UA296NEW - A N ew Perspective on the Foundations of...

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A New Perspective on the Foundations of Analysis A New Perspective on the Foundations of Analysis by Chappell Brown Author: Chappell Brown First created: June , 1993 Last revision :February, 1996 Page: 1 Revised version of a paper origninally published in Physics Essays , Sept. 1995
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A New Perspective on the Foundations of Analysis ABSTRACT T.E.Phipps, Jr.’s new definition of discrete infinite process value is examined in the light of asymptotic analysis. The analysis suggests a more general approach to limit operations, which are conventionally used to define these values. An analogy between imaginary numbers and divergent processes is proposed, which helps to explain the current state of analysis, where many infinite processes lack numerical content due to problems created by divergence. A general method whereby any real function can be resolved into convergent and divergent parts, just as complex numbers are resolved into real and imaginary parts is described. By defining a generalized limit as the limit of the convergent part of a function, a path to a fully general formulation of limits that would finally banish divergence from analysis is described. Key Words : divergent series, summability, terminal summation, asymptotics, numerical methods, set theory. Author: Chappell Brown First created: June , 1993 Last revision :February, 1996 Page: 2 Revised version of a paper origninally published in Physics Essays , Sept. 1995
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A New Perspective on the Foundations of Analysis Resume La nouvelle definition donnee par T.E. Phipps au traitement des valeurs discretes et infinies est examinee par la voie de l’analyse asymptotique. Cette analyse propose une methode general pour traiter les limites qui sont utilisees pour la definition des valeurs. Une similitude entre les nombres imaginaires et la divergence du processus est proposee. Cette derniere va aider a expliquer l’etat actuel de l’analyse on plusieur traitements manquent le contenu numerique pose par le probleme de divergence. Une method pour une complete et generale formulation des limites qui pourraient finalement faire disparaitre la divergence est decrite. 1. Introduction After more than three centuries of development, it might seem that little could be added to established methods for assigning values to infinite sums, and yet T.E.Phipps, Jr. asks us to reconsider the fundamental definition, first formalized in the early 19th century by Cauchy and Bolzano whereby the value of the expression a 0 + a 1 + a 2 + (1) Author: Chappell Brown First created: June , 1993 Last revision :February, 1996 Page: 3 Revised version of a paper origninally published in Physics Essays , Sept. 1995
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A New Perspective on the Foundations of Analysis is obtained as the limit lim n { a 0 + a 1 + a 2 + + a n - 1 } (2) when it exists. This definition has an intuitive feel that gives it a certain validity even outside the
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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UA296NEW - A N ew Perspective on the Foundations of...

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