Find a power series for the function, centered at c and determine the interval of convergence.

A power series centered at is follows the formula:

The given function resembles the power series:

or

For better comparison, we let . The function becomes:

Apply Law of exponents: .

Apply the aforementioned formula for power series on   , we may replace "x" with " " and " " with " ". We let:

 

 

 

Applying   we get:

                      

                     

The power series of the function centered at is:

or 

To determine the interval of convergence, we may apply geometric series test wherein the series   is convergent if   or . If then the geometric series diverges.

Applying on the series  , we get:

By comparing  with   , we determine: .

Apply the condition for convergence of geometric series:  .

Multiply each sides by :

Check the convergence at endpoints that may satisfy .

Let on  , we get:

Using geometric series test,  the  satisfy . Thus, the series diverges at .

 Let on  , we get:

 

 Using geometric series test,  the satisfy . Thus, the series diverges at .

 Thus, the power series  has an interval of convergence:  .