1043 11.1 Sequences and Series (b) The graph of a sequence is a set of discrete points. Plot the points 11, 12, 12, 2.662, 13, 6.242, c , 110, 9.982, as shown in Figure 5(a). At first, the insect population increases rapidly, and then it oscillates about the line y = 9.7. (See the Note following this example.) The oscillations become smaller as n increases, indicating that the population density converges to near 9.7 thousand per acre. In Figure 5(b), the first 20 terms have been plotted with a calculator. NOTE The insect population converges to the value k = 9.7 thousand per acre in Example 3. This value of k can be found by solving the quadratic equation k = 2.85k - 0.19k2, which equates the values of a n for consecutive years. LOOKING AHEAD TO CALCULUS An infinite series converges if the sequence of partial sums S1, S2, S3 , c converges. For example, it can be shown that 1 + 1 2 + 1 3 + 1 4 + gdiverges, while 1 - 1 2 + 1 3 - 1 4 + gconverges. Series and Summation Notation Suppose a person has a starting salary of $30,000 and receives a $2000 raise each year. Then 30,000, 32,000, 34,000, 36,000, 38,000 are terms of the sequence that describe this person’s salaries over a 5-year period. The total earned is given by the finite series 30,000 + 32,000 + 34,000 + 36,000 + 38,000, whose sum is $170,000. A sequence can be used to define a series. For example, the infinite sequence 1, 1 3 , 1 9 , 1 27 , 1 81 , 1 243 , c defines the terms of the infinite series 1 + 1 3 + 1 9 + 1 27 + 1 81 + 1 243 + g. If a sequence has terms a1, a2, a3, c , then Sn is defined as the sum of the first n terms. That is, Sn = a1 + a2 + a3 + g+ an. 0 2 4 6 8 10 12 2 4 6 8 10 12 n an Year Insect Density (in thousands/acre) (a) 0 0 14 21 (b) Figure 5 S Now Try Exercise 95.
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