MAT1026 CALCULUS II 21.02.2012 Dokuz Eylül Üiversitesi Fe Fakültesi Matematik Bölümü Istructor: Egi Mermut web: http://kisi.deu.edu.tr/egi.mermut/ TRUE/FALSE QUESTIONS FOR SEQUENCES Write TRUE or FALSE for each of the followig statemets. If you claim TRUE, prove that. If you claim FALSE, give a couter example ad give a correspodig statemet that becomes true if you correct some hypothesis. By a sequece i the followig statemets, we mea a sequece of real umbers. For all of the questios below: Let ( ) ad (b ) be sequeces. Let L R. If ( ) coverges to a real umber L, we write L as, or we write just L, or we write lim = L. If ( ) diverges to, we write as, or we write just, or we write lim =. Similarly for divergece to. 1. A sequece of real umbers is a real-valued fuctio whose domai is R. 2. The set R + of all positive real umbers has a smallest elemet. 3. A oempty subset A of R is said to be bouded from above if there exists M R such that a M for some a A. 4. A oempty subset A of R is said to be bouded if there exists M R such that a M for all a A. 5. A real umber L is ot a lower boud of oempty subset A of R if there exists a A such that a > L. 6. A oempty subset A of R is bouded if ad oly if there exists a real umber K > 0 such that a K for all a A. 7. if(r + ) = 0, that is, 0 is the greatest lower boud of the set R + of all positive real umbers. 8. The set Z + of positive itegers is bouded from above. 9. There exists a real umber M such that M for every positive iteger. 10. For each real umber a > 0 ad for each real umber b, there exists Z + such that a > b. 11. There exists a real umber ε > 0 such that 1 ε for every Z+. 12. A sequece ( ) of real umbers is said to be bouded if there exists a real umber M such that M for all Z +. 13. A sequece ( ) of real umbers is bouded if ad oly if there exists a real umber K > 0 such that K for all Z +. ({ }) ( ) 1 1 14. if Z+ = 0, that is, 0 is the greatest lower boud of the sequece. 15. The set R of all egative real umbers has a greatest elemet. 16. ({ }) 1 sup Z+ = 0, that is, 0 is the least upper boud of the sequece ( ) 1. 17. A sequece ( ) is said to coverge to a real umber L if there exists a real umber ε > 0 ad N Z + such that for all Z +, > N L < ε. 1
18. The sequece ( ) 1 coverges to 1. 19. Let c R. Let = c for every Z +. The the costat sequece ( ) = (c) coverges to c. 20. A sequece ( ) may coverge to a real umber L 1 ad to also aother real umber L 2 such that L 2 L 1. 21. A sequece ( ) is said to be a coverget sequece if if for every real umber ε > 0, there exists N Z + such that for all Z +, > N L < ε. 22. A sequece ( ) is said to be a diverget sequece if there exists ε > 0 ad there exists N Z + ad L R such that for all Z +, 23. Every diverget sequece is ubouded. > N ad L ε. 24. Every subsequece of a coverget sequece is coverget. 25. Every coverget sequece is ubouded. 26. Every decreasig sequece is bouded. 27. Every ubouded sequece is mootoe. 28. If ( ) is a ubouded sequece, the () is diverget. 29. A sequece ( ) is said to be a icreasig sequece if < +1 for some Z +. 30. A sequece ( ) is said to be oicreasig sequece if +1 for all Z +. 31. A sequece ( ) is said to be a decreasig sequece if > +1 for some Z +. 32. A sequece ( ) is said to be odecreasig sequece if +1 for all Z +. 33. A sequece ( ) is said to be a mootoe sequece if it is a icreasig sequece. 34. A icreasig sequece ( ) is bouded from below. 35. A decreasig sequece ( ) is bouded from above. 36. Let ( ) be a icreasig sequece. The () ( ) is bouded from above. 37. Let ( ) be a decreasig sequece. The () ( ) is bouded from below. is a bouded sequece if ad oly if is a bouded sequece if ad oly if 38. A odecreasig sequece that is bouded from below is coverget. 39. A oicreasig sequece that is bouded from above is coverget. 40. If ( ) is a mootoe sequece, the () is coverget. 41. If ( ) is a bouded sequece, the () is a coverget sequece. 42. If ( ) is a decreasig sequece ad 0 for all Z +, the ( ) is coverget. 43. Every bouded mootoe sequece is coverget. 44. If A ad b B for some real umbers A ad B, ad if > b for all Z +, the A > B. 2
