Infinite Series 9 Copyright © Cengage Learning. All rights reserved.

Alternating Series Copyright © Cengage Learning. All rights reserved. 9.5

3 Use the Alternating Series Test to determine whether an infinite series converges. Use the Alternating Series Remainder to approximate the sum of an alternating series. Classify a convergent series as absolutely or conditionally convergent. Rearrange an infinite series to obtain a different sum. Objectives

4 Alternating Series

5 The simplest series that contain both positive and negative terms is an alternating series, whose terms alternate in sign. For example, the geometric series is an alternating geometric series with Alternating series occur in two ways: either the odd terms are negative or the even terms are negative. Alternating Series

6 =

7 Determine the convergence or divergence of Solution: Note that So, the 1 st condition of Theorem 9.14 is satisfied. Also note that the 2 nd condition of Theorem 9.14 is satisfied because for all n. So, applying the Alternating Series Test, you can conclude that the series converges. NOTE that the same series without sign alternation is the classic harmonic series, which diverges: Example 1 – Using the Alternating Series Test

8

9 not monotonically decreasing => diverges

10 Alternating Series Remainder

11

12

13 Note that this problem is kind of like the previous one except that it is harder because we do NOT know in advance how many terms to use and have to use while loop to determine when bN’s get small enough for the user specified error. n = 6 an = – still too big n = 7 an = – 1 st one which will not affect 3 rd digit

14 My example Harmonic series. Converges very slowly terms are needed before error gets below 0.001

15 Absolute and Conditional Convergence

16 Occasionally, a series may have both positive and negative terms and not be an alternating series. For instance, the series has both positive and negative terms, yet it is not an alternating series. One way to obtain some information about the convergence of this series is to investigate the convergence of the series By direct comparison, you have for all n, so Therefore, by the Direct Comparison Test, the series converges. Absolute and Conditional Convergence

17 Note that even though an <= |an| for any n, we can NOT use the comparison test right away because it was formulated for series with all positive terms.

18

19 52) Absolutely convergent 54) Absolutely convergent 56) Conditionally convergent 58) Absolutely convergent 60) diverges, an does not go to 0 62) Absolutely convergent (compare to convergent geometric series (1/e)^n ) 64) Absolutely convergent 66)Conditionally convergent 68) diverges an -> Pi/2 70) Conditionally convergent, alternating series

20 Rearrangement of Series

21 A finite sum such as (1 + 3 – – 4) can be rearranged without changing the value of the sum. This is NOT necessarily true of an infinite series—it depends on whether the series is absolutely convergent (every rearrangement has the same sum) or conditionally convergent.

22 Chegg solution: