3. 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

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

5. Testing for Monotonicity: The Ratio Method If all terms of a sequence 〈 S n 〉 are positive, we can investigate whether it is monotonic or not by investigating the value of the ratio S n+1 / S n. 1. S n+1 / S n ≥ 1 ; n ε IN then the sequence is increasing 2. S n+1 / S n ≤ 11 ; n ε IN then the sequence is decreasing What about when: 1. S n / S n+1 ≥ 1 ; n ε IN 2. S n / S n+1 ≤ 1 ; n ε IN

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)