MTH 201 Homework 4 Solution, Fall 2016

MATH 201. Homework IV Solution. Section 10.4: 590 Chapter 10 Infinite Sequences and Series 10.4 COMPARISON TESTS 1. Compare with , which is a convergent p-series, since p 2 1. Both series have nonnegative terms for n 1. For ! _ œ n 1 " n2 œ � n 1, we have n n 30 . Then by Comparison Test, converges. Ÿ � Ê 2 2 1 1 1 n n 30 n 30 2 2 2 � � ! _ œ n 1 2. Compare with , which is a convergent p-series, since p 3 1. Both series have nonnegative terms for n 1. For ! _ œ n 1 " n3 œ � n 1, we have n n 2 . Then by Comparison Test, Ÿ � Ê Ê Ê 4 4 1 1 n n 1 n n 1 n 1 n n 2 n n 2 n n 2 n 2 n 2 4 4 4 4 3 4 4 4 � � � � � � � ! _ œ n 1 converges. 3. Compare with , which is a divergent p-series, since p 1. Both series have nonnegative terms for n 2. For ! _ œ n 2 " " # Èn œ Ÿ n 2, we have n 1 n . Then by Comparison Test, diverges. � Ÿ Ê È È ! 1 1 1 n 1 n n 1 È È È � � _ œ n 2 4. Compare with , which is a divergent p-series, since p 1 1. Both series have nonnegative terms for n 2. For ! _ œ n 2 " n œ Ÿ n 2, we have n n n . Thus diverges. � Ÿ Ê Ê œ Ê 2 2 1 1 n n 1 n 2 n 1 n 2 n n n n n n n n n n n n n n 2 2 2 2 2 2 2 � � � � � � � ! _ œ n 2 5. Compare with , which is a convergent p-series, since p 1. Both series have nonnegative terms for n 1. ! _ œ n 1 " n 3 2 3 2 œ � For n 1, we have 0 cos n 1 . Then by Comparison Test, converges. Ÿ Ÿ Ê Ÿ 2 cos n 1 cos n n n n 2 2 3 2 3 2 3 2 ! _ œ n 1 6. Compare with , which is a convergent geometric series, since r 1. Both series have nonnegative terms for ! ¹ ¹ _ œ n 1 " 3 3 1 n l l œ � n 1. For n 1, we have n 3 3 . Then by Comparison Test, converges. † Ê Ÿ n n 1 1 1 n 3 3 n 3 † † n n n ! _ œ n 1 7. Compare with . The series is a convergent p-series, since p 1, and the series ! ! ! _ _ _ œ œ œ n 1 n 1 n 1 È È 5 5 n n n 1 3 2 3 2 3 2 3 2 œ � 5 converges by Theorem 8 part 3. Both series have nonnegative terms for n 1. For n 1, we have œ È ! _ œ n 1 1 n3 2 n n 4n 4n n 4n n 4n 5n n 4n 5n 20 5 n 4 5. 3 4 3 4 4 3 4 4 4 4 3 4 4 n 4n n 4 Ÿ Ê Ÿ Ê � Ÿ � œ Ê � Ÿ � œ � Ê Ÿ a b 4 3 4� � 5 Then by Comparison Test, converges. Ê Ÿ Ê Ÿ Ê Ÿ œ n n 4 n 4 n 4 n n 4 n n 4 n 4 5 n 4 5 n 4 5 n 3 4 4 3 4 3 4 3 2 a b È � � � � � � � � É É É ! _ œ n 1 8. Compare with , which is a divergent p-series, since p 1. Both series have nonnegative terms for n 1. For ! _ œ n 1 " " # Èn œ Ÿ n 1, we have n 1 2 n 2 2 n 1 3 n 2 n 1 3n 3 2 n n n 3 Ê Ê � Ê � Ê � È È È È È ˆ ‰ n 2 n n n n 3 1 Ê � � � Ê Ê Ê Ê 2 2 n n 2 n 1 n 1 n 1 n 3 n 3 n n 3 n n 3 n n 2 n 1 1 1 1 È Ê É ˆ ‰ È ˆ ‰ ˆ ‰ È È È � � � � � � � � � � 2 2 2 2 2 2 . Then by Comparison Test, diverges. Ê È È È È È n 1 n 1 n 3 n 3 1 n � � � � 2 2 ! _ œ n 1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.4 Comparison Tests 591 9. Compare with , which is a convergent p-series, since p 2 1. Both series have positive terms for n 1. ! _ œ n 1 " Ä_ n b n a 2 n n œ � lim 1 0. Then by Limit Comparison Te œ œ œ œ œ œ � lim lim lim lim lim n n n n n 1 n n n 3 3n 2n 6n 2 6 n 2n 3n 4n 6n 4 6 Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � � � n 2 n n 3 3 2 2 3 2 2 3 2 2 st, converges. ! _ œ n 1 n 2 n n 3 � � � 3 2 10. Compare with , which is a divergent p-series, since p 1. Both series have positive terms for n 1. ! _ œ n 1 " " # Ä_ Èn n a b œ Ÿ lim n n 1 1 0. Then by Limit Comparison œ œ œ œ œ œ œ � lim lim lim lim lim n n n n n 1 n n n n n 2n 1 2 n 2 n 2 2n 2 Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � É È n 1 n 2 2 2 2 2 2 É É É É È Test, diverges. ! É _ œ n 1 n 1 n 2 � � 2 11. Compare with , which is a divergent p-series, since p 1 1. Both series have positive terms for n 2. ! _ œ n 2 " Ä_ n b n a œ Ÿ lim n n 1 0. Then by Limit Comp œ œ œ œ œ œ � lim lim lim lim lim n n n n n 1 n n n n 1 3n 2n 1 6n 2 6 n + n 3n 2n 6n 2 6 Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � � � � n n 1 n 1 n 1 2 3 2 2 3 2 2 arison Test, diverges. ! _ œ n 2 n n 1 n 1 n 1 a b a ba b � � � 2 12. Compare with , which is a convergent geometric series, since r 1. Both series have positive terms for ! ¹ ¹ _ œ n 1 " 2 2 1 n l l œ � n 1. 