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Real Sequences

Real Sequences . Monotonic Sequences . Increasing and Decreasing Sequences. 1) A sequence 〈 S n 〉 is said to be : increasing if : S n+1 ≥ S n ; n ε IN 2) A sequence 〈 S n 〉 is said to be : decreasing if : S n+1 ≤ S n ; n ε IN.

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Real Sequences

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  1. Real Sequences

  2. Monotonic Sequences

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

  4. 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

  5. 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 εIN then the sequence is increasing 2. Sn+1 / Sn≤11 ; n ε IN then the sequence is decreasing What about when: 1. Sn/ Sn+1≥ 1 ; n εIN 2. Sn/ Sn+1≤ 1 ; n εIN

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

  7. Another Method

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

  9. Another Method

  10. Example 3

  11. Example 4

  12. Example 5

  13. Example 6

  14. Example 7

  15. Another Method

  16. Example (8)

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