By Ekkehard Kopp
Development at the uncomplicated ideas via a cautious dialogue of covalence, (while adhering resolutely to sequences the place possible), the most a part of the publication matters the important subject matters of continuity, differentiation and integration of genuine services. all through, the ancient context during which the topic used to be constructed is highlighted and specific cognizance 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 techniques and concepts.
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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. Also, Completeness and Convergence 23 < 2, and if x; < 2 then Xn+l = JI + X n < Jf+2 = ,J3 < 2 also. So (x n ) is bounded above by 2. We have shown that (x n ) is increasing and bounded above, hence x = limn X n exists in ~.
Now 0 ::s an + lanl ::s 2lanl, so that the series Ln>l(a n + lanl) converges by comparison with 2Ln>1Ianl. Write k . Uk = Ln=l(an + lanl), so that the senes Ln>l(a n + lanl) has sum u = hmk--+oouk. But Sk = Uk - tk; therefore (Sk) converges to = u - t, so that L:l an = s. s We can now see immediately that the Ratio and Root Tests are really tests for absolute convergence; we reformulate them as follows, using the above Proposition: Ratio Test Suppose we are given a real series Ln an. (i) If there exists r < 1 and no E N such that la~:l converges (absolutely).
Now test your program on several examples. 24 Analysis Subsequences The above examples should convince you of the importance of monotone sequences. But by no means all sequences are monotone, of course! For example, -1,1 - 1,1 - 1,1 ... jumps 'up' and 'down' alternately, and has no limit, while X n = (-:t defines another non-monotone sequence which does have a limit, namely O. , in the second it is the decreasing I I I sequence '2 ' 4 ' 6' ... The meaning of 'subsequence' should be clear from the above: we 'pick out' some of the elements of the sequence (x n ) , usually according to some fixed rule.