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

2. Series and Convergence Copyright © Cengage Learning. All rights reserved. 9.2

3. 3 Understand the definition of a convergent infinite series. Use properties of infinite geometric series. Use the nth-Term Test for Divergence of an infinite series. Objectives

4. 4 Sequences A sequence is defined as a function whose domain is the set of positive integers. Although a sequence is a function, it is common to represent sequences by subscript notation rather than by the standard function notation. For instance, in the sequence 1 is mapped onto a 1, 2 is mapped onto a 2, and so on. The numbers a 1, a 2, a 3,..., a n,... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by {a n }.

5. 5 Limit of a Sequence Sequences whose terms approach to limiting values, are said to converge. For instance, the sequence {1/2 n } converges to 0. We are not going to spend much time on theory of sequences because sequence is just a function, so we can apply all the limit theory we learned in Calculus 1 Converges to L crossing it only once Converges to L oscillating

6. 6 Infinite Series

7. 7 One important application of infinite sequences is in representing “infinite summations.” Informally, if {a n } is an infinite sequence, then is an infinite series (or simply a series). The numbers a 1, a 2, a 3, are the terms of the series. For some series it is convenient to begin the index at n = 0 (or some other integer). As a typesetting convention, it is common to represent an infinite series as simply In such cases, the starting value for the index must be taken from the context of the statement. To find the sum of an infinite series, consider the following sequence of partial sums. If this sequence of partial sums converges, the series is said to converge.

8. 8 Maxima fails

9. 9 Infinite Series The series is a telescoping series of the form Note that b 2 is canceled by the second term, b 3 is canceled by the third term, and so on. Because the nth partial sum of this series is S n = b 1 – b n + 1 => telescoping series will converge if and only if b n approaches a finite number as Moreover, if the series converges, its sum is

10. 10 Apart[ ] breaks rational functions to partial fractions Maxima fails

11. 11 Geometric Series

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14. 14 Example 3(a) – Convergent and Divergent Geometric Series The geometric series has a ratio of with a = 3. Because 0 < |r| < 1, the series converges and its sum is

15. 15 Example 3(b) – Convergent and Divergent Geometric Series The geometric series has a ratio of Because |r| ≥ 1, the series diverges. cont’d

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17. 17 My example:

18. 18 nth-Term Test for Divergence

19. 19 The contrapositive of Theorem 9.8 provides a useful test for divergence. This nth-Term Test for Divergence states that if the limit of the nth term of a series does not converge to 0, the series must diverge.

20. 20 Because the limit of the nth term is 0, the nth-Term Test for Divergence does not apply and you can draw no conclusions about convergence or divergence.

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