1. Copyright © Cengage Learning. All rights reserved. 11.2 Series

2. 22 If we try to add the terms of an infinite sequence we get an expression of the form a 1 + a 2 + a 3 +... + a n +... which is called an infinite series (or just a series) and is denoted, for short, by the symbol A series is the sum of all the terms of an infinite sequence.

3. 33 Series It would be impossible to find a finite sum for the series 1 + 2 + 3 + 4 + 5 +... + n +... because if we start adding the terms we get the cumulative sums 1, 3, 6, 10, 15, 21,... and, after the nth term, we get n(n + 1)/2, which becomes very large as n increases. However, if we start to add the terms of the series we get

4. 44 Series The table shows that as we add more and more terms, these partial sums become closer and closer to 1.

5. 55 Series In fact, by adding sufficiently many terms of the series we can make the partial sums as close as we like to 1. So it seems reasonable to say that the sum of this infinite series is 1 and to write We use a similar idea to determine whether or not a general series has a sum.

6. 66 Series We consider the partial sums s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 s 4 = a 1 + a 2 + a 3 + a 4 and, in general, s n = a 1 + a 2 + a 3 +... + a n = These partial sums form a new sequence {s n }, which may or may not have a limit.

7. 77 Try This: Write a formula for the nth partial sum of the series.

8. 88 Series If lim n  s n = s exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series  a n.

9. 99 Series

10. 10 Series: Geometric An important example of an infinite series is the geometric series a + ar + ar 2 + ar 3 +... + ar n–1 +... = a  0 Each term is obtained from the preceding one by multiplying it by the common ratio r.

11. 11 Series We summarize the results of Example 2 as follows. Copy

12. 12 Series: Harmonic Show that the harmonic series is divergent. Solution: For this particular series it’s convenient to consider the partial sums s 2, s 4, s 8, s 16, s 32,... and show that they become large. Therefore the harmonic series diverges.

13. 13 Series The converse of Theorem 6 is not true in general. If lim n  a n = 0, we cannot conclude that  a n is convergent. Copy

14. 14 Series The Test for Divergence follows from Theorem 6 because, if the series is not divergent, then it is convergent, and so lim n  a n = 0. Copy

15. 15 Series Copy

16. 16 Try These: Determine if each series converges or diverges.

17. 17 Telescoping Series: Example 7 pg753

18. 18 Video Examples: Harmonic Series : Telescoping Series: https://www.youtube.com/watch?v=H8HVYiEqzBo https://www.youtube.com/watch?v=5LMzbgfZ8cA Geometric Series: https://www.youtube.com/watch?v=xjmy5hkZccY https://www.youtube.com/watch?v=4yyLfrsSXQQ

19. 19 Homework: Page 757 # 4-6 even, 18-36 even