section11-2 - Supplemental Material for Section 11.2:...

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Unformatted text preview: Supplemental Material for Section 11.2: Convergence of an Infinite Series The notion of a convergent infinite series is central to the rest of the material in Chapter 8. The idea is given a sequence of numbers a0 , a1 , a2 , a3 . . . investigate the sum of these numbers; that is, a0 + a1 + a2 + a3 + . . . . What should be clear is that if such a sum exists , then it can be approximated by k a0 + a1 + + ak = n=0 an . For each positive integer k. So as k increases, the approximation gets better and better. The formal definition makes all of this precise using the notion of limit of a sequence. Definition. Let {an } be a sequence (of terms). Then an converges means k that the sequence has a limit that is a number. Otherwise we say n=0 an that an diverges. k The sequence {sk } defined by sk = n=0 an is called the sequence of partial sums of the infinite series an . So said another way an converges means the sequence {sk } has a numerical limit. If an converges, then we let n=0 an = k limk n=0 an . To better understand the meaning of " an converges", let {an } be a sequence of terms. Define a function f on the infinite interval [0, ) by f (x) = an for x in the interval [n, n + 1) for each positive integer n. The graph of f is given in Figure 1. L L a1 a5 a0 a3 L L a7 1 a2 a4 2 3 L 4 5 L 6 7 L 8 a6 Figure 1: Graph of y = f (x) 1 L For any positive integer k, sk = n=0 an = 0 f (x) dx. Thus an con verges; that is, limk sk exists and is a nimber is equivalent to 0 f (x) dx converges. When each an 0, an converges means that the area under the graph of y = f (x) is finite. k k 2 ...
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This note was uploaded on 09/04/2008 for the course MATH 133 taught by Professor Wei during the Spring '07 term at Michigan State University.

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section11-2 - Supplemental Material for Section 11.2:...

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