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

              an  

to mean the nth term of the sequence.

Example:

If 

                     1
        an  =                     
                   n + 2

then we have

        a1 = 1/3,     a2 = 1/4,     etc.

Exercise:  

Write the general term an 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        3        4
                ,         ,         ,         , ...                               
           2        3        4        5

We see that as n becomes large the numbers approach 1.  In particular if any small error number e is given, we can find an N such that for n > N, |an -1| < e.  We say that the limit of the sequence approaches 1

In general, 

If an is a sequence that converges to a limit L then for any e > 0

we can find an N such that for all n > N

        |an - L| < e  

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 + 1
        an  =                      
                     n - 3

by considering the function

                      2n + 1
        f(n)  =                      
                       n - 3

We note 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.


The Squeeze Theorem

Suppose that 

        lim an  =  lim bn  =  L

and that there is an N such that for any n > N,

        an   <   cn   <   bn

then 

        lim cn = L

Example

Show that 

        

Note that

        -1             sin n            1
                 <                <              
        n                n!              n

both the left hand and right hand sides converge to 0 hence 

       


Monotonic and Bounded Sequences


          Definition of Monotonicity and Boundedness

A sequence is monotonically decreasing (increasing) if

          an  >  an+1   (an  <  an+1for all n.

A sequence is bounded from above (below) if there is a number M such that

          an < M  ( an > M) for all n.

 


Example

Determine the montonicity and boundedness of the following sequences

  1. cos(n)

  2. 1/n3

Solution

  1. cos(n) is not monotonic since, for example

            cos 1  >  cos 2

    and 

            cos 3  <  cos 4

    However, cos(n) is bounded above by 1 and below by -1 since

            -1  <  cos(n)  <  1

  2. 1/n3 is monotonic since for n > 0

            (1/n3)' = -3/n4  <  0

    so 1/n3 is monotonically decreasing.


Exercises:
 

Classify the monotonicity and boundedness of the following sequences:

  1. an = sin(n)

  2. an = 1/n

  3.            n + 1
    an  =                
               n + 2

  4. an = n2 + 1

 

Theorem

A bounded monotonic sequence converges



Example

Show that that sequence

                    n
        an =              
                   en 

converges.



Solution

If

                      x
        f(x)  =             =  xe-x 
                     ex 

Then

        f '(x) = (1 - x) e-x

Which is always negative for x > 1.  Hence an is monotonic.

for x > 0,

        xe-x  <  1

so we can conclude that the sequence converges.

 



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