Power Series

Definition of a Power Series

Definition of a Power Series Let f(x) be the function represented by the series         Then f(x) is called a power series function.

More generally, if f(x) is represented by the series

       

Then we call f(x) a power series centered at x = c.  The domain of f(x) is called the Interval of Convergence and half the length of the domain is called the Radius of Convergence.

The Radius of Convergence

To compute the radius of convergence, we use the ratio test.


Example:  Find the radius of convergence of

       

Solution:   We use the Ratio Test:

       
We solve     
   
       

or 

        |x - 3| < 2 

so that  

        1 < x < 5

Since 

           1
                (5 - 1) = 2
           2

the radius of convergence is 2.  Notice that we could have use the geometric series test and obtained the same result.  The ratio test is the most likely test to work, but occasionally another test such as the geometric series test or the root test is easier to use.

Exercise:  

Find the radius of convergence of

       

Interval of Convergence  

To find the interval of convergence we follow the three steps:

1. Use the ratio test to find the interval where the series is absolutely convergent.
   

2. Plug in the left endpoint to see if it converges at the left endpoint.  (AST may be useful).
   

3. Plug in the right endpoint to see if it converges at the right endpoint.  (AST may be useful).

Example:

Find the interval of convergence for the previous example:

       

Solution:  

1. We have already  done this step and found that the series converges absolutely 
   for 1 < x < 5.
   

2. We plug in x = 1 to get
   
          
   
   This series diverges by the limit test.
   

3. We plug in x = 5 to get
   
          
   
   This series also diverges by the limit test.

Hence the endpoints are not included in the interval of convergence.  We can conclude that the interval of convergence is

         1 < x < 5

Exercise

Find the interval of convergence of the previous exercise:

       

Differentiation  and Integration of Power Series

Since a power series is a function, it is natural to ask if the function is continuous, differentiable or integrable.  The following theorem answers this question.

Theorem Suppose that a function is given by the power series            and that the interval of convergence is            (c - R, c + R) (plus possible endpoints)   then f(x) is continuous, differentiable, and integrable on that interval (not necessarily including the endpoints).  To obtain the derivative or the integral of f(x) we can pass the derivative or integral through the S.  In other words                and             Furthermore, the radius of convergence for the derivative and integral is R.

Example:

Consider the series

       

by the GST this series converges for |x| < 1,  hence the center of convergence is 0 and the radius is 1.  By the above theorem

       

has center of convergence 0 and radius of convergence 1 also.  We can also say that

       

also has center of convergence 0 and radius of convergence 1.

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