Real Sequences

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 }

〈 Sn 〉 =

Example 1

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

Domain 〈 Sn 〉 = IN ={1 , 2, 3, 4 , 5 , ……… }

〈 1 /n 〉

Graphing Sequences in R2

Example:Graph the sequence:〈 Sn 〉 = 〈 1 /n 〉

Compare the graph of the sequence sn= 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

〈 Sn 〉 = 〈 1 /n 〉

Increasing and Decreasing Sequences1) A sequence 〈 Sn 〉 is said to be :increasing if : Sn+1 ≥ Sn ; n ε

INstrictly increasing if : Sn+1 > Sn ; n ε

IN

2) A sequence 〈 Sn 〉 is said to be :decreasing if : Sn+1 ≤ Sn ; n ε INstrictly decreasing if : Sn+1 < Sn ; n ε

IN 3) A sequence 〈 Sn 〉 is said to be constant

if :Sn+1 = Sn ; n ε IN

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

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

Example 1

This sequence is increasing ( also strictly increasing ).

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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.

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 .

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