1 0. Then by Limit Comparison Test, conver œ œ œ œ � lim lim lim lim n n n n a b 1 2 3 4 4 ln 4 3 4 4 4 ln 4 2 Ä_ Ä_ Ä_ Ä_ Î � � n n 2n 3 4n n n n n n n n ! _ œ n 1 ges. 13. Compare with , which is a divergent p-series, since p 1. Both series have positive terms for n 1. ! _ œ n 1 " Ä_ Èn 1 2 b n a œ Ÿ lim n n . Then by Limit Comparison Test, diverges. œ œ œ œ _ lim lim lim n n n 1 n n 4 5 5 5 4 4 n Ä_ Ä_ Ä_ Î † 5n n 4n n n n n È È ˆ ‰ ! _ œ n 1 14. Compare with , which is a convergent geometric series, since r 1. Both series have positive terms for ! ˆ ‰ ¹ ¹ _ œ n 1 2 2 5 5 n l l œ � n 1. exp ln exp n ln œ œ œ œ lim lim lim lim lim n n n n n a b 10n 8 10n 8 10n 8 2 5 10n 15 10n 15 10n 15 n n Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � � n n 2n 3 5n 4 n n ˆ ‰ a b ˆ ‰ ˆ ‰ ˆ ‰ exp exp exp exp œ œ œ œ lim lim lim lim n n n n ln 1 n 1 n 10n 15 10n 8 100n 230n 120 70n 70n Ä_ Ä_ Ä_ Ä_ Î � Î � � � � � ˆ ‰ a ba b 10n 15 10 10 10n 8 10n 15 10n 8 2 2 2 2 exp exp e 0. Then by Limit Comparison Test, converges. œ œ œ � lim lim n n 140n 140 2n 3 200n 230 200 5n 4 7 10 n Ä_ Ä_ � � Î � ! ˆ ‰ _ œ n 1 15. Compare with , which is a divergent p-series, since p 1 1. Both series have positive terms for n 2. ! _ œ n 2 " Ä_ n b n a œ Ÿ lim n n n . Then by Limit Comparison Test, diverges. œ œ œ œ œ _ lim lim lim lim n n n n 1 n ln n 1 n ln n n 1 Ä_ Ä_ Ä_ Ä_ Î Î " ln n ! _ œ n 2 16. Compare with , which is a convergent p-series, since p 2 1. Both series have positive terms for n 1. ! _ œ n 1 " Ä_ n b n a 2 n n œ � lim 1 0. Then by Limit Comparison Test, ln 1 converg œ œ œ œ � � lim lim lim n n n ln 1 1 n n 1 1 Ä_ Ä_ Ä_ � Î � � � " Š ‹ Š ‹ Š ‹ n2 2 2 1 2 1 n2 n3 2 n3 n2 ! ˆ ‰ _ œ n 1 es. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.4 Comparison Tests 591 9. Compare with , which is a convergent p-series, since p 2 1. Both series have positive terms for n 1. ! _ œ n 1 " Ä_ n b n a 2 n n œ � lim 1 0. Then by Limit Comparison Te œ œ œ œ œ œ � lim lim lim lim lim n n n n n 1 n n n 3 3n 2n 6n 2 6 n 2n 3n 4n 6n 4 6 Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � � � n 2 n n 3 3 2 2 3 2 2 3 2 2 st, converges. ! _ œ n 1 n 2 n n 3 � � � 3 2 10. Compare with , which is a divergent p-series, since p 1. Both series have positive terms for n 1. ! _ œ n 1 " " # Ä_ Èn n a b œ Ÿ lim n n 1 1 0. Then by Limit Comparison œ œ œ œ œ œ œ � lim lim lim lim lim n n n n n 1 n n n n n 2n 1 2 n 2 n 2 2n 2 Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � É È n 1 n 2 2 2 2 2 2 É É É É È Test, diverges. ! É _ œ n 1 n 1 n 2 � � 2 11. Compare with , which is a divergent p-series, since p 1 1. Both series have positive terms for n 2. ! _ œ n 2 " Ä_ n b n a œ Ÿ lim n n 1 0. Then by Limit Comp œ œ œ œ œ œ � lim lim lim lim lim n n n n n 1 n n n n 1 3n 2n 1 6n 2 6 n + n 3n 2n 6n 2 6 Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � � � � n n 1 n 1 n 1 2 3 2 2 3 2 2 arison Test, diverges. ! _ œ n 2 n n 1 n 1 n 1 a b a ba b � � � 2 12. Compare with , which is a convergent geometric series, since r 1. Both series have positive terms for ! ¹ ¹ _ œ n 1 " 2 2 1 n l l œ � n 1. 