13.7 Sums of Infinite Series

1. 13.7 Sums of Infinite Series

2. The sumofaninfiniteseries of numbers (or infinite sum) is defined to be the limit of its associated sequence of partial sums. One way to find a sum is to examine the partial sums. Often the pattern will be one from the math induction section (13-4). Ex 1) Find the sum, if it exists, of the infinite series sum exists looks like pg 702 #6 ⁞

3. Ex 2) Determine whether the series converges or diverges. If it diverges, tell why. write out a few partial sums diverges, since sequence of partial sums diverges If a series is a geometric series, it is VERY easy to determine its sum if it exists.

4. Ex 3) Determine whether each series converges or diverges. If it converges, find the sum. If it diverges, tell why. a) b) 0.1 + 0.2 + 0.4 + 0.8 + … geometric, r = 2 2 > 1 geometric, diverges converges c) 5 + 0.2 + 0.04 + 0.008 + 0.0016 + … geometric a1 = 0.2 r = 0.2

5. The sum of a geometric series may be used to express a repeating decimal as a number in fraction form. Ex 4) Express 1.6413413413413… as a ratio of two integers. can be written as 1.6 + 0.0413 + 0.0000413 + 0.0000000413 + … geometric a1 = 0.0413 r = 0.001 so

6. The sum of a geometric series may be used to analyze some physical situations. Ex 5) A ball is dropped from a height of 8 ft. Each time it strikes the ground, it bounces back to a height of 70% of the distance it fell. Find the total distance the ball traveled. there is an initial 8 ft and then twice .7 times the previous height 8(.70) 8(.70) 8 8 + 2(.7)(8) + 2(.7)2(8) + … a1 = 11.2 r = .7

7. Homework #1308 Pg 725 #1–13 all, 16, 17, 22, 23, 29–32, 34, 35