Series

Definition of a Series

Let an be a sequence then we define the nth partial sum of an as

 sn = a1 + a2 + ... + an

In other words, we define sn by adding up the first n terms of an.  We define the series as the limit of  the sn 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
an  =                            -
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 nth term test.  Hence the harmonic series diverges.