1 0. Then by Limit Comparison Test, conver œ œ œ œ � lim lim lim lim n n n n a b 1 2 3 4 4 ln 4 3 4 4 4 ln 4 2 Ä_ Ä_ Ä_ Ä_ Î � � n n 2n 3 4n n n n n n n n ! _ œ n 1 ges. 13. Compare with , which is a divergent p-series, since p 1. Both series have positive terms for n 1. ! _ œ n 1 " Ä_ Èn 1 2 b n a œ Ÿ lim n n . Then by Limit Comparison Test, diverges. œ œ œ œ _ lim lim lim n n n 1 n n 4 5 5 5 4 4 n Ä_ Ä_ Ä_ Î † 5n n 4n n n n n È È ˆ ‰ ! _ œ n 1 14. Compare with , which is a convergent geometric series, since r 1. Both series have positive terms for ! ˆ ‰ ¹ ¹ _ œ n 1 2 2 5 5 n l l œ � n 1. exp ln exp n ln œ œ œ œ lim lim lim lim lim n n n n n a b 10n 8 10n 8 10n 8 2 5 10n 15 10n 15 10n 15 n n Ä_ Ä_ Ä_ Ä_ Ä_ Î � � � � � � n n 2n 3 5n 4 n n ˆ ‰ a b ˆ ‰ ˆ ‰ ˆ ‰ exp exp exp exp œ œ œ œ lim lim lim lim n n n n ln 1 n 1 n 10n 15 10n 8 100n 230n 120 70n 70n Ä_ Ä_ Ä_ Ä_ Î � Î � � � � � ˆ ‰ a ba b 10n 15 10 10 10n 8 10n 15 10n 8 2 2 2 2 exp exp e 0. Then by Limit Comparison Test, converges. œ œ œ � lim lim n n 140n 140 2n 3 200n 230 200 5n 4 7 10 n Ä_ Ä_ � � Î � ! ˆ ‰ _ œ n 1 15. Compare with , which is a divergent p-series, since p 1 1. Both series have positive terms for n 2. ! _ œ n 2 " Ä_ n b n a œ Ÿ lim n n n . Then by Limit Comparison Test, diverges. œ œ œ œ œ _ lim lim lim lim n n n n 1 n ln n 1 n ln n n 1 Ä_ Ä_ Ä_ Ä_ Î Î " ln n ! _ œ n 2 16. Compare with , which is a convergent p-series, since p 2 1. Both series have positive terms for n 1. ! _ œ n 1 " Ä_ n b n a 2 n n œ � lim 1 0. Then by Limit Comparison Test, ln 1 converg œ œ œ œ � � lim lim lim n n n ln 1 1 n n 1 1 Ä_ Ä_ Ä_ � Î � � � " Š ‹ Š ‹ Š ‹ n2 2 2 1 2 1 n2 n3 2 n3 n2 ! ˆ ‰ _ œ n 1 es. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 592 Chapter 10 Infinite Sequences and Series 17. diverges by the Limit Comparison Test (part 1) when compared with , a divergent p-series: ! _ œ n 1 " Èn lim lim lim n n n Ä _ Ä _ Ä _ Œ � Š ‹ n n n œ œ œ È È È n 2 n n n � " " # � # ˆ ‰ 1 6 18. diverges by the Direct Comparison Test since n n n n n 0 , which is the nth � � � � � Ê � È 3 n n n � " È term of the divergent series or use Limit Comparison Test with b ! n 1 " " n n n œ 19. converges by the Direct Comparison Test; , which is the nth term of a convergent geometric series sin n 2n n Ÿ " # 20. converges by the Direct Comparison Test; and the p-series converges 1 cos n 2 n n n � " Ÿ ! 21. diverges since lim 0 n Ä _ 2n 2 3n 1 3 � œ Á 22. converges by the Limit Comparison Test (part 1) with , the nth term of a convergent p-series: " n lim lim 1 n n Ä _ Ä _ Š ‹ Š ‹ n n n n œ œ ˆ ‰ n n � " 23. converges by the Limit Comparison Test (part 1) with " , the nth term of a convergent p-series: n Š ‹ 10n n(n 1)(n 2) 10 20 1 20 � � 19. converges by the Direct Comparison Test; , which is the nth term of a convergent geometric series sin n 2n n Ÿ " # 20. converges by the Direct Comparison Test; and the p-series converges 1 cos n 2 n n n � " Ÿ ! 21. diverges since lim 0 n Ä _ 2n 2 3n 1 3 � œ Á 22. converges by the Limit Comparison Test (part 1) with , the nth term of a convergent p-series: " n lim lim 1 n n Ä _ Ä _ Š ‹ Š ‹ n n n n œ œ ˆ ‰ n n � " 23. converges by the Limit Comparison Test (part 1) with " , the nth term of a convergent p-series: n lim lim lim lim 10 n n n n Ä _ Ä _ Ä _ Ä _ Š ‹ Š ‹ 10n n(n 1)(n 2) n œ œ œ œ 10n n 20n 1 20 n 3n 2 2n 3 2 � � � � � 24. converges by the Limit Comparison Test (part 1) with " , the nth term of a convergent p-series: n lim lim lim lim 5 n n n n Ä _ Ä _ Ä _ Ä _ � � Š ‹ 5n 3n n (n 2) n 5 n œ œ œ œ 5n 3n 15n 3 30n n 2n 5n 10 3n 4n 5 6n 4 � � � � � � � � 25. converges by the Direct Comparison Test; , the nth term of a convergent geometric series ˆ ‰ ˆ ‰ ˆ ‰ n n 3n 1 3n 3 n n n � " � œ 26. converges by the Limit Comparison Test (part 1) with , the nth term