SalasSV_10_03_ex - 602 CHAPTER 10 SEQUENCES; INDETERMINATE...

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EXERCISES 10.3 State whether the sequence converges and, if it does, ±nd the limit. 1. 2 n . 2. 2 n . 3. ( 1) n n . 4. n . 5. n 1 n . 6. n + ( 1) n n . 7. n + 1 n 2 . 8. sin π 2 n . 9. 2 n 4 n + 1 . 10. n 2 n + 1 . 11. ( 1) n n . 12. 4 n n 2 + 1 . 13. ( 1 2 ) n . 14. 4 n 2 n + 10 6 . 15. tan n π 4 n + 1 . 16. 10 10 n n + 1 . 17. (2 n + 1) 2 (3 n 1) 2 . 18. ln ³ 2 n n + 1 ´ . 19. n 2 2 n 4 + 1 . 20. n 4 1 n 4 + n 6 . 21. cos n π . 22. n 5 17 n 4 + 12 . 23. e 1 / n . 24. 4 1 n . 25. ln n ln ( n + 1). 26. 2 n 1 2 n . 27. n + 1 2 n . 28. 1 n 1 n + 1 . 29. ³ 1 + 1 n ´ 2 n . 30. ³ 1 + 1 n ´ n / 2 . 31. 2 n n 2 . 32. 2ln3 n ln ( n 2 + 1).
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10.3 LIMIT OF A SEQUENCE ± 603 ± c In Exercises 33 and 34, use technology (graphing utility or CAS) to determine whether the sequence converges, and if it does, give the limit. 33. (a) sin n n . (b) tan 1 ± n n + 1 ² . (c) n 2 + n n . 34. (a) n n . (b) 3 n n ! (c) n n n ! . 35. If 0 < a < b , show that n a n + b n b . 36. (a) Determine the values of r for which r n converges. (b) Determine the values of r for which nr n converges. 37. Prove that if a n L and b n M , then a n + b n L + M . 38. Let α be a real number. Prove that if a n L , then α a n α L . 39. Given that ± 1 + 1 n ² n e show that ± 1 + 1 n ² n + 1 e . 40. Determine the convergence or divergence of a rational sequence a n = α k n k + α k 1 n k 1 +···+ α 0 β j n j + β j 1 n j 1 +···+ β 0 given that: (a) k = j ; (b) k < j ; (c) k > j . Justify your answers. Assume that α k ±= 0 and β j ±= 0. 41. Prove that a bounded nonincreasing sequence converges to its greatest lower bound. 42. From a sequence with terms a n , collect the even-numbered terms e n = a 2 n and the odd-numbered terms o n = a 2 n 1 . Show that a n L iff e n L and o n L . 43. Prove the pinching theorem for sequences. 44. Let { a n } and { b n } be sequences such that a n 0 and { b n } is bounded. Prove that a n b n 0. 45.
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SalasSV_10_03_ex - 602 CHAPTER 10 SEQUENCES; INDETERMINATE...

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