# IJPAM: Volume 14, No. 1 (2004)

**A SIMPLE UNIFYING FORMULA FOR**

TAYLOR'S THEOREM AND CAUCHY'S

MEAN VALUE THEOREM

TAYLOR'S THEOREM AND CAUCHY'S

MEAN VALUE THEOREM

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 to functional terms
, 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