IJPAM: Volume 14, No. 1 (2004)

A SIMPLE UNIFYING FORMULA FOR
TAYLOR'S THEOREM AND CAUCHY'S
MEAN VALUE THEOREM

Jochen Einbeck
Department of Statistics
University of Munich
Akademiestrasse 1, D-80799 Munich, GERMANY
e-mail: einbeck@stat.uni-muenchen.de


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.

Received: April 8, 2004

AMS Subject Classification: 41A58

Key Words and Phrases: Taylor's Formula, Generalized Mean Value Theorem, Widder's Theorem, nonparametric smoothing

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 14
Issue: 1