TAILIEUCHUNG - kaplan 2
Infinite infinite series is an indicated sum of the form:.going on to infinitely many terms. Such series are familiar even in the with numbers. Thus one writes:this is the same as sayingOf course, we do not interpret this as an ordinary addition problem, which forever to carry out. Instead we say, for example, 113 equals to accuracy. We "round off" after a certain number of decimal places and use rational a sufficiently good approximation to the procedure just used applies to the general series a1 a2 + . . a, . . . To evaluate it, we round off after k terms and replace the series by the finite sum++ +.376Advanced Calculus, Fifth , the rounding-off procedure must be justified; we must be sure that than k terms would not significantly affect the result. For the a justification is impossible. For one term gives 1 as sum, two terms give 2, give 3, and so on; rounding off is of no help here. This series is an example divergent serieso n t h e other hand, for the -+-+-+.+-+.-4 9 seems safe to round off. Thus one has as sums of the first k terms:.+a+The sums do not appear to change much, and one would hazard the guess that of 50 terms would not differ from that of four terms by more than, say, $While it will be seen below that such appearances can be misleading, in this instinct happens to be right. This series is an example of a convergent seriesIt is the purpose of the present chapter to systematize the procedure to formulate tests that enable one to decide when rounding off is meaningful.(convergent case) or meaningless (divergent case). As the example of the of shows, the notion of infinite series lies right at the heart of the the real number system. Accordingly, a complete theory of series would profound analysis of the real number system. It is not our purpose to cany this , so some of the rules will be justified in an intuitive manner. For a the reader is referred to the books by Hardy and Knopp listed at the end chapter. Sections , and 1 cover real variable theory in more provide proofs for some of the key results of this chapterBecause of the large number of theorems appearing in this chapter, the be numbered serially from 1 to 59 throughout the to each positive integer n there is assigned a number sn,then the numbers s, to form an infinite sequence. The numbers are thought of as arranged in order,.according to subscript:.Examples of such sequences are the following:.377Chapter 6 Infinite Series.'These are formed by the rules: times it is convenient to number the members of the sequence starting with 0,.with 2, or with some other integerA sequence s, is said to converge to the numbers s or to have the l m t s:. s, = s()n+wif to each number t > 0 a value N can be found such thatI S , - S ( N.();This is illustrated in Fig. (a). If s, does not converge, it is said to divergeThe limit s is clearly unique. For if s' is a limit different from s , we let 6 =.1 - s 1 ) / 2Then as in (), Is, - sl N , and similarly, Is, - s'l N'. Therefore if No is the larger of N and N', for n > No we have both Is, -s I < Is, - s'l c t , so that+ Sn - 8'1 5 I S26 = I S - s I I = ( S - 8,- s,I+ IS,- s'I < 6+€=2t,1and that is impossible.,.BAa. ConvergentI.,.IIs.,I.: . . IIIIS2IIIIIFigure $4$5BAe. D~
đang nạp các trang xem trước
