By Ekkehard Kopp

ISBN-10: 0340645962

ISBN-13: 9780340645963

Construction at the uncomplicated recommendations via a cautious dialogue of covalence, (while adhering resolutely to sequences the place possible), the most a part of the publication issues the principal issues of continuity, differentiation and integration of actual features. all through, the ancient context within which the topic was once built is highlighted and specific realization is paid to exhibiting how precision permits us to refine our geometric instinct. The purpose is to stimulate the reader to mirror at the underlying strategies and ideas.

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Historical comment In the eighteenth century, when the development of Calculus techniques led mathematicians to exploit the properties of power series (especially Taylor and Maclaurin series) which arose naturally in the Calculus, most mathematicians had few qualms about treating power series as "true' polynomials, and using them as such. 2. Example I Perhaps the most consistent advocate of treating infinite series as polynomials was Leonhard Euler (1707-83). Here is his remarkable calculation of the sum of the series Ln~l ~.

We have x = j(X), which in turn can enable us to find the value of x. The following basic examples illustrate the technique - note that in these cases it is usually simpler to use induction to compare X n and Xn+l directly, rather than considering their ratio. Example 3 Suppose that J J Xl = 1, and for n :::: 1, X n+l = ,JI + X n. /2, etc. It is certainly not immediately clear what the limit will be ifit exists! However, the first few terms do suggest that (x n ) is increasing. /2 =X2. Now if Xn-l < X n, then X n = ,JI + Xn-l < ,JI + X n = Xn+l, hence (x n ) is increasing, by induction.

N+4)! (21i)! 2. 3 Alternati ng series Much more care is needed when we deal with series whose terms are not necessarily positive: for example, is there a difference in the behaviour of 111 1+"2+)++;;+00. and 1 1 1 n-l 1 1-"2+)-4+···+(-1) ;;+oo.? The former series diverges, but might there be 'enough cancellation' between the terms of the latter to allow it to converge? Also: in the former case it seems plausible that the order in which we add terms together should not matter. _~-~) 5 10 which we cap apparently write as 12 + ...

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