8.4. Click here for solutions. Click here for answers. OTHER CONVERGENCE TESTS. 3 n. 2n 1! sn 3. 2 n n 2. 3n n 1. 1 n 1 5 n 1 n n 2

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1 SECTION OTHER CONVERGENCE TESTS OTHER CONVERGENCE TESTS A Click here for aswers. S Click here for solutios. 4 Test the series for covergece or divergece s s 3 l 5 8 Approximate the sum of the series to the idicated accuracy. 5. 2! 6. 2! ! 0 2 s 3 2 l Determie whether the series is absolutely coverget, coditioally coverget, or diverget s ! arcta ! !! ! si 2 2 cos6 s 8 3! 2 2 2! ! (five decimal places) 6 Copyright 203, Cegage Learig. All rights reserved.

2 2 SECTION OTHER CONVERGENCE TESTS ANSWERS E Click here for exercises. S Click here for solutios.. Coverget 2. Coverget 3. Diverget 4. Coverget 5. Coverget 6. Diverget 7. Coverget 8. Coverget 9. Diverget 0. Diverget. Diverget 2. Coverget 3. Coverget 4. Coverget Absolutely coverget 20. D ive rg e t 2. Absolutely coverget 22. Absolutely coverget 23. Absolutely coverget 24. Absolutely coverget 25. D ive rg e t 26. D ive rg e t 27. Absolutely coverget 28. D ive rg e t 29. Absolutely coverget 30. Absolutely coverget 3. Absolutely coverget 32. Absolutely coverget 33. D ive rg e t 34. Absolutely coverget 35. D ive rg e t 36. Absolutely coverget 37. Absolutely coverget 38. D ive rg e t Copyright 203, Cegage Learig. All rights reserved.

3 SECTION OTHER CONVERGENCE TESTS 3 SOLUTIONS E Click here for exercises. Copyright 203, Cegage Learig. All rights reserved.. ( ) 3 +4.b = 3 > 0 ad b+ <b for +4 all ; b =0so the series coverges by the Alteratig Series Test =0 ( ) b = 5 is decreasig ad positive for all, ad =0so the series 3 +2 coverges by the Alteratig Series Test. ( ) +. =so + ( ) + does ot exist ad the series diverges by the Test for Divergece. 4. ( ) 2.b = > 0 ad b+ <b for all, 2 ad =0, so the series coverges by the Alteratig 2 Series Test. 5. ( ) +3.b = +3 is positive ad decreasig, ad +3 =0, so the series coverges by the Alteratig Series Test. 6. ( ) = 5 so ( )+ does ot exist ad the series diverges 5 + by the Test for Divergece. ( ) 7. l.b = is positive ad decreasig for l =2 2, ad =0so the series coverges by the l Alteratig Series Test. 8. ( ) 2 +.b = > 0 for all b + <b ( +) 2 + < 2 + ( +) ( 2 + ) < [ ( +) 2 + ] < < 2 +, which is true for all. Also 2 + = / =0. Therefore the series +/2 coverges by the Alteratig Series Test. 9. ( ) =,so 2 ( ) does ot exist. Thus the series diverges 2 + by the Test for Divergece. 0. =( ) 2 4 +,so a = as. 2 Therefore, a 0(i fact the it does ot exist) ad the series ( ) 2 diverges by the Test for 4 + Divergece.. =( ) ,so a = as. Therefore, 0(i fact the it does ot exist) ad the series ( ) 2 2 diverges by the Test for Divergece. 2. ( ) +4. b = > 0 for all. Let +4 x f (x) = x +4. The f 4 x (x) = 2 x (x +4) 2 < 0 if x>4, so{b } is decreasig after =4. +4 = =0. So the series +4/ coverges by the Alteratig Series Test. 3. ( ) + 2. b = > 0 ad b b which is 2 + certaily true. (/2 )=0by l Hospital s Rule, so the series coverges by the Alteratig Series Test l decreases ad 3 l =0,sobythe Alteratig Series Test the series coverges. ( ) 5. (2 )!. b 5 = (2 5 )! = 362,880 < , so ( ) 4 (2 )! ( ) (2 )! b 4 = (2 4)! = ad 40,320 s 3 = , so, correct to four 720 ( ) decimal places, (2)! =0

4 4 SECTION OTHER CONVERGENCE TESTS Copyright 203, Cegage Learig. All rights reserved. 7. b 6 = 2 6 6! = < 0.000, so 46,080 ( ) 5 ( ) ! 2! =0 =0 8. b 8 =/ < ad s 7 = ,625 46, , so correct to five decimal places, = ( ) /2 is a coverget p-series (p = 3 2 > ), so the give series is absolutely coverget. 20. Usig the Ratio Test, + = ( 3)+ / ( +) 3 ( 3) / 3 ( ) 3 =3 =3> + so the series diverges. 2. Usig the Ratio Test, a+ = ( 3)+ / ( +)! ( 3) /! = 3 =0<,so + the series is absolutely coverget. 22. a+ =! ( +)! = =0<,so + ( ) the series is absolutely coverget by the Ratio! 23. Test. 2 + < ad coverges (p =2> ), so 2 2 coverges absolutely by the Compariso Test a+ = /[(2 +)!] /[(2 )!] = (2 +)2 =0 so by the Ratio Test the series is absolutely coverget = 2 3,so ( ) Test for Divergece. 26. ( ) does ot exist, so diverges by the Test for Divergece. 2 diverges by the 3 4 ( ) a+ = 2 /[ + ( +)3 +2] 2 /(3 + ) = = 2 3 < so the series coverges absolutely by the Ratio Test. 28. a+ = 5 /[ ( +2) ] 5 /[ ( +) ] = 5 ( ) = > 29. a+ = ( +2)5 + /[ ( +)3 2(+) ] ( +)5 /(3 2 ) 5 ( +2) = 9( +) 2 = 5 9 < so the series coverges absolutely by the Ratio Test. so si 2 2 ad 2 si 2 2 coverges (p-series, p =2> ), 2 coverges absolutely by the Compariso Test. arcta < π/2 3 ad π/2 coverges (p =3> ), so 3 3 ( ) arcta coverges absolutely by the 3 Compariso Test. cos π 6, so sice coverges (p = 3 2 > ), the give series coverges absolutely by the Compariso Test. 33. a+ = ( +)!/0 + + =!/0 0 = 34. a+ [ = 8 ( +) 3 ]/ [( +)!] (8 3 )/ (!) = 8 ( +) =0< so the series coverges absolutely by the Ratio Test. 35. a+ = ( +) + / /5 2+3 ( ) + = ( +)= 25

5 SECTION OTHER CONVERGENCE TESTS = ( 2)+ ( +) 2 /[( +3)!] ( 2) 2 /[( +2)!] 2( +) 2 = 2 ( +3) =0< so the series coverges absolutely by the Ratio Test. 37. a+ = ( +3)! /[ ( +)!0 +] ( +2)!/(!0 ) = = 0 < so the series coverges absolutely by the Ratio Test. 4 7 (3 2) (3 +) 38. a+ = (2 +)(2 +3) 4 7 (3 2) (2 +) 3 + = 2 +3 = 3 2 > Copyright 203, Cegage Learig. All rights reserved.

4.3. The Integral and Comparison Tests

4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece