A Simple Unifying Formula for Taylor's Theorem and Cauchy's Mean Value Theorem

Einbeck, Jochen

Abstract

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.

Citation

Einbeck, J. (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

• Journal Article Type: Article
• Publication Date: Apr 1, 2004
• Deposit Date: May 11, 2016
• Publicly Available Date: May 13, 2016
• Journal: International journal of pure and applied mathematics : IJPAM.
• Print ISSN: 1311-8080
• Electronic ISSN: 1314-3395
• Publisher: Academic Publications
• Peer Reviewed: Peer Reviewed
• Volume: 14
• Issue: 1
• Pages: 69-74
• Keywords: Taylor's formula, Generalized Mean Value Theorem, Widder's Theorem, Nonparametric smoothing.
• Public URL: https://durham-repository.worktribe.com/output/1548267
• Publisher URL: https://ijpam.eu/contents/2004-14-1/8/index.html