Sec 8.4. Alternating Series Test. A. Before Class Video Examples. Example 1: Determine whether the following series is convergent or divergent.

Transcription

1 Sec 8.4 Alteratig Series Test A. Before Class Video Examples Example 1: Determie whether the followig series is coverget or diverget. a) ( 1)+1 =1 b) ( 1) 2 1 =1 Example 2: Determie whether the followig series is absolutely coverget, coditioally coverget, or diverget. a) ( 1) + 4 =1 b) ( 1)+1 =1 1

2 c) ( 1) + =1 Example : Determie whether the followig series is coverget or diverget. a) 4! =1 b) ( ) =1 2

3 B. Alteratig Series a 1 Alteratig Series: b cos b b b b b i. If b1 b for all (i.e. sequece is decreasig), ad ii. If lim b 0 1 The 1 1 b coverges. If ot, it diverges. Examples: Determie whether each of the followig series is coverget or diverget. 1.) ) ) 1 cos 1

4 C. The Alteratig Series Estimatio Theorem 1 S b of the alteratig series that satisfies 1 If is the sum 1 i. If b1 b for all (i.e. sequece is decreasig), ad ii. If lim b 0 R = S S b 1 Remaider after terms True Sum Partial Sum Examples: a. Determie whether the series is coverget or diverget. b. Fid the partial sums, S5 through S 8. c. Give a upper boud for R 5. (i.e. give the maximum value for the differece betwee R 5 ad the true sum.) d. For the differece betwee R 5 ad the true sum to be withi , to which term should calculated? (i.e. what is the smallest value of to guaratee the error less tha ?) S be 4

5 a. Determie whether the series is coverget or diverget. b. Fid the partial sum S 4 c. Give the true sum S if possible. d. Fid the true differece betwee S ad the partial sum S 4. Compare the value to R4 calculated by the estimatio formula. e. For the differece betwee R 5 ad the true sum to be withi 0.001, to which term should (i.e. what is the smallest value of to guaratee the error less tha 0.001?) S be calculated? 5

6 . 0 1! 1 a. Determie whether the series is coverget or diverget. b. Fid the partial sum S 4. c. Fid R 4. d. What should be for R 4 to be withi ? 6

7 D. Absolute Covergece Alteratig Series may: Diverge Coditioally coverge Absolutely coverge If 1 a is coverget, the 1 1 a is ABSOLUTELEY coverget. 1 If you have show that 1 1 If you have show that 1 absolute or coditioal. a is diverget, it you are doe. a is coverget, it remais to be determied whether the covergece is o If you sequetially show that 1 a is coverget, you have ABSOLUTE covergece. o Alteratively: Examples: 1. If you sequetially show that 1 If, i your first step, you are able to show that 1 coverget. If, i your first step, you show that 1 coverget or diverget a is diverget, you have CONDITIONAL covergece. a is coverget, the 1 a is diverget, a is ABSOLUTELY a might still be CONDITIONALY

8 E. Ratio Test For 1 a 1 a (a geeral or alteratig series) If lim L, ad a L 1 a 1 will Absolutely Coverge L 1 a 1 will Diverge L 1 a 1 Icoclusive (Try aother test) * Use this test wheever the sequece cotais a factorial. * Be careful ot to cofuse this with the Limit Compariso Test! Examples: !

9 . 1 Geeral Examples (Determie which test to use)

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11 F. Root Test For 1 a (a geeral or alteratig series) If lim a L, ad L 1 a 1 will Absolutely Coverge L 1 a 1 will Diverge L 1 a 1 Icoclusive (Try aother test) Examples: l 11

12 F. Calculator 1.) Use your calculator to evaluate TI 8/84: 2d STAT (List) Scroll to Math Choose 5: sum( 2d STAT Scroll to OPS Choose 5: seq( Eter the fx, x, lowerboud, upperboud ) (For ifiity, eter 100) ENTER TI 89: F Choose 4: sum Eter the fx, x, lowerboud, upperboud ) (For ifiity, use the ifiity key secod fuctio o CATALOG key) ENTER Webwork: 1. Match each of the followig series with the correct statemet: A. The series is absolutely coverget. C. The series is coditioally coverget. D. The series diverges. 12

13 For # 6 ad 8: Cosider the series. Attempt the Ratio Test to determie whether the series coverges. Give, Which of the followig statemets is true? A. The Ratio Test says that the series coverges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series coverges coditioally. D. The Ratio Test is icoclusive, but the series coverges absolutely by aother test or tests. E. The Ratio Test is icoclusive, but the series diverges by aother test or tests. F. The Ratio Test is icoclusive, but the series coverges coditioally by aother test or tests

14 11. Match each of the followig series with the first correct statemet. A. The series is absolutely coverget usig compariso with a p-series. B. The series is absolutely coverget usig compariso with a geometric series. C. The series is absolutely coverget usig the Ratio Test. D. The series diverges