Creating Power Series From Functions

The Geometric Power Series

Recall that


Substituting x for r, we have 

We write



Milking the Geometric Power Series

By using substitution, we can obtain power series expansions from the geometric series.

Example 1  

Substituting x2 for x, we have 


Example 2  

Multiplying by x we have


Example 3  

Suppose we want to find the power series for 

        f(x)  =                    
                      2x - 3

centered at x = 4.  We rewrite the function as 

                    1                            1
          2(x - 4) + 8 - 3          2(x - 4) + 5


Example 4  

Substituting -x for x, we have


Example 5  

Substituting x2 in for x in the previous example, we have


Example 6  

Taking the integral of the previous example, we have


Exercise  Find the power series that represents the following functions:

  1. ln(1 + x)

  2. tanh-1x

  3. -(1 - x)-2 

Integrating Impossible Functions

We can use power series to integrate functions where there are no standard techniques of integration available.


Use power series to find the integral 


Then use this integral to approximate 



Notice that this is a very difficult integral to solve.  We resort to power series.  First we use the series expansion from Example 6, replacing x with x2.  


Integrating we arrive at the solution 


Now to solve the definite integral, notice that when we plug in 0 we get 0, hence the definite integral is

Using the first 5 terms to approximate this we get 0.300

Notice that the error is less than the next term (which comes from x23/253)

        E < 1/253  =  .004.


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