of a convergent p-series: " n lim lim lim 1 1 n n n Ä _ Ä _ Ä _ Š ‹ Š ‹ n n 2 œ œ � œ É É n 2 2 n n � 27. diverges by the Direct Comparison Test; n ln n ln n ln ln n and diverges � Ê � Ê � � " " " " n ln n ln (ln n) n ! _ œ n 3 28. converges by the Limit Comparison Test (part 2) when compared with , a convergent p-series: ! _ œ n 1 " n lim lim lim 2 lim 0 n n n n Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ (ln n) n n n œ œ œ œ (ln n) n 1 n 2(ln n) ln n Š ‹ 29. diverges by the Limit Comparison Test (part 3) with , the nth term of the divergent harmonic series: " n lim lim lim lim n n n n Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ ˆ ‰ ˆ ‰ 1 n ln n 2 n n n œ œ œ œ _ È È n n ln n 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 19. converges by the Direct Comparison Test; , which is the nth term of a convergent geometric series sin n 2n n Ÿ " # 20. converges by the Direct Comparison Test; and the p-series converges 1 cos n 2 n n n � " Ÿ ! 21. diverges since lim 0 n Ä _ 2n 2 3n 1 3 � œ Á 22. converges by the Limit Comparison Test (part 1) with , the nth term of a convergent p-series: " n lim lim 1 n n Ä _ Ä _ Š ‹ Š ‹ n n n n œ œ ˆ ‰ n n � " 23. converges by the Limit Comparison Test (part 1) with " , the nth term of a convergent p-series: n lim lim lim lim 10 n n n n Ä _ Ä _ Ä _ Ä _ Š ‹ Š ‹ 10n n(n 1)(n 2) n œ œ œ œ 10n n 20n 1 20 n 3n 2 2n 3 2 � � � � � 24. converges by the Limit Comparison Test (part 1) with " , the nth term of a convergent p-series: n lim lim lim lim 5 n n n n Ä _ Ä _ Ä _ Ä _ � � Š ‹ 5n 3n n (n 2) n 5 n œ œ œ œ 5n 3n 15n 3 30n n 2n 5n 10 3n 4n 5 6n 4 � � � � � � � � 25. converges by the Direct Comparison Test; , the nth term of a convergent geometric series ˆ ‰ ˆ ‰ ˆ ‰ n n 3n 1 3n 3 n n n � " � œ 26. converges by the Limit Comparison Test (part 1) with , the nth term of a convergent p-series: " n lim lim lim 1 1 n n n Ä _ Ä _ Ä _ Š ‹ Š ‹ n n 2 œ œ � œ É É n 2 2 n n � 27. diverges by the Direct Comparison Test; n ln n ln n ln ln n and diverges � Ê � Ê � � " " " " n ln n ln (ln n) n ! _ œ n 3 28. converges by the Limit Comparison Test (part 2) when compared with , a convergent p-series: ! _ œ n 1 " n lim lim lim 2 lim 0 n n n n Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ (ln n) n n n œ œ œ œ (ln n) n 1 n 2(ln n) ln n Š ‹ 29. diverges by the Limit Comparison Test (part 3) with , the nth term of the divergent harmonic series: " n lim lim lim lim n n n n Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ ˆ ‰ ˆ ‰ 1 n ln n 2 n n n œ œ œ œ _ È È n n ln n 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 24. converges by the Limit Comparison Test (part 1) with " , the nth term of a convergent p-series: n lim lim lim lim 5 n n n n Ä _ Ä _ Ä _ Ä _ � � Š ‹ 5n 3n n (n 2) n 5 n œ œ œ œ 5n 3n 15n 3 30n n 2n 5n 10 3n 4n 5 6n 4 � � � � � � � � 25. converges by the Direct Comparison Test; , the nth term of a convergent geometric series ˆ ‰ ˆ ‰ ˆ ‰ n n 3n 1 3n 3 n n n � " � œ 26. converges by the Limit Comparison Test (part 1) with , the nth term of a convergent p-series: " n lim lim lim 1 1 n n n Ä _ Ä _ Ä _ Š ‹ Š ‹ n n 2 œ œ � œ É É n 2 2 n n � 27. diverges by the Direct Comparison Test; n ln n ln n ln ln n and diverges � Ê � Ê � � " " " " n ln n ln (ln n) n ! _ œ n 3 28. converges by the Limit Comparison Test (part 2) when compared with , a convergent p-series: ! _ œ n 1 " n lim lim lim 2 lim 0 n n n n Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ (ln n) n n n œ œ œ œ (ln n) n 1 n 2(ln n) ln n Š ‹ 29. diverges by the Limit Comparison Test (part 3) with , the nth term of the divergent harmonic series: " n lim lim lim lim n n n n Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ ˆ ‰ ˆ ‰ 1 n ln n 2 n n n œ œ œ œ _ È È n n ln n 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.4 Comparison Tests 593 30. converges by the Limit Comparison Test (part 2) with , the nth term of a convergent p-series: " n lim lim lim 8 lim 8 lim 32 l n n n n n Ä _ Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ Š ‹ Š ‹ ˆ ‰ ˆ ‰ (ln n) n n 4n 4n 2 ln n n n œ œ œ œ œ (ln n) n n ln n im 32 0 0 n Ä _ " n œ œ † 31. diverges by the Limit Comparison Test (part 3) with , the nth term of the divergent harmonic series: " n lim lim lim lim n n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ 1 ln n n n œ œ œ œ _ n 1 ln n � " Š ‹ 32. diverges by the Integral Test: dx u du lim u lim b ln 3 ' ' 2 ln 3 ln (x 1) x 1 2 � � # " " # # # œ œ œ � œ _ b b Ä _ Ä _ � ‘ a b b ln 3 33. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n for " # n � � n 2 n n 1 n n n 1 n or use Limit Comparison Test with . Ê � � Ê � � Ê � # # $ $Î# # " " � a b È n n n 1 1 n È 34. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n " # # n � � n 1 nn n or use Limit Comparison Test with . Ê � � Ê � Ê � # $Î# $Î# � " " � È n 1 n n n 1 n n È È 35. converges because which is the sum of two convergent series: ! ! ! _ _ _ œ œ œ n 1 n 1 n 1 " � " �" # n n2 n2 n n n œ � converges by the Direct Comparison Test since , and is a convergent geometric series ! ! _ _ œ œ n 1 n 1 " " " �" # # n2 n 2 n n n n � 36. converges by the Direct Comparison Test: and , the sum of ! ! ˆ ‰ _ _ œ œ n 1 n 1 n 2 n 2 n2 n n2 n n � " " " " " " # n n n n n œ � � Ÿ � the nth terms of a convergent geometric series and a convergent p-series 37. converges by the Direct Comparison Test: , which is the nth term of a convergent geometric series " " � 3 1 3 n 1 n 1 � 38. diverges; lim lim 0 n n Ä _ Ä _ Š ‹ ˆ ‰ 3 3 3 3 3 n 1 n n �" " " " œ � œ Á 39. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 1 1 5 5 n l l œ � lim lim lim 0. n n n Ä _ Ä _ Ä _ Š ‹ a b n 1 1 n 3n 2 5n n 2 † Î � � � 1 5 n 1 1 n 3n 2n 3 œ œ œ 40. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 3 1 4 5 n l l œ � lim lim lim 1 0. n n n Ä _ Ä _ Ä _ Š ‹ a b ˆ ‰ ˆ ‰ 2 3 n n 3 4 n n n n n n n n 8 12 n 9 12 3 4 8 12 1 9 12 1 1 1 Î � � � � œ œ œ œ � 41. diverges by Limit Comparison Test: compare with , which is a divergent p-series, lim lim ! _ œ n 1 1 2 n 1 n n n Ä _ Ä _ Š ‹ 2 n n n 2n n Î œ � n 2n lim lim 1 0. œ œ œ � n n Ä _ Ä _ 2 ln 2 1 2 ln 2 2 ln 2 2 ln 2 n n 2 n 2 n � a b a b 42. diverges by the definition of an infinite series: ln ln n ln n 1 , s ln 1 ln 2 ln 2 ln 3 ! ! ˆ ‰ � ‘ a b a b a b _ _ œ œ n 1 n 1 n n 1 k � œ � � œ � � � ln k 1 ln k ln k ln k 1 ln k 1 lim s � Þ Þ Þ � � � � � � œ � � Ê œ �_ a b a b a b a b a b k Ä _ k Section 10.4 Comparison Tests 593 30. converges by the Limit Comparison Test (part 2) with , the nth term of a convergent p-series: " n lim lim lim 8 lim 8 lim 32 l n n n n n Ä _ Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ Š ‹ Š ‹ ˆ ‰ ˆ ‰ (ln n) n n 4n 4n 2 ln n n n œ œ œ œ œ (ln n) n n ln n im 32 0 0 n Ä _ " n œ œ † 31. diverges by the Limit Comparison Test (part 3) with , the nth term of the divergent harmonic series: " n lim lim lim lim n n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ 1 ln n n n œ œ œ œ _ n 1 ln n � " Š ‹ 32. diverges by the Integral Test: dx u du lim u lim b ln 3 ' ' 2 ln 3 ln (x 1) x 1 2 � � # " " # # # œ œ œ � œ _ b b Ä _ Ä _ � ‘ a b b ln 3 33. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n for " # n � � n 2 n n 1 n n n 1 n or use Limit Comparison Test with . Ê � � Ê � � Ê � # # $ $Î# # " " � a b È n n n 1 1 n È 34. