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

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slide2

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 }

slide3

Example 1

〈 1 /n〉

〈Sn〉=

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

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

= {1/n : nεN}

graphing sequences in r 2

Graphing Sequences in R2

Example:

Graph the sequence:

〈Sn〉= 〈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

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

increasing and decreasing sequences
Increasing and Decreasing Sequences

1) A sequence 〈Sn〉 is said to be :

increasing if : Sn+1 ≥ Sn ; n ε IN

strictly increasing if : Sn+1 > Sn ; n ε IN

2) A sequence 〈Sn〉 is said to be :

decreasing if : Sn+1 ≤ Sn ; n ε IN

strictly 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
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
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

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Example 1

This sequence is increasing ( also strictly increasing ).

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Example 2

This sequence is decreasing ( also strictly decreasing )

eventually increasing or decreasing sequences
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.
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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 .