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SEQUENCES

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

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SEQUENCES

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

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

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

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

  5. SEQUENCES Convergence or Divergence 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 Example: 2 1)Sandwich Thm: 2)Cont. Func. Thm: 3 3)L’Hôpital’s Rule:

  6. SEQUENCES

  7. SEQUENCES Example Note:

  8. SEQUENCES Factorial; NOTE Example

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

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

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

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

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

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

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

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

  20. SEQUENCES

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  27. SEQUENCES TERM-082

  28. SEQUENCES TERM-082

  29. SEQUENCES TERM-092

  30. SEQUENCES TERM-092

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

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