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n " # # n � � n 1 nn n or use Limit Comparison Test with . Ê � � Ê � Ê � # $Î# $Î# � " " � È n 1 n n n 1 n n È È 35. converges because which is the sum of two convergent series: ! ! ! _ _ _ œ œ œ n 1 n 1 n 1 " � " �" # n n2 n2 n n n œ � converges by the Direct Comparison Test since , and is a convergent geometric series ! ! _ _ œ œ n 1 n 1 " " " �" # # n2 n 2 n n n n � 36. converges by the Direct Comparison Test: and , the sum of ! ! ˆ ‰ _ _ œ œ n 1 n 1 n 2 n 2 n2 n n2 n n � " " " " " " # n n n n n œ � � Ÿ � the nth terms of a convergent geometric series and a convergent p-series 37. converges by the Direct Comparison Test: , which is the nth term of a convergent geometric series " " � 3 1 3 n 1 n 1 � 38. diverges; lim lim 0 n n Ä _ Ä _ Š ‹ ˆ ‰ 3 3 3 3 3 n 1 n n �" " " " œ � œ Á 39. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 1 1 5 5 n l l œ � lim lim lim 0. n n n Ä _ Ä _ Ä _ Š ‹ a b n 1 1 n 3n 2 5n n 2 † Î � � � 1 5 n 1 1 n 3n 2n 3 œ œ œ 40. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 3 1 4 5 n l l œ � lim lim lim 1 0. n n n Ä _ Ä _ Ä _ Š ‹ a b ˆ ‰ ˆ ‰ 2 3 n n 3 4 n n n n n n n n 8 12 n 9 12 3 4 8 12 1 9 12 1 1 1 Î � � � � œ œ œ œ � 41. diverges by Limit Comparison Test: compare with , which is a divergent p-series, lim lim ! _ œ n 1 1 2 n 1 n n n Ä _ Ä _ Š ‹ 2 n n n 2n n Î œ � n 2n lim lim 1 0. œ œ œ � n n Ä _ Ä _ 2 ln 2 1 2 ln 2 2 ln 2 2 ln 2 n n 2 n 2 n � a b a b 42. diverges by the definition of an infinite series: ln ln n ln n 1 , s ln 1 ln 2 ln 2 ln 3 ! ! ˆ ‰ � ‘ a b a b a b _ _ œ œ n 1 n 1 n n 1 k � œ � � œ � � � ln k 1 ln k ln k ln k 1 ln k 1 lim s � Þ Þ Þ � � � � � � œ � � Ê œ �_ a b a b a b a b a b k Ä _ k Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 30. converges by the Limit Comparison Test (part 2) with , the nth term of a convergent p-series: " n lim lim lim 8 lim 8 lim 32 l n n n n n Ä _ Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ Š ‹ Š ‹ ˆ ‰ ˆ ‰ (ln n) n n 4n 4n 2 ln n n n œ œ œ œ œ (ln n) n n ln n im 32 0 0 n Ä _ " n œ œ † 31. diverges by the Limit Comparison Test (part 3) with , the nth term of the divergent harmonic series: " n lim lim lim lim n n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ 1 ln n n n œ œ œ œ _ n 1 ln n � " Š ‹ 32. diverges by the Integral Test: dx u du lim u lim b ln 3 ' ' 2 ln 3 ln (x 1) x 1 2 � � # " " # # # œ œ œ � œ _ b b Ä _ Ä _ � ‘ a b b ln 3 33. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n for " # n � � n 2 n n 1 n n n 1 n or use Limit Comparison Test with . Ê � � Ê � � Ê � # # $ $Î# # " " � a b È n n n 1 1 n È 34. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n " # # n � � n 1 nn n or use Limit Comparison Test with . Ê � � Ê � Ê � # $Î# $Î# � " " � È n 1 n n n 1 n n È È 35. converges because which is the sum of two convergent series: ! ! ! _ _ _ œ œ œ n 1 n 1 n 1 " � " �" # n n2 n2 n n n œ � converges by the Direct Comparison Test since , and is a convergent geometric series ! ! _ _ œ œ n 1 n 1 " " " �" # # n2 n 2 n n n n � 36. converges by the Direct Comparison Test: and , the sum of ! ! ˆ ‰ _ _ œ œ n 1 n 1 n 2 n 2 n2 n n2 n n � " " " " " " # n n n n n œ � � Ÿ � the nth terms of a convergent geometric series and a convergent p-series 37. converges by the Direct Comparison Test: , which is the nth term of a convergent geometric series " " � 3 1 3 n 1 n 1 � 38. diverges; lim lim 0 n n Ä _ Ä _ Š ‹ ˆ ‰ 3 3 3 3 3 n 1 n n �" " " " œ � œ Á 39. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 1 1 5 5 n l l œ � lim lim lim 0. n n n Ä _ Ä _ Ä _ Š ‹ a b n 1 1 n 3n 2 5n n 2 † Î � � � 1 5 n 1 1 n 3n 2n 3 œ œ œ 40. