1. SEQUENCES A sequence is a list of numbers in a given order: DEF: Example first termsecond term n-th term index

2. SEQUENCES A sequence is a list of numbers in a given order: DEF: Example

3. SEQUENCES Find a formula for the general term of the sequence Example Find a formula for the general term of the sequence Example the digit in the th decimal place of the number pi This sequence arose when the 13th-century Italian mathematician known as Fibonacci Recursive Definitions

4. SEQUENCES Example Representing Sequences LIMIT OF THE SEQUENCE as We say the sequence convg Remark: If converges to L, we write or simply and call L the limit of the sequence Remark: If there exist no L then we say the sequence is divergent.

5. SEQUENCES Example Convergence or Divergence 1 2 3 How to find a limit of a sequence Example: (IF you can) use Math-101 to find the limit. Use other prop. To find the limit abs,r^n,bdd+montone 1)Sandwich Thm: 2)Cont. Func. Thm: 3)L’Hôpital’s Rule:

6. SEQUENCES

7. Example Note:

8. SEQUENCES Factorial; Example NOTE

9. SEQUENCES Example Find where Sol: by sandw. limit is 0

10. SEQUENCES Example For what values of r is the sequence convergent?

11. SEQUENCES

12. DEFINITION bounded from above Example Is bounded above by any number greater than one Upper bound Least upper bound If M is an upper bound but no number less than M is an upper bound then M is the least upper bound. DEFINITION bounded from below Example Is bounded below Lower bound greatest 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 If is bounded from above and below, If is not bounded bounded we say that unbounded

13. SEQUENCES If is bounded from above and below, If is not bounded bounded we say that unbounded Example: bounded unbounded

14. SEQUENCES DEFINITION non-decreasing DEFINITION non-increasing Example Is the sequence inc or dec Sol_1 Sol_2

15. SEQUENCES DEFINITION non-decreasing DEFINITION non-increasing Example Is the sequence inc or dec

16. SEQUENCES if it is either nonincreasing or nondecreasing. DEFINITION monotonic DEFINITION non-decreasing DEFINITION non-increasing

17. SEQUENCES THM_part1 non-decreasing bounded by above convg THM6 1) bounded 2) monotonic convg THM_part2 non-increasing bounded by below convg

18. SEQUENCES THM6 1) bounded 2) monotonic convg Example Is the sequence inc or dec

19. SEQUENCES How to find a limit of a sequence (convg or divg) Example: (IF you can) use Math-101 to find the limit. Use other prop. To find the limit abs,r^n,bdd+montone 1)Sandwich Thm: 2)Cont. Func. Thm: 3)L’Hôpital’s Rule: 1)Absolute value: 2)Power of r: 3)bdd+montone: Bdd + monton  convg Example:

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27. TERM-082

28. SEQUENCES TERM-082

29. SEQUENCES TERM-092

30. SEQUENCES TERM-092

45. SEQUENCES If is bounded from above and below, If is not bounded bounded we say that unbounded Example: bounded unbounded