Real Sequences

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)