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

Einbeck, Jochen

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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 Mar 28, 2024
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.
Publisher URL https://ijpam.eu/contents/2004-14-1/8/index.html

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