11.5 Alternating Series; 11.6 Convergence, Ratio, Root Tests

Transcription

1 Alteratig Series; Covergece, Ratio, Root Tests 11.5 Alteratig Series; 11.6 Covergece, Ratio, Root Tests Goals: 1. Recogize Alteratig Series: terms alterate betwee positive ad egative. 2. Apply the Alteratig Series Test: coditios for covergece of alteratig series 3. Coditioal ad Absolute Covergece: compare covergece of alteratig series to covergece of series of absolute values 4. Approximate a series' sum, S, by its th partial sum, s. 5. For s approx, Remaider: R N S s N < a N+1 6. Use Ratio Test, esp. for series with factorials. 7. Use Root test, esp. for series with powers of. Study 11.5 # 1 17, 23 Study 11.6 # 1 13, 19 29, 33, 35, Alteratig Series Alteratig Series: Let a > 0 ( 1) a a 1 + a 2 a 3 + a 4... ( 1) +1 a a 1 a 2 + a 3 a G. Battaly

2 Alteratig Series; Covergece, Ratio, Root Tests 11.5 Alteratig Series Alteratig Series: Let a > 0 ( 1) a a 1 + a 2 a 3 + a 4... ( 1) +1 a a 1 a 2 + a 3 a Alteratig Series Test: ( 1) a ( 1) +1 a a > 0 Coverge if: a 0 a +1 a for all Alteratig Series Alteratig Series Test: ( 1) a ( 1) +1 a a > 0 Coverge if: 1. a 0 2. a +1 a for all ( 1) 3 Coverge? or diverge? G. Battaly

3 Alteratig Series; Covergece, Ratio, Root Tests 11.5 Alteratig Series Alteratig Series Test: ( 1) a ( 1) +1 a a > 0 Coverge if: 1. a 0 2. a +1 a for all ( 1) 3 Coverge? or diverge? a 1 > (+1) 3 3 ( 1) 3 Coverges by alteratig series test retur 11.5 Alteratig Series Alteratig Series Test: ( 1) a ( 1) +1 a a > 0 Coverge if: 1. a 0 2. a +1 a for all ( 1) Similar to harmoic series, but is alteratig. Harmoic series diverges. Does this? Coverge? or diverge? G. Battaly

4 Alteratig Series; Covergece, Ratio, Root Tests 11.5 Alteratig Series Alteratig Series Test: ( 1) a ( 1) +1 a a > 0 Coverge if: 1. a 0 2. a +1 a for all ( 1) +1 Coverge? or diverge? a 1 > for all ( 1) Coverges by alteratig series test retur 11.5 Alteratig Series; 11.6 Covergece, Ratio, Root Tests Defiitios: Coditioally Coverget: The series Ʃa is coditioally coverget if the series coverges, but the series Ʃ a diverges. Absolutely Coverget: The series Ʃa is absolutely coverget if the series Ʃ a coverges. Theorum: If Ʃ a coverges, the Ʃ a coverges. Remaider: Give a coverget alteratig series with sum S, the absolute value of the remaider satisfies: R N S S N a N+1 1 G. Battaly

5 Alteratig Series; Covergece, Ratio, Root Tests 11.5 Alteratig Series; 11.6 Covergece, Ratio, Root Tests Defiitios: Coditioally Coverget: The series Ʃa is coditioally coverget if the series coverges, but the series Ʃ a diverges. Absolutely Coverget: The series Ʃa is absolutely coverget if the series Ʃ a coverges. Series Ʃ a Ʃ a Type, Cov p lik ( 1) 3 3 lik lik ( 1) +1 ( 1) C C C C DD DD Coditioal Covergece Coditioal Covergece Absolute Covergece Absolute Covergece 1 1 ½ 11.5 Alteratig Series; 11.6 Covergece, Ratio, Root Tests Defiitios: Coditioally Coverget: The series Ʃa is coditioally coverget if the series coverges, but the series Ʃ a diverges. Absolutely Coverget: The series Ʃa is absolutely coverget if the series Ʃ a coverges. Series ( 1) 3 ( 1) +1 ( 1) Ʃ a Ʃ a Type C C C C D D Absolute Covergece Coditioal Covergece Coditioal Covergece p 3 1 ½ 1 G. Battaly

6 Alteratig Series; Covergece, Ratio, Root Tests 11.3 Itegral Test The Itegral Test: Let f(x) > 0, cotiuous, ad decreasig, x > 0, ad a f() The, 2 3 Coverge? or diverge? f(x) 1 > 0, cotiuous, decreasig for x > 0 x x 1/2 dx 1 a ad f(x)dx 1 either both coverge or both diverge Both itegral ad Series Diverge ( 1) +1 a 1 > for all ( 1) Coverges by alteratig series test retur 11.5 Alteratig Series 36 Remaider: Give a coverget alteratig series with sum S, the absolute value of the remaider satisfies: R N S S N a N+1 ( 1) +1 Coverges by alteratig series test What is the sum, S, of the series? Ca we estimate S by usig s, eg: s 999? What error is associated with the estimate? sum(seq(y1,x,1,999)) with Y1 ( 1) x+1 /x results i Ca use s 999 as estimate of S: S The error geerated by usig the estimate is bouded by the value of s /1000 R N S S N < a N+1 1/1000 usig 1000 caused a overflow o my calculator G. Battaly

7 Alteratig Series; Covergece, Ratio, Root Tests 11.6 Covergece, Ratio, Root Tests Ratio Test: Let Ʃa be a series with ozero terms. a +1 a a +1 a < 1 > 1 The the series coverges absolutely The the series diverges The test is icoclusive if a +1 a 1 Especially useful for series havig factorials Covergece, Ratio, Root Tests 2! Coverge? or diverge? all terms ozero, has factorial, use ratio test G. Battaly

8 Alteratig Series; Covergece, Ratio, Root Tests 11.6 Covergece, Ratio, Root Tests 2! Coverge? or diverge? all terms ozero, has factorial, use ratio test a +1 a 2 +1 /(+1)! 2 /! 2 +1! 2 (+1)! 2 (+1) 0 < 1 (+1)! (+1)! The series coverges absolutely 11.6 Covergece, Ratio, Root Tests 6 5 Coverge? or diverge? all terms ozero, ot obvious which, try ratio test a +1 a G. Battaly

9 Alteratig Series; Covergece, Ratio, Root Tests 11.6 Covergece, Ratio, Root Tests 6 5 Coverge? or diverge? all terms ozero, ot obvious which, try ratio test a +1 a (+1)/(6/5) +1 / (6/5) (+1) > 1 The series diverges 11.6 Covergece, Ratio, Root Tests Root Test: Let Ʃa be a series a < 1 The the series coverges absolutely a > 1 The the series diverges The test is icoclusive if 1 a Especially useful for series ivolvig th roots. G. Battaly

10 Alteratig Series; Covergece, Ratio, Root Tests 11.6 Covergece, Ratio, Root Tests e 2 Coverge? or diverge? has powers of try root test 11.6 Covergece, Ratio, Root Tests e 2 Coverge? or diverge? has powers of try root test e 2 e2 1/ e 2 0 < 1 The series coverges absolutely G. Battaly

11 Alteratig Series; Covergece, Ratio, Root Tests 11.6 Covergece, Ratio, Root Tests Coverge? or diverge? has powers of try root test 11.6 Covergece, Ratio, Root Tests Coverge? or diverge? has powers of try root test > 1 The series diverges G. Battaly

12 Alteratig Series; Covergece, Ratio, Root Tests 11.6 Covergece, Ratio, Root Tests Note: Neither the Ratio Test or The Root Test work for p series. G. Battaly