math12722011-11-07-111203212650-phpapp02.pptx

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1. Alternati ng Series Ratio a nd Roo t T e s ts ∞ 11 ( - 1. ) 1 1 1 1 = 1 - 2 + .3 - 4 + 5 · · · ∑ n=1 n=1 An alternating series is a series whose terms are alternately positive and negative. For example:

2. The Alternating Series Test : If the alternating series satisfies a nd t he n the series is convergent. ( - 1 ) b n = b 1 - b 2 + b 3 - b 4 · · · b n > b n +1 b n Lim b n = ∞ ∑ n=1 n=1 ≤ n ∞

3. Ex: Test series for convergence. ( - 1 ) n 2. n 3 + 1 ∑ n=1 n=1 ∞

4. Ex: Test series for convergence . ( - 1 ) n 2. n 3 + 1 = n 3 + 1 > o ∑ ∞ n=1 n=1 b n n 2.

5. Ex: Test series for convergence . ( - 1 ) n 2. n 3 + 1 = n 3 + 1 > o ∑ ∞ n=1 n=1 b n n 2. Lim b n = 0 ∞ n

6. Ex: Test series for convergence . ( - 1 ) n 2. n 3 + 1 = n 3 + 1 > o ∑ ∞ n=1 n=1 b n n 2. Lim b n = 0 ∞ n b n decreasing?

7. Ex: Test series for convergence . ( - 1 ) n 2. n 3 + 1 = n 3 + 1 > o ∑ ∞ n=1 n=1 b n n 2. Lim b n = 0 ∞ n b n decreasing? When n ≥ 2 , it is decreasing (check by derivative)

8. Ex: Test series for convergence . ( - 1 ) n 2. n 3 + 1 = n 3 + 1 > o ∑ ∞ n=1 n=1 b n n 2. Lim b n = 0 ∞ n b n decreasing? When n ≥ 2 , it is decreasing (check by derivative) ( - 1 ) n 2 n 3 + 1 n=1 ∑ ∞ n=1 So is convergent.

9. Notice that i s divergen t but i s convergent. convergent if 1 = 1 + 1 1 1 1 2 3 4 5 + + + · · · ( - 1 ) n = 1 - 1 + 1 1 1 2 3 4 + 5 · · · A series is called absolutely | a n | is convergent . I t is called conditionall y convergent if it is C onvergent but | a n | is divergent. ∑ n=1 ∞ 2 ∑ ∞ n=1 n=1 ∑ ∞ n=1 ∑ ∞ n=1 a n ∑ ∞ n=1

10. Theorem: Absolutely convergent series is convergent. i s convergent because | n 2 | | cos n | < 1 | n 2 | < | cos n | 1 n 2 ∑ n=1 ∞ n 2

11. Theorem: Absolutely convergent series is convergent. i s convergent because | n 2 | | cos n | < 1 | n 2 | < | cos n | 1 n 2 ∑ n=1 ∞ n 2 Ex: Determine if is convergent. ∑ ∞ n=1 cos n n 2

12. Theorem: Absolutely convergent series is convergent. i s convergent because | n 2 | | cos n | < 1 | n 2 | < | cos n | 1 n 2 ∑ n=1 ∞ n 2 Ex: Determine if is convergent. ∑ ∞ n=1 cos n n 2 It has both positive and negative terms, but not alternating.

13. Theorem: Absolutely convergent series is convergent. i s convergent because | n 2 | | cos n | < 1 | n 2 | < | cos n | 1 n 2 ∑ cos n ∞ n 2 Ex: Determine if is convergent. ∑ ∞ n=1 cos n n 2 It has both positive and negative terms, but not alternating. so ∑ ∞ n=1 n 2 is absolutely convergent, n=1

14. Theorem: Absolutely convergent series is convergent. i s convergent because | n 2 | | cos n | < 1 | n 2 | < | cos n | 1 n 2 ∑ cos n ∞ n 2 Ex: Determine if is convergent. ∑ ∞ n=1 cos n n 2 It has both positive and negative terms, but not alternating. so ∑ ∞ n=1 n 2 is absolutely convergent, n=1

15. The Ratio Test: If , then the series is absolutely convergent (and thus convergent). If or , then the series is divergent. If , no conclusion can be drawn. a n + 1 a n | a n = L > 1 | = Lim n ∞ = L 1 < ∑ ∞ n=1 Lim n | | a n + 1 a n = ∞ ∑ ∞ n=1 ∞ Lim n ∞ a n + 1 a n | | 1

16. The Root Test : If is absolutely convergent (and thus convergent). If , then the series is divergent . If | | = L < 1 , then the series | | = 1 , no conclusion can be drawn. Lim √ a n n ∑ n=1 ∞ a n n ∞ Lim n ∞ √ n a n = L > 1 or = ∞ ∑ a n ∞ n=1 Lim n ∞ √ n a n

17. Ex: Test the series for convergence. n 3 3 n ∑

18. Ex: Test the series for convergence. 00 L " 3 " = :1 3 " 11\ + 1. 11\ = ( 11\ + 1. ) 3 3 11\ + 1. 3 11\ 1. 11\ + 1. 3 11\ = 3 11\ 3 1. --- 3 So is (absolutely) convergent. 00 L " 3 " = :1 3 "