Taylor's Theorem: The Elusive c is Not So Elusive

Summary

For a suitably nice, real-valued function f defined on an open interval containing [a, b], f (b) can be expressed as p_n(b) (the nth Taylor polynomial of f centered at a) plus an error term of the (Lagrange) form f⁽ⁿ⁺¹⁾(c)/(n + 1)! for some c in (a, b). This article is for those who think that not much more can be said about where in (a, b) c is located. Supplementary materials, including further numerical illustrations and sketches of some proofs, are available at http://faculty.colostate-pueblo.edu/rick.kreminski.

Additional information

Notes on contributors

Rick Kreminski

Rick Kreminski ([email protected]) currently serves as Dean of the College of Science and Mathematics at Colorado State University-Pueblo, having relocated there after 17 years at Texas A&M-Commerce. While his family barely tolerated a midlife crisis that led to yet another degree (Dedman School of Law, JD 2008), his wife has given him permission to train for a possible 5th marathon: running up and then down a hill visible from both home and office, namely, Pikes Peak.