By Ekkehard Kopp

Development at the simple options via a cautious dialogue of covalence, (while adhering resolutely to sequences the place possible), the most a part of the publication matters the relevant themes of continuity, differentiation and integration of genuine features. all through, the historic context within which the topic used to be constructed is highlighted and specific consciousness is paid to exhibiting how precision permits us to refine our geometric instinct. The goal is to stimulate the reader to mirror at the underlying recommendations 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 ~.

1 1. (x~ + 6) for n 2: 1. Show that for n 2: 1 1 = S(xn - 2)(xn - 3) Show by induction that each element of the sequence lies between 2 and 3. Is this sequence monotone? Find its limit as n - t 00. 2. We have shown that a bounded sequence (xn ) either converges or has at least two subsequences which converge to different limits. In this latter case we say 28 Analysis that (x n) oscillates boundedly. Investigate the various possibilities for unbounded sequences by considering the following examples: (i) x; = (_n)n (ii) X n = 2n (iii) x; = - 2n .

This set is either bounded or unbounded above. We consider both cases separately. If C is bounded above, then it contains only finitely many elements of N. If C is empty, let nl = 1. Otherwise we can find nl E N greater than the maximum of C, so none of the numbers nl, ni + 1, nl + 2, ... belong to C. But if ns is not in C there must exist nz E N such that ni > nl and X n 2 ~ x n 1 • Now nz is greater than nl, so it cannot belong to C either. } constructed in this way is increasing. On the other hand, if C is unbounded above, we can find an infinite sequence of natural numbers nl < n2 < n3 < ...