(9/1/08)Math 10B. Lecture Examples.Section 9.2. Geometric series†Example 1 Suppose you want to go from a point A toward a second point B two miles away. Firstyou go one mile (Figure 1). Then you go a half mile further for a total of 1 +12miles(Figure 2). Next you go h alf that distance ( Figure 3), and so forth, so that at each stageyou go half as far as you did in the previous stage. (a) How far h ave you gone afterone, two, t h ree, five, and eight stages? (b) Predict the limit of your distance from Aas the number of stages tends to ∞.s0 1 2A Bs0 1 2A Bs0 1 2A BFIGURE 1 FIGURE 2 FIGURE 3Answer: (a) See t he table below. (b) The distance seems to be approaching11 −12= 2 miles.Stages Distance (miles) Decimal approximation1 1 11–2 1 +121.51–3 1 +12+ (12)21.751–5 1 +12+ (12)2+ (12)3+ (12)41.93751–81 +12+ (12)2+ (12)3+ (12)4+ (12)5+(12)6+ (12)71.9921875†Lecture notes to accompany Section 9.2 of Calculus by Hughes-Hallett et al.1Math 10B. Lecture Examples. (9/1/08) Section 9.2, p. 2Example 2 This time supp ose you go one mile from A toward B in the first stage, but then14mileback toward A in the second stage, (14)2=116mile away from A in the third stage,(14)3=164mile toward A in the fourth stage, and so on. (a) How far you are fromA after one , two, three, five, and eight stages? (b) Predict the limit of your distancefrom A as the number of stages tends t o ∞.Answer: (a) See t he table below. (b) The distance seems to be approaching11 +14= 0.8.Stages Total distance Decimal approximation1 1 11–2 1 −140.751–3 1 −14+ (14)20.81251–5 1 −14+ (14)2− (14)3+ (14)40.80078131–81 −14+ (14)2− (14)3+ (14)4− (14)5+(14)6− (14)70.7999878Example 3 Give a concise formula for511Xn=2(0.99)nand fi n d its approximate decimal value.Answer:511Xn=2(0.99)n= (0.99)21 − (0.99)5101 − 0.99.= 97.427602Example 4 Evaluate400Xj=0(−1)jAnswer:400Xj=0(−1)j= 1Section 9.2, p. 3 Math 10B. Lecture Examples. (9/1/08)Example 5 Does the infinite geometric series∞Xj=0(0.95)jconverge? If so, give its value.Answer: The Geometric Series converges because |0 .95| < 1. •∞Xj=0(0.95)j= 20Example 6 Give the exact value of the infinite geometic series∞Xj=3(−34)j.Answer:∞Xj=3(−34)j= −21112Example 7 A woman opens a new savings account on January 1, 2000 and deposits $5000 in it.She deposits $5000 on each subsequent January 1 through the year 1020. The accountpays 8% annual interest compounded annually, based on the December 31 balance. Shedoes not make any additional withdrawals or deposits. How much is in the account onJanuary 2, 2020?Answer: [Balance] = 50001 − (1.08)211 − 1.08.= $252,115Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 10.2: Examples 1–3‡The chapter and section numbers o n Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof t he textbook for the