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(9/1/08)
Math 10B. Lecture Examples.
Section 9.2. Geometric seriesโ
Example 1 Suppose you want to go from a point A toward a second point B two miles away. First you go one mile (Figure 1). Then you go a half mile further for a total of 1 + 12 miles (Figure 2). Next you go half that distance (Figure 3), and so forth, so that at each stage you go half as far as you did in the previous stage. (a) How far have you gone after one, two, three, five, and eight stages? (b) Predict the limit of your distance from A as the number of stages tends to โ.
0 1 2 s
A B
0 1 2 s
A B
0 1 2 s
A B
FIGURE 1 FIGURE 2 FIGURE 3
Answer: (a) See the table below. (b) The distance seems to be approaching 1 1 โ (^12)
= 2 miles.
Stages Distance (miles) Decimal approximation
1 1 1
1โ2 1 + 12 1. 5
1โ3 1 + 12 + ( 12 )^2 1. 75
1โ5 1 + 12 + ( 12 )^2 + ( 12 )^3 + ( 12 )^4 1. 9375
1โ8 1 +^
1 2 + (^
1 2 )
5 +( 21 )^6 + ( 12 )^7
โ Lecture notes to accompany Section 9.2 of Calculus by Hughes-Hallett et al.
Math 10B. Lecture Examples. (9/1/08) Section 9.2, p. 2
Example 2 This time suppose you go one mile from A toward B in the first stage, but then 14 mile back toward A in the second stage, ( 14 )^2 = 161 mile away from A in the third stage, ( 14 )^3 = 641 mile toward A in the fourth stage, and so on. (a) How far you are from A after one, two, three, five, and eight stages? (b) Predict the limit of your distance from A as the number of stages tends to โ.
Answer: (a) See the table below. (b) The distance seems to be approaching 1 1 + (^14)
= 0.8.
Stages Total distance Decimal approximation
1 1 1
1โ2 1 โ 14 0. 75
1โ3 1 โ 14 + ( 14 )^2 0. 8125
1โ5 1 โ 14 + ( 14 )^2 โ ( 14 )^3 + ( 14 )^4 0. 8007813
1โ8 1 โ^
1 4 + (^
1 4 )
5
+( 41 )^6 โ ( 14 )^7
Example 3 Give a concise formula for
โ^511
n=
(0.99)n^ and find its approximate decimal value.
Answer:
โ^511
n=
(0.99)n^ = (0.99)^2
1 โ (0.99)^510 1 โ 0. 99
= 97.. 427602
Example 4 Evaluate
โ^400
j=
(โ1)j
Answer:
โ^400
j=
(โ1)j^ = 1
