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.  

Example

If 

        a_n  =  n² - 5

find S₄.

Solution

We just plug in 1 through 5 for n and add

        S₄  =  (1² - 5) + (2² - 5) + (3² - 5) + (4² - 5)

        =  -4 + (-1) + 4 + 11  =  10

We can also write the problem in sigma notation as

We define the infinite series S as the limit of  the s_n that is

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: 

        (1/4 - 1/1) + (1/9 - 1/4) + (1/16 - 1/9) + (1/25 - 1/16) + ...

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

        =   -1

Such a series is called a telescoping series.

Geometric Series

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

The series below is an example of a geometric series.

       S  =  3/2 + 3/4 + 3/8 + ...

For geometric series we have the following theorem.

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

The formula comes directly from the formula for the sum of a finite geometric series.

If  |r| < 1, then rⁿ⁺¹ goes to zero as n goes to infinity, and the formula follows.

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.

Application

Recently, the federal government gave a tax refund of a total of 10 billion dollars.  Economists predict that Americans will spend 70% of this refund on items that directly go back into the wallets of Americans.  Again 70% of the spent money will again be used toward money that can be spent in America.  If this process continues forever at the 70% rate, how much money will exchanged from this original 10 billion dollars?

Solution

This is an infinite geometric series with 

        a  =  10        and        r  =  0.7

We have that the sum is 

                       10
        S  =                       =  30       
                    1 - 0.7

There refund will result in 30 billion dollars being exchanged in the US.

The Harmonic Series

The last example is important enough to deserve its own theorem.  We call it the "Harmonic Series Test".

Harmonic Series Test  The series with terms 1/n diverges

Proof:  we write

        1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 
        1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 + 1/17 + ...

        < 1/1 + 1/2 + 1/4 + 1/4 + 1/8 + /18 + 1/8 + 1/8 
        + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/32 + ...

        = 1/2 + 1/2 + 1/2 + ...

which diverges.  Hence the harmonic series diverges.

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