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 3 1 4 5 n l l œ � lim lim lim 1 0. n n n Ä _ Ä _ Ä _ Š ‹ a b ˆ ‰ ˆ ‰ 2 3 n n 3 4 n n n n n n n n 8 12 n 9 12 3 4 8 12 1 9 12 1 1 1 Î � � � � œ œ œ œ � 41. diverges by Limit Comparison Test: compare with , which is a divergent p-series, lim lim ! _ œ n 1 1 2 n 1 n n n Ä _ Ä _ Š ‹ 2 n n n 2n n Î œ � n 2n lim lim 1 0. œ œ œ � n n Ä _ Ä _ 2 ln 2 1 2 ln 2 2 ln 2 2 ln 2 n n 2 n 2 n � a b a b 42. diverges by the definition of an infinite series: ln ln n ln n 1 , s ln 1 ln 2 ln 2 ln 3 ! ! ˆ ‰ � ‘ a b a b a b _ _ œ œ n 1 n 1 n n 1 k � œ � � œ � � � ln k 1 ln k ln k ln k 1 ln k 1 lim s � Þ Þ Þ � � � � � � œ � � Ê œ �_ a b a b a b a b a b k Ä _ k Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 30. converges by the Limit Comparison Test (part 2) with , the nth term of a convergent p-series: " n lim lim lim 8 lim 8 lim 32 l n n n n n Ä _ Ä _ Ä _ Ä _ Ä _ ’ “ Š ‹ Š ‹ Š ‹ ˆ ‰ ˆ ‰ (ln n) n n 4n 4n 2 ln n n n œ œ œ œ œ (ln n) n n ln n im 32 0 0 n Ä _ " n œ œ † 31. diverges by the Limit Comparison Test (part 3) with , the nth term of the divergent harmonic series: " n lim lim lim lim n n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ 1 ln n n n œ œ œ œ _ n 1 ln n � " Š ‹ 32. diverges by the Integral Test: dx u du lim u lim b ln 3 ' ' 2 ln 3 ln (x 1) x 1 2 � � # " " # # # œ œ œ � œ _ b b Ä _ Ä _ � ‘ a b b ln 3 33. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n for " # n � � n 2 n n 1 n n n 1 n or use Limit Comparison Test with . Ê � � Ê � � Ê � # # $ $Î# # " " � a b È n n n 1 1 n È 34. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n " # # n � � n 1 nn n or use Limit Comparison Test with . Ê � � Ê � Ê � # $Î# $Î# � " " � È n 1 n n n 1 n n È È 35. converges because which is the sum of two convergent series: ! ! ! _ _ _ œ œ œ n 1 n 1 n 1 " � " �" # n n2 n2 n n n œ � converges by the Direct Comparison Test since , and is a convergent geometric series ! ! _ _ œ œ n 1 n 1 " " " �" # # n2 n 2 n n n n � 36. converges by the Direct Comparison Test: and , the sum of ! ! ˆ ‰ _ _ œ œ n 1 n 1 n 2 n 2 n2 n n2 n n � " " " " " " # n n n n n œ � � Ÿ � the nth terms of a convergent geometric series and a convergent p-series 37. converges by the Direct Comparison Test: , which is the nth term of a convergent geometric series " " � 3 1 3 n 1 n 1 � 38. diverges; lim lim 0 n n Ä _ Ä _ Š ‹ ˆ ‰ 3 3 3 3 3 n 1 n n �" " " " œ � œ Á 39. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 1 1 5 5 n l l œ � lim lim lim 0. n n n Ä _ Ä _ Ä _ Š ‹ a b n 1 1 n 3n 2 5n n 2 † Î � � � 1 5 n 1 1 n 3n 2n 3 œ œ œ 40. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 3 1 4 5 n l l œ � lim lim lim 1 0. n n n Ä _ Ä _ Ä _ Š ‹ a b ˆ ‰ ˆ ‰ 2 3 n n 3 4 n n n n n n n n 8 12 n 9 12 3 4 8 12 1 9 12 1 1 1 Î � � � � œ œ œ œ � 41. diverges by Limit Comparison Test: compare with , which is a divergent p-series, lim lim ! _ œ n 1 1 2 n 1 n n n Ä _ Ä _ Š ‹ 2 n n n 2n n Î œ � n 2n lim lim 1 0. œ œ œ � n n Ä _ Ä _ 2 ln 2 1 2 ln 2 2 ln 2 2 ln 2 n n 2 n 2 n � a b a b 42. diverges by the definition of an infinite series: ln ln n ln n 1 , s ln 1 ln 2 ln 2 ln 3 ! ! ˆ ‰ � ‘ a b a b a b _ _ œ œ n 1 n 1 n n 1 k � œ � � œ � � � ln k 1 ln k ln k ln k 1 ln k 1 lim s � Þ Þ Þ � � � � � � œ � � Ê œ �_ a b a b a b a b a b k Ä _ k Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ n n 1 � ln n Š ‹ 32. diverges by the Integral Test: dx u du lim u lim b ln 3 ' ' 2 ln 3 ln (x 1) x 1 2 � � # " " # # # œ œ œ � œ _ b b Ä _ Ä _ � ‘ a b b ln 3 33. