UA196GEN - G eneralized Limits An Asymptotic View of Limit...

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Generalized Limits An Asymptotic View of Limit Operations Suppose then that s n is a sequence with limit c : {s n } = c lim n →∞ We would analyze this statement in terms of asymptotics by observing that s n could be written in the form: s n = c + r n where the sequence r n tends to zero. Intuitively, the asymptotic view tells us that s n “looks” more like the constant c as we move toward the right side of the interval ( 0, ) . This asymptotic analysis is potentially more general since we could replace the constant c with a sequence which varies with n and still make an informative statement about s n even when it doesn’t converge to a limit. In this case, performing a resolution into constant and variable terms is equivalent to finding the limit, since we can rewrite (1) as: s n - r n = c which will give us the value of the limit if we use any value of n. Normally, we don’t use asymptotics to define limits since historically, the limit operator came first. Also, in the logical development of the calculus, limits are introduced first as a fundamental definition, and asymptotic operators are later defined in terms of limits. But there is an important reason for reversing the historical and logical roles of the two approaches to infinite processes . That will become clear after consideration of some specific examples. Author: Chappell Brown First created: June 22, 1993 Last revision: January, 1996 Page: 1 © Copyright 1996 All rights reserved. Permission to copy and distribute for non-profit scholarly purposes granted.
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Generalized Limits It may happen that the sequence s n is given in an asymptotic form. For example if s n = 1 + 1 n then we know immediately that its limit is 1, since the added varying term tends to zero as n tends to infinity. However, more interesting examples arise when we have simple rational processes that depend on the variable n as it tends to infinite. For example, it is well known that: 1 - 1 2 + 1 3 - 1 4 (-1) n n = log2 lim n but here it is not at all obvious how we could write the nth term as the sum of log (2) and some sequence which tends to zero. But asymptotic methods do provide an answer, in fact an infinite sequence of answers which run something like this: 1 - 1 2 + 1 3 - 1 4 (-1) n n = log2 + (-1) n-1 2n + r n 1 1 - 1 2 + 1 3 - 1 4 (-1) n n = log2 + (-1) n-1 2n - (-1) n-1 4n(n-1) + r n 2 1 - 1 2 + 1 3 - 1 4 (-1) n n = Author: Chappell Brown First created: June 22, 1993 Last revision: January, 1996 Page: 2 © Copyright 1996 All rights reserved. Permission to copy and distribute for non-profit scholarly purposes granted.
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Generalized Limits = log2 + (-1) n-1 2n - (-1) n-1 4n(n-1) + (-1) n-1 4n(n-1) 2 + r n 3 The terms that we see developing here all tend to zero, including the remainder sequences, so they have no ultimate effect on the limit of the sequence of sums. Also, we know very little about the remainder sequences save that they tend to zero more quickly than any of the terms that precede them. While the exact nature of the remainder sequences is left somewhat vague, it is their rate of convergence toward zero that provides us with essential insight into the limit. The specific information we have
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