Sequences

Definition of a Sequence

A sequence is a list of numbers, or more formally, a function f(n) from the natural numbers to the real numbers.  We write

              a_n  

to mean the n^th term of the sequence.

Example:

If 

                        1
        a_n  =                       
                     n + 2   

then we have

        a₁ = 1/3, a₂ = 1/4, etc.

Exercise:  

Write the general term a_n for the following sequences:

1.  -1,1,-1,1,-1,1,...
   

2. 1,4,9,16,...
   

3. 1/2, -1/6, 1/24,-1/120,...
   

4. 1, 1/2, -1/4, -1/8, 1/16, 1/32, -1/64, -1/128, ...

The Limit of a Sequence

Consider the sequence

1/2, 2/3, 3/4, 4/5, ...

We see that as n becomes large the numbers approach 1.  

If there is no such L then we say that the sequence diverges.

Theorem              Let f(n) = an  be a sequence, then an -> L if and only if

Example:  We find the limit of the sequence 

                     2n + 3
        a_n =                    
                      n - 3

by considering the function

                         2n + 3
        f(n)  =                    
                          n - 3


We not that as 

        n ->

we get 

        /  

hence we can use L'Hopital's Rule:  Taking derivatives of the top and bottom, we have 2/1 hence the limit is 2.


Exercises

Find the limit of the following sequences or state that the sequences diverge.

1. 1/2, 1/4, 1/8, 1/16, ...
   

2. 5/6, 6/8, 7/10, 8/12, .
   

3. 1/2, 2/5, 4/8, 8/16, ...
   

4. 1/cos(1), 2/cos(2), 3/cos(3), ...

Application

Currently there are 5,000 of a certain species of monkeys left in a jungle.  Each year, because of poachers and loss of habitat, the population decreases by 5%.  Write a sequence that models this and find the limit of the sequence.

Solution

We have 

        a₀  =  5000        a₂  =  5000 - 5000(.05)  =  5000(.95)

        a₂  =  5000(.95) - 5000(.95)(.05)  =  5000(.95)(.95)  =  5000(.95)²

Following this pattern, we get

        a_n  =  5000(.95)ⁿ 

We find the limit of this sequence to be 0, that is the population will eventually become extinct.

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