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n for " # n � � n 2 n n 1 n n n 1 n or use Limit Comparison Test with . Ê � � Ê � � Ê � # # $ $Î# # " " � a b È n n n 1 1 n È 34. converges by the Direct Comparison Test with , the nth term of a convergent p-series: n 1 n " # # n � � n 1 nn n or use Limit Comparison Test with . Ê � � Ê � Ê � # $Î# $Î# � " " � È n 1 n n n 1 n n È È 35. converges because which is the sum of two convergent series: ! ! ! _ _ _ œ œ œ n 1 n 1 n 1 " � " �" # n n2 n2 n n n œ � converges by the Direct Comparison Test since , and is a convergent geometric series ! ! _ _ œ œ n 1 n 1 " " " �" # # n2 n 2 n n n n � 36. converges by the Direct Comparison Test: and , the sum of ! ! ˆ ‰ _ _ œ œ n 1 n 1 n 2 n 2 n2 n n2 n n � " " " " " " # n n n n n œ � � Ÿ � the nth terms of a convergent geometric series and a convergent p-series 37. converges by the Direct Comparison Test: , which is the nth term of a convergent geometric series " " � 3 1 3 n 1 n 1 � 38. diverges; lim lim 0 n n Ä _ Ä _ Š ‹ ˆ ‰ 3 3 3 3 3 n 1 n n �" " " " œ � œ Á 39. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 1 1 5 5 n l l œ � lim lim lim 0. n n n Ä _ Ä _ Ä _ Š ‹ a b n 1 1 n 3n 2 5n n 2 † Î � � � 1 5 n 1 1 n 3n 2n 3 œ œ œ 40. converges by Limit Comparison Test: compare with , which is a convergent geometric series with r 1, ! ˆ ‰ _ œ n 1 3 1 4 5 n l l œ � lim lim lim 1 0. n n n Ä _ Ä _ Ä _ Š ‹ a b ˆ ‰ ˆ ‰ 2 3 n n 3 4 n n n n n n n n 8 12 n 9 12 3 4 8 12 1 9 12 1 1 1 Î � � � � œ œ œ œ � 41. diverges by Limit Comparison Test: compare with , which is a divergent p-series, lim lim ! _ œ n 1 1 2 n 1 n n n Ä _ Ä _ Š ‹ 2 n n n 2n n Î œ � n 2n lim lim 1 0. œ œ œ � n n Ä _ Ä _ 2 ln 2 1 2 ln 2 2 ln 2 2 ln 2 n n 2 n 2 n � a b a b 42. diverges by the definition of an infinite series: ln ln n ln n 1 , s ln 1 ln 2 ln 2 ln 3 ! ! ˆ ‰ � ‘ a b a b a b _ _ œ œ n 1 n 1 n n 1 k � œ � � œ � � � ln k 1 ln k ln k ln k 1 ln k 1 lim s � Þ Þ Þ � � � � � � œ � � Ê œ �_ a b a b a b a b a b k Ä _ k Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 594 Chapter 10 Infinite Sequences and Series 43. converges by Comparison Test with which converges since , and ! ! ! ’ “ _ _ _ œ œ œ n 2 n 2 n 2 1 1 1 1 n n 1 n n 1 n 1 n a b a b � � � œ � s 1 1 lim s 1; for n 2, n 2 ! 1 k k 1 1 1 1 1 1 1 1 2 2 3 k 2 k 1 k 1 k k œ � � � � Þ Þ Þ � � � � œ � Ê œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ a b � � � k Ä _ n n 1 n 2 ! n n 1 n! n n 1 Ê � � � Ê � Ê Ÿ a ba b a b a b 1 1 n! n n 1 a b � 44. converges by Limit Comparison Test: compare with , which is a convergent p-series, lim ! _ œ n 1 1 n 1 n 3 3 n 1 n 2 n Ä _ Î lim lim lim lim 1 0 œ œ œ œ œ � n n n n Ä _ Ä _ Ä _ Ä _ n n 1 n 2 n 1 n n 1 n 3n 2 2n 3 2 n 2n 2 3 2 2 a b a ba b a b � x � � � x � � � 45. diverges by the Limit Comparison Test (part 1) with , the nth term of the divergent harmonic series: " n lim lim 1 n x 0 Ä _ Ä ˆ ‰ ˆ ‰ sin n n œ œ sin x x 46. diverges by the Limit Comparison Test (part 1) with , the nth term of the divergent harmonic series: " n lim lim lim 1 1 1 n n x 0 Ä _ Ä _ Ä ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ tan sin cos n n n n n œ œ œ œ Š ‹ ˆ ‰ ˆ ‰ " " cos x x sin x † 47. converges by the Direct Comparison Test: and is the product of a tan n n n n n 1.1 1.1 1.1 1.1 � œ ! ! _ _ œ œ n 1 n 1 1 # " convergent p-series and a nonzero constant 48. converges by the Direct Comparison Test: sec n and is the �" # # " � Ê � œ 1 1 sec n n n n n 1 3 1 3 1 3 1 3 ˆ ‰ ! ˆ ‰ ! _ _ œ œ n 1 n 1 product of a convergent p-series and a nonzero constant 49. converges by the Limit Comparison Test (part 1) with : lim lim coth n lim " � � n e e e e n n n Ä _ Ä _ Ä _ Š ‹ Š ‹ coth n n n œ œ n n n n lim 1 œ œ n Ä _ " � � e 1 e 2n 2n 594 Chapter 10 Infinite Sequences and Series 43. converges by Comparison Test with which converges since , and ! ! ! ’ “ _ _ _ œ œ œ n 2 n 2 n 2 1 1 1 1 n n 1 n n 1 n 1 n a b a b � � � œ � s