1064 CHAPTER 11 Further Topics in Algebra The sums are getting closer and closer to the number 4. LOOKING AHEAD TO CALCULUS In the discussion of lim nS∞ Sn = 4, we used the phrases “large enough” and “as close as desired.” This description is made more precise in a standard calculus course. EXAMPLE 6 Finding the Sum of the First n Terms Evaluate a 6 i=1 2 # 3i. SOLUTION This series is the sum of the first six terms of a geometric sequence having a1 = 2 # 31 = 6 and r = 3. Sn = a111 - r n2 1 - r Formula for Sn S6 = 611 - 362 1 - 3 Let n = 6, a1 = 6, and r = 3. S6 = 2184 Evaluate. S Now Try Exercise 41. Infinite Geometric Series We extend our discussion of sums of sequences to include infinite geometric sequences such as 2, 1, 1 2 , 1 4 , 1 8 , 1 16 , c , with first term 2 and common ratio 1 2 . Evaluating Sn gives the following sequence of sums. S1 = 2 S2 = 2 + 1 = 3 S3 = 2 + 1 + 1 2 = 7 2 = 3.5 S4 = 2 + 1 + 1 2 + 1 4 = 15 4 = 3.75 S5 = 2 + 1 + 1 2 + 1 4 + 1 8 = 31 8 = 3.875 S6 = 63 16 = 3.9375, and so on For no value of n is Sn = 4. However, if n is large enough, then Sn is as close to 4 as desired. We say the sequence converges to 4. This is expressed as lim nS∞ Sn = 4. Read this as: “The limit of Sn as n increases without bound is 4 .” Because lim nS∞ Sn = 4, the number 4 is the sum of the terms of the infinite geometric sequence 2, 1, 1 2 , 1 4 , c , and 2 + 1 + 1 2 + 1 4 + g= 4. The calculator graph in Figure 10 supports this. 0 0 5 6.1 Figure 10 As n gets larger, Sn approaches 4.
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