By Ekkehard Kopp
Development at the simple thoughts via a cautious dialogue of covalence, (while adhering resolutely to sequences the place possible), the most a part of the ebook matters the crucial subject matters of continuity, differentiation and integration of actual services. all through, the old context during which the topic was once built is highlighted and specific recognition 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 strategies and ideas.
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PROOF Suppose we have proved this for series with positive terms. Then we can use the convergence ofLm laml and Ln Ibnl to deduce that Lk Ck converges absolutely. As in Theorem 7 in Chapter 3 we can then use the fact that am = a~ - a; and b, = b~ - b;; and the validity of our claim for series with positive terms to show thatE~o Ck = st. Hence we can assume without loss that our series have positive terms. Now we only need to show that Lk Ck converges to the sum st. j ~ k, Functions Defined by Power Series 45 then we obtain: Uo = aobo = solo and, in general, Uk = st.
Consider the sequence given by Xo = a + 1, Xn+l = x n (1 + ak-~n)X for a n fixed a > 0 and fixed kEN. Then: x; > 0, Xn+l < X n and ~ > a. To see this, we again use induction. All three claims are true with n = 0, and we assume them for some fixed n. Now since X n > 0, a < ~ means that Xn+l < X n; however Xn+l remains positive, since kx~ + a - x~ > 0 for all k ~ 1; and finally we can use the (which is greater than -1 because Bernoulli inequality with x = ak-X; Xn kx~ + a - x~ > 0) to conclude that k k x n+!
Lk bk converges absolutely, as claimed. Finally we need to show that the sums of Ln an and Lk bk are equal in general. (Ixl ~ x) = max(-x, 0). Note that x+ and x- are non-negative and that x = x+ when x > 0, while x = _x- when x < O. Thus x == x+ - x- and Ixl = x+ + x- by definition. Now the series Ln a:, Ln a; converge by comparison with Ln Ian I, and the series Lk bt, Lk bl; converge by comparison with Lk Ibkl, since all terms are nonnegative, Moreover, Ln an == Ln a~ - Ln a; and Lk bk = Lk bt - Lk bk , since all the series involved converge, so that we can apply Theorem 1 of Chapter 2 to the partial sums in each case.
Analysis by Ekkehard Kopp