Series

Definition of a Series

Let a_n be a sequence then we define the n^th partial sum of a_n as

sn = a1 + a2 + ... + an

In other words, we define s_n by adding up the first n terms of a_n.  We define the series as the limit of  the s_n that is

S =  San   =  a1 + a2 + a3 + ...

If the limit exists then we say that the series converges.  Otherwise, we say that the series diverges.  

Example 

consider 

                            1                    1
        a_n  =                            -          
                     n² + 2n + 1            n²

Evaluate 

       

Solution

We write out the first four terms: 

            1           1               1           1                1           1                1           1
                  -              +             -              +             -              +             -              +  ...
            4           1               9           4               16          9               25         16

                  1           1           1           1            1           1           1           1
         =  -         +            -           +            -           +           -           +              +  ...
                  1           4           4           9            9          16         16         25

        =   -1

Such a series is called a telescoping series.

Geometric Series

We define a geometric series to be a series of the form

        Sarⁿ

For example:

        3/2 + 3/4 + 3/8 + ...

Geometric Series Test For 0 < |r| < 1  we have and for |r| > 1 the series diverges.

Proof:

Let

        s  =  a + ar + ar² + ar³ + ar⁴ + ...  

Then

        rs  =  ar + ar² + ar³ + ar⁴ + ...    

subtracting the second equation from the first we get

        s - rs  =  a 

or

        s(1 - r)  =  a,

                      a
        s  =                
                   1 - r

The Limit Test

The Limit Test  If S an converges then

Note:  The contrapositive says that if the limit is nonzero, then the series does not converge.

Caution:  If the limit goes to zero then the series still may diverge.

Examples

1.  diverges by the limit test since the limit is 1 not 0.
   

2. does not converge even though the limit goes to 0.  This series is called the harmonic series.

The Harmonic Series

Harmonic Series Test  The series with terms 1/n diverges.

Proof:  we write

            1        1        1         1         1         1         1        1
                 +        +        +        +         +        +        +        +
            1        2        3         4         5         6         7        8

               1        1        1         1         1         1         1        1
          +        +        +        +        +         +        +        +        + ...
               9       10       11      12       13        14       15      16

              1        1        1         1         1         1         1        1
        >         +        +        +        +         +        +        +        +
              1        2        4         4         8         8         8        8

               1        1        1         1         1         1         1        1
          +        +        +        +        +         +        +        +        + ...
              16      16       16       16       16        16       16      16

              1        1        1         1       
        =         +        +        +        + ...
              1        2        2         2     

which diverges by the n^th term test.  Hence the harmonic series diverges.

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