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        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, |a_n -1| < e.  We say that the limit of the sequence approaches 1

In general, 

If a_n 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

        |a_n - 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
        a_n  =                      
                     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 a_n  =  lim b_n  =  L

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

        a_n   <   c_n   <   b_n

then 

        lim c_n = 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+1 )  for 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. a_n = sin(n)
   

2. a_n = 1/n
   

3.            n + 1
   a_n  =                
              n + 2
   

4. a_n = n² + 1
   
   

 

Theorem A bounded monotonic sequence converges

Example

Show that that sequence

                    n
        a_n =              
                   eⁿ 

converges.

Solution

If

                      x
        f(x)  =             =  xe^-x 
                     e^x 

Then

        f '(x) = (1 - x) e^-x

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

for x > 0,

        xe^-x  <  1

so we can conclude that the sequence converges.

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