Publication details for Jochen EinbeckEinbeck, Jochen (2004). A Simple Unifying Formula for Taylor's Theorem and Cauchy's Mean Value Theorem. International Journal of Pure and Applied Mathematics 14(1): 69-74.
- Publication type: Journal Article
- ISSN/ISBN: 1311-8080, 1314-3395
- Keywords: Taylor's formula, Generalized Mean Value Theorem, Widder's Theorem, nonparametric smoothing
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
We introduce a formula which generalizes Taylor's theorem from powers of linear terms z-x to functional terms \phi(z)-\phi(x), leading to a formula which reduces in a special case to Cauchy's generalized mean value theorem. In other words, regarding Cauchy's mean value theorem as an extension of the simple mean value theorem, we provide the analogous extension of Taylor's theorem. The filling of this gap is easy and requires only mathematics on an undergraduate level, so that the mentioned analogy might be a useful tool for illustration at schools and universities.