MATHPOWER TM 12, WESTERN EDITION Chapter 6 Sequences and Series

6.6.2 Infinite Geometric Series A group of runners passes by a water station where there is only one bucket of water. The first runner drinks half of the water. The second runner drinks half of what is left. The third runner drinks half of what is left. This continues for all of the runners. Number of Runners Amount of water The graph illustrates that, as more runners pass the water station, each runner receives less water. As the number of runners increases, the amount of water each runner receives approaches no water. The limiting value of the sequence is 0. This could be written as a sequence:

6.6.3 Consider the sum of the series Infinite Geometric Series [cont’d] For very large values of n, approaches 0. S n = -1(0 - 1) S n = 1

Infinite Geometric Series In general, for the sum of a geometric series where | r | < 1: Where | r | < 1, the value of r n has a limiting value of 0 since the terms become increasingly close to 0 as n increases indefinitely. For the geometric series n +..., where | r | > 1, the sum of the first n terms increases without bound and therefore, the series has no sum.

6.6.5 Finding the Sum of an Infinite Series Find the sum of the each geometric series: S n = 32 a) b) c) For this series, r = -1.5; since | r | > 1, the series has no sum.

6.6.6 A pile driver pounds a metal post into the ground. With the first impact, the post moves 30 cm; with the second impact, it moves 27 cm. Predict the total distance the post will be driven into the ground if a) S 8 = 156 b) S 8 = c) S n = 300 Finding the Sum - An Application a) the distances form an arithmetic sequence and the post is pounded eight times. b) the distances form a geometric sequence and the post is pounded eight times. c) the distances form a geometric sequence and the post is pounded until it has essentially stopped moving.

6.6.7 Suggested Questions: Pages 316 and odd, 18, 25 a