice properties of the Taylor polynomials are explored including the use of Taylor polynomials in evaluating l'Hopital-type limits whose derivatives are gross to evaluate directly. The article then discusses Lagrange's form of the Taylor polynomial remainder and derives the integral form of the remainder. Unfortunately, I cannot yet embed in-line equations to google sites... maybe that will change... or maybe I'll move to another website platform.