# Analysis (Modular Mathematics Series) by Ekkehard Kopp

By Ekkehard Kopp

Development at the simple recommendations via a cautious dialogue of covalence, (while adhering resolutely to sequences the place possible), the most a part of the e-book issues the crucial subject matters of continuity, differentiation and integration of actual services. all through, the historic context during which the topic was once built is highlighted and specific recognition is paid to displaying how precision permits us to refine our geometric instinct. The purpose is to stimulate the reader to mirror at the underlying suggestions and concepts.

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**Example text**

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 < ... all of which lie in C. But by definition of C, ni > nl 26 Analysis means that x n 2 ~ X n l , n3 > n2 implies x n 3 this way is decreasing. ~ x n 2 ' etc. So the subsequence found in • Theorem 2-Bolzano-Weierstrass Theorem - - - Every bounded monotone sequence in IR has a convergent subsequence. PROOF Given the sequence (x n ) choose a monotone subsequence (x n, ) .

To deal with the variety of possible limits of subsequences of divergent sequences we introduce a new concept and some notation: • Definition 2 (i) A real number a is an accumulation point of the sequence (x n ) if some subsequence of (x n ) converges to a. (ii) Suppose (x n ) is a given real sequence. If (x n ) converges, then x = limn~oo x; is its only accumulation point, since for each e > 0 there are at most finitely many n for which IXn - x] 2: e. We shall write N(x, e) for the open interval (x - e, x + e) of radius e centred at x (we call this the e-neighbourhood of x), and say that nearly all elements of (x n ) belong to N(x, e) if the above statement is true.

Clearly, if r ::s 0 the terms ~ do not converge to 0, so the series must diverge. For r > 0 we can use the Condensation Test, which means we need to look at the geometric series Lk 2k(2 = Lk(2 1- r)k. This series converges if and only if2 1- r < 1, and this occurs exactly ifr > 1. Hence we have proved the series Ln ~ converges if and only if r > 1. Note how this distinguishes between the two special cases (r = 1 or 2) which we discussed earlier. t) Example 9 Of the above tests, the Ratio Test is much the easiest one to use.