Proving Taylor's Theorem from the Fundamental Theorem of Calculus by Fixed-point Iteration Article Swipe
Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most general form can be proved simply as an immediate consequence of the Fundamental Theorem of Calculus (FTOC). The proof shows the deep connection between the Taylor expansion and fixed-point iteration, which is a foundational concept in numerical and functional analysis. One elegant variant of the proof also demonstrates the use of combinatorics and symmetry in proofs in mathematical analysis. Since the proof emphasizes concepts and techniques that are widely used in current science and industry, it can be a valuable addition to the undergraduate mathematics curriculum.
Related Topics
Concepts
Mathematical proof
Fundamental theorem
Analytic proof
Calculus (dental)
Mathematics
Differential calculus
Taylor series
Fixed-point theorem
Taylor's theorem
Automated theorem proving
Brouwer fixed-point theorem
Connection (principal bundle)
Fundamental theorem of calculus
Point (geometry)
Algebra over a field
Discrete mathematics
Pure mathematics
Mathematical analysis
Algorithm
Geometry
Medicine
Dentistry
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2211.01318
- https://arxiv.org/pdf/2211.01318
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4308241905
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W4308241905Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.2211.01318Digital Object Identifier
- Title
-
Proving Taylor's Theorem from the Fundamental Theorem of Calculus by Fixed-point IterationWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2022Year of publication
- Publication date
-
2022-07-18Full publication date if available
- Authors
-
Christopher ThronList of authors in order
- Landing page
-
https://arxiv.org/abs/2211.01318Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2211.01318Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2211.01318Direct OA link when available
- Concepts
-
Mathematical proof, Fundamental theorem, Analytic proof, Calculus (dental), Mathematics, Differential calculus, Taylor series, Fixed-point theorem, Taylor's theorem, Automated theorem proving, Brouwer fixed-point theorem, Connection (principal bundle), Fundamental theorem of calculus, Point (geometry), Algebra over a field, Discrete mathematics, Pure mathematics, Mathematical analysis, Algorithm, Geometry, Medicine, DentistryTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
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