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SEQUENCES DEF: A sequence is a list of numbers in a given order: a1 , Example first term Example Example a2 , , an , 1 1 1 1 1, , , , , 2 3 4 5 second term 1 2 3 4 , , , , 2 3 4 5 1 n n 1 an n-th term 1 n index n n 1 n 1 2 3 4 5 n 1 , , , , n 3 9 27 81 3 n 1 OR n n 1 3 n 2 SEQUENCES DEF: A sequence is a list of numbers in a given order: a1 , a2 , , an , Example 1 2 3 4 5 , , , , , 4 5 6 2 3 Example 2 3 4 5 , , , , 27 81 3 9 (1) n n n 1 n 1 n 1 n ( 1 ) n 1 3 n 2 SEQUENCES Find the n-term TERM-092 1 2n an n 3 2n 1 n 1,2,3, TERM-131 an (n)( n 2) (n 1) n 1,2,3, SEQUENCES Example Find a formula for the general term of the sequence 3.14159265358979 1, 4, 1, 5, 9, 2, 6, the digit in the th decimal place of the number pi Recursive Definitions Example 1, 1, Find a formula for the general term of the sequence 2, 3, 5, 8, 13, 21 a1 1, a2 1, an an 1 an 2 This sequence arose when the 13th-century Italian mathematician known as Fibonacci SEQUENCES Representing Sequences Example 1 2 3 4 , , , , 2 3 4 5 n n 1 n 1 LIMIT OF THE SEQUENCE as Remark: If an n1 n We say the sequence n or simply an L and call L the limit of the sequence 1 n lim 1 n n 1 converges to L, we write lim an L n n 1 Remark: an n n 1 convg If there exist no L then we say the sequence is divergent. SEQUENCES Convergence or Divergence Example 1 2 n n 1n 1 2 n n 1 How to find a limit of a sequence (IF you can) Use other prop. use Math-101 to find the limit. To find the limit abs,r^n,bdd+montone Example: n n n 1 lim lim x x x 1 1)Sandwich Thm: cos n n (1) n 1 n 2)Cont. Func. Thm: 3 1,1,1,1, an L f (an ) f ( L) n 1 n 1n 2 3)L’Hôpital’s Rule: ln n n n n 1 n 1 SEQUENCES SEQUENCES Example n 1 lim (1) n n Note: n n lim (1) n n 1 SEQUENCES Factorial; n! 1 2 3 (n 1) n Example 3! 3 2 1 6 5! 5 4 3 2 1 120 NOTE 10! 10 (9!) n! n (n 1)! SEQUENCES Example Find where Example n! 1 2 3 (n 1) n Find where 1 lim n n! 9n 2 5 lim n n! n! 1 2 3 (n 1) n SEQUENCES Example For what values of r is the sequence convergent? n {r } n The sequence { r } is lim r n n conv div 1 r 1 other valu es SEQUENCES 1 r 1 other valu es conv div n The sequence { r } is Example: 9.7 n 1 n 0.99 0.5 n n 1 n SEQUENCES DEFINITION { an } DEFINITION { an } bounded from above an M for all n M Upper bound an M for all n M If M is an upper bound but no number less than M is an upper bound then M is the least upper bound. If m is a lower bound but no number greater than m is a lower bound then m is the greatest lower bound Lower bound Example 3 1 Is bounded below Example n n 1 bounded from below Is bounded above by any number greater than one an 1.1 an 1.001 M 1 Least upper bound n If an is bounded from above and below, an bounded an 3 If an greatest upper bound = ?? is not bounded we say that an unbounded SEQUENCES If an is bounded from above and below, an Example: If an is not bounded we say that an bounded n 3 1 n n 1 bounded unbounded n 2 unbounded SEQUENCES DEFINITION DEFINITION { an }non-decreasing { an }non-increasing an an1 for all n 1 an an1 for all n 1 a1 a2 a3 a4 Example 1 3 n Is the sequence inc or dec a1 a2 a3 a4 n 1 n 1 1 n 1 n 1 1 3 3 n 1 n an 1 an Sol_1 Sol_2 f ( x) 3 1x f ' ( x) x12 0 ( x 1) SEQUENCES DEFINITION DEFINITION { an }non-decreasing { an }non-increasing an an1 for all n 1 a1 a2 a3 a4 Example n 2 n 1 Is the sequence inc or dec an an1 for all n 1 a1 a2 a3 a4 SEQUENCES DEFINITION { an } non-decreasing an an1 for all n 1 a1 a2 a3 a4 DEFINITION { an } non-increasing an an1 for all n 1 a1 a2 a3 a4 DEFINITION { an } monotonic if it is either nonincreasing or nondecreasing. SEQUENCES DEFINITION { an } non-decreasing an an1 for all n 1 a1 a2 a3 a4 Is the sequence non-decreasing? SEQUENCES THM6 { an } 1) bounded convg 2) monotonic THM_part1 { an } non-decreasing bounded by above THM_part2 convg { an } non-increasing bounded by below convg SEQUENCES THM6 { an } 1) bounded convg 2) monotonic Example 1 3 n Is the sequence inc or dec SEQUENCES How to find a limit of a sequence (convg or divg) (IF you can) Use other prop. use Math-101 to find the limit. To find the limit abs,r^n,bdd+montone Example: n lim n n 1 Example: x lim x x 1 1)Sandwich Thm: cos n n (1) n 1 n (1) n n! 1)Absolute value: an 0 then an 0 2)Cont. Func. Thm: an L f (an ) f ( L) n 1 n 1n 2 3)L’Hôpital’s Rule: ln n n 2)Power of r: n n 1 n 1 3)bdd+montone: Bdd + monton convg SEQUENCES SEQUENCES SEQUENCES TERM-082 SEQUENCES TERM-082 SEQUENCES TERM-092 SEQUENCES TERM-092 SEQUENCES Example n! lim n n n Find where n! 1 2 3 (n 1) n Sol: 0 n! 1 2 3 n n n n n n n n! 1 2 3 n 0 n n nnn n n! 1 2 3 n 1 0 n n nn n n less than one 0 n! 1 n n n by sandw. limit is 0 SEQUENCES Multiple-Choice Problems SEQUENCES SEQUENCES SEQUENCES SEQUENCES SEQUENCES SEQUENCES SEQUENCES SEQUENCES If an is bounded from above and below, an Example: If an is not bounded we say that an bounded n 3 1 n n 1 bounded unbounded n 2 unbounded