UA496SUM - Formal Sums Is it possible to assign a unique...

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Formal Sums Is it possible to assign a unique value to every infinite sum in a way that is consistent with the axioms of the real numbers? This question will no doubt elicit diverse responses . For example, one could answer in the affirmative by assigning the value zero to every infinite sum. There is no inherent contradiction in making such an assignment. But obviously, such a definition trivializes the potential of infinite processes as a tool of analysis. Another common answer is negative, due to problems that arise with divergent sums. An example would be the following: Suppose we assign a value , say x , to the sum 1 + 1 + 1 + 1 . .. then we could write x = 1 + 1 + 1 + ... = 1 + ( 1 + 1 + 1 + . .. ), or x = 1 + x . Subtracting x from both sides, we find that 0 = 1, a contradiction. Consequently, this divergent sum cannot be assigned any finite value consistent with the axioms of arithmetic. This example bolsters a commonly held belief that certain infinite sums are, by their very nature, outside the realm of real arithmetic. If we look at the partial sums of the above example: 1=1 1+1 = 2 1+1+1 = 3 1+1+1+ ... +1 = n it seems evident that x has a value that is greater than any finite number. This judgement is reinforced by the equation x = 1 + x which could reasonably be construed to have the solution " ". Then, by labelling operations such as " - Author: Chappell Brown First created: September , 1992 Last revision: April, 1996 Page: 1 © Copyright 1996 All rights reserved. Permission to copy and distribute for non-profit scholarly purposes granted. http://www.limit.com
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Formal Sums " as undefined, we avoid the contradictory result 0 = 1 . This common textbook example seems to force us to the conclusion that mathematicians arrived at in the last century, that infinite sums by their nature divide into two classes: convergent sums for which meaningful finite values exists, and divergent sums, for which no reasonable value can be found. As compelling as these, and other examples like them seem, they do not actually constitute a rigorous proof of this belief. Indeed, an examination of our introductory question will reveal that the dichotomy between convergent and divergent processes is more an article of faith than an artifact of logic. How exactly do we get at this question in a comprehensive and rigorous fashion? The intellectual development of the calculus over the past four centuries could be viewed as an experiment to nail down a definitive answer. Newton introduced infinite series as a tool for studying the behavior of complex processes in terms of simple and well understood functions. A statement such as: e x = 1 + x + x 2 2! + x 3 3! + was viewed as a means for characterizing a complex function such as e x in terms of a series of simpler polynomials. Newton and his contemporaries found that the smaller the value of x, the more quickly the series of polynomials: 1 , 1 + x , 1 + x + x 2 2! , 1
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UA496SUM - Formal Sums Is it possible to assign a unique...

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