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

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

Graphing Sequences in R 2 Example: Graph the sequence: 〈 S n 〉 = 〈 1 /n 〉

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

F(x)= 1/x ; x ε [1,∞)

Representing Sequences on The Real Line 〈 S n 〉 = 〈 1 /n 〉

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

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

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 increasing 2. S n+1 / S n ≤ 1 ; n ε IN decreasing

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

Another Method

Example 2 This sequence is decreasing ( also strictly decreasing )

Another Method

Example 3

Example 4

Example 5

Example 6

Example 7

Question

Eventually increasing or decreasing sequences A sequence may have “odd” behavior at first, but eventually behaves monotonically. S n : 5, 7 -6, 22, 13, 1, 2, 3, 4, 5,6,7,8, …. t n : 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.

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

Example (1)