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Real Sequences

Real Sequences . A real sequence is a real function S whose domain is the set of natural numbers IN . The range of the sequence < S n > is simply the range of the function S . range 〈 S n 〉 = { S(n) : n ε IN }

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Real Sequences

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  1. Real Sequences

  2. A real sequence is a real function S whose domain is the set of natural numbers IN . The range of the sequence < Sn> is simply the range of the function S . range 〈Sn〉= { S(n) : n ε IN } = { Sn : n ε IN }

  3. Example 1 〈 1 /n〉 〈Sn〉= Domain 〈Sn〉= IN ={1 , 2, 3, 4 , 5 , ……… } Range〈Sn〉= {1 , 1/2 , 1/3 , 1/4 ,….…….} = {1/n : nεN}

  4. Graphing Sequences in R2 Example: Graph the sequence: 〈Sn〉= 〈1 /n 〉

  5. Compare the graph of the sequence sn= 1/n with the part of the graph of f(x) = 1/x in the interval [1,∞)

  6. F(x)= 1/x ; x ε [1,∞)

  7. Representing Sequences on The Real Line 〈Sn〉= 〈1 /n 〉

  8. Increasing and Decreasing Sequences 1) A sequence 〈Sn〉 is said to be : increasing if : Sn+1 ≥ Sn ; n ε IN strictly increasing if : Sn+1 > Sn ; n ε IN 2) A sequence 〈Sn〉 is said to be : decreasing if : Sn+1 ≤ Sn ; n ε IN strictly decreasing if : Sn+1 < Sn ; n ε IN 3) A sequence 〈Sn〉 is said to be constant if : Sn+1 = Sn ; n ε IN

  9. Testing for Monotonicity: The difference Method • 〈Sn〉is increasing if Sn+1 - Sn≥ 0 ; n ε IN (Why?) • 〈Sn〉is decreasing if Sn+1 - Sn≤0 ;n ε IN (Why?) What about if Sn – Sn+1≤0 ; n ε IN ? What about if Sn – Sn+1≥ 0 ; n ε IN

  10. Testing for Monotonicity: The Ratio Method • If all terms of a sequence〈Sn〉are positive, we can investigate whether it is monotonic or not by investigating the value of the ratio Sn+1 / Sn . 1. Sn+1 / Sn≥ 1 ; n ε IN increasing 2. Sn+1 / Sn≤ 1 ; n ε IN decreasing

  11. Example 1 This sequence is increasing ( also strictly increasing ).

  12. Another Method

  13. Example 2 This sequence is decreasing ( also strictly decreasing )

  14. Another Method

  15. Example 3

  16. Example 4

  17. Example 5

  18. Example 6

  19. Example 7

  20. Question

  21. Eventually increasing or decreasing sequences • A sequence may have “odd” behavior at first, but eventually behaves monotonically. • Sn: 5, 7 -6, 22, 13, 1, 2, 3, 4, 5 ,6,7,8, …. • tn: 2 , 2 , 2 , 2 , 2 , 8 , 7, 6 , 5, 4,3,2,1,0,-1,-2, ….. • Such a sequence is said to increase or decrease eventually.

  22. Example 5 Starting from the 5-th term , we have a sequence 〈S5+(n-1)〉 , that is monotonic . notice that 〈S5+(n-1)〉can be expressed as follows : S5+(n-1) : S5 , S6 , S7 , S8 , S9 , ……., and more precisely : S5+(n-1) : 2 , 6 , 7 , 8 , 9 ,10 , …… Thus 〈Sn〉 is eventually monotonic .

  23. Example (1)

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