45. If ( ) coverges to L ad ( k ) k=1 is a subsequece of (), the ( k ) k=1 coverges to L, too. 46. If ( ) has two coverget subsequeces ( k ) k=1 ad ( m ) m=1 lim m m, the ( ) is diverget. such that lim k k { k 47. For all k Z +, let k = 2, if k is eve; k, if k is odd. The (k ) k=1 is a subsequece of (). 48. If ( ) coverges to L ad ( k ) k=1 is a subsequece of (), the k k for all k Z +. 49. lim = 0 lim = 0. 50. lim = L lim L = 0. 51. lim = L lim = L. 52. If L b for all Z + beyod some idex N 0 ad lim b = 0, the lim = L. 53. If ( ) is a diverget sequece, the ( ) is also diverget. 54. If 0 ad (b ) is bouded, the b 0. [ ] [ ] 55. lim (b ) = lim lim b for all sequeces ( ) ad (b ). 56. If ( ) ad (b ) are coverget sequeces ad b for all Z +, the lim lim b. 57. If ( + b ) is a coverget sequece, the both () ad (b ) are coverget. 58. If c R ad (c ) is a coverget sequece, the () is a coverget sequece. 59. If b 0, the 0 or b 0. 60. If ( b ) is a coverget sequece, the both () ad (b ) are coverget. [ ] 61. lim (c) = c lim for every real umber c. 62. If ( ) is a coverget sequece, the the sequece ( 1 ) is also coverget. 63. ( If (a ) ) is a coverget sequece of ozero real umbers with ozero limit, the the 1 sequece is bouded. 64. If L ad if f(x) is a real-valued fuctio of a real variable which is cotiuous at L ad which is defied at for all Z +, the the sequece (f( )) coverges to f(l). 65. A sequece ( ) is said to diverge to if it is a icreasig sequece that is ot bouded from above. 66. A sequece ( ) is said to diverge to if it is a decreasig sequece that is ot bouded from below. 67. If ( ), (b ) ad (c ) are sequeces of real umbers that satisfy b c for all Z +, ad if ( ) ad (c ) are coverget, the (b ) is also coverget. 3
68. If ( ) is a icreasig sequece of real umbers, the either () diverges to (if it is ot bouded from above), or ( ) is bouded from above ad coverges to sup({ Z + }). 69. If ( ) is a decreasig sequece of real umbers, the either () diverges to (if it is ot bouded from below), or ( ) is bouded from below ad coverges to if({ Z + }). [ ] [ ] 70. lim ( + b ) = lim + lim b for all sequeces ( ) ad (b ). 71. Let f be a fuctio defied o the iterval [N 0, ) for some N 0 Z + ad let ( ) be a = f() for all itegers N 0. If lim f(x) = L for some real umber L, the () coverges to L, that is, lim = L. 72. Let f be a fuctio defied o the iterval [N 0, ) for some N 0 Z + ad let ( ) be a = f() for all itegers N 0. If lim f(x) =, the () diverges to, that is, lim =. 73. Let f be a fuctio defied o the iterval [N 0, ) for some N 0 Z + ad let ( ) be a = f() for all itegers N 0. If lim f(x) =, the () diverges to, that is, lim =. 74. Let f be a fuctio defied o the iterval [N 0, ) for some N 0 Z + ad let ( ) be a = f() for all itegers N 0. If the sequece ( ) coverges to L, that is, if lim = L for some real umber L, the we also have that lim f(x) = L. 75. If ( ) is a diverget sequece ad c ozero real umber, the the sequece (c) is diverget, too. [ ] 76. If ( ) is a coverget sequece, the lim ( ) = lim. [ ] [ ] 77. lim ( b ) = lim lim b for all sequeces ( ) ad (b ). 78. Let f be a fuctio defied o [d, ) for some real umber d. Let ( ) be a sequece of real umbers that diverges to. If lim f(x) = L for some real umber L, ad if [d, ) for all Z +, the the sequece (f( )) coverges to L, that is, lim f() = L. 79. Let L R. Let f be a fuctio that is defied at L. If f is ot cotiuous at L, the there exists a sequece ( ) that coverges to L such that is i Domai(f) for every Z + but (f( )) does ot coverge to f(l) (that is, either (f()) is a diverget sequece or (f()) is a coverget sequece but its limit is ot equal to f(l)). 80. If ( ) is a there exists a real umber L ad both of the subsequeces (a 2k 1 ) k=1 ad (a 2k) k=1 coverge to L, the () coverges to L. 81. If 0 for all Z + ad 1 0, the. 82. If > 0 for all Z + ad 1 0, the. 4
83. If < 0 for all Z + ad 1 0, the. 1 84. Let p R. The lim = 0 if ad oly if p > 0. p 85. For a positive real umber a, lim a1/ = 0. 5 86. lim! = 1. 87. lim =. 88. lim 2 = 1. p 89. lim = 0 if a > 1 is a real umber ad p R is arbitrary. a l() 90. lim c = 1 for every positive real umber c. 91. Let r R. The the sequece (r ) is coverget if ad oly if r > 1 or r = 1. 92. If r < 1, the lim r = 1. 93. lim 94. lim! 10 = 1. ( 1 ) 1 = e. ( si 95. The sequece 2 ) is a diverget sequece. 96. Let ( ) be a sequece of positive real umbers: For every Z +, let s = The (s ) is a icreasig sequece. > 0 for all Z +. a k = a 1 + a 2 +... +, k=1 97. Let ( ) be a sequece of positive real umbers: > 0 for all Z +. For every Z +, let The (s ) is coverget. s = a k = a 1 + a 2 +... +, k=1 98. If L ad b, the b. 99. If 0, b 1, the b 0. 100. If ( ) is diverget, the (a2 ) is diverget. 101. If ( ) is coverget ad (b ) is bouded, the (b ) is coverget. 102. If 0 ad b L, the b 0. 103. If L ad b, the + b. 5
104. If ad (b ) is bouded, the b. 105. If or, the. 106. If ad b for all Z +, the b. Solve the 23 problems that I have give i Homework 1. Be sure that you ca prove all the theorems i Sectio 7.1 of your textbook (pages 410 419). See the lecture otes. Solve all the exercises of Sectio 7.1 of your textbook (pages 419 422). Sequeces of real umbers is a fudametal cocept that you must lear ad do well. As also for a review of limits from the previous term, be sure that you ca fid the limits of the sequeces i the exercises of Sectio 7.1 of your textbook (pages 419 422). 6