Example 4.22.   The infinite series    has radius of convergence by the Cauchy-Hadamard formula.  We see this by calculating   

            ,  so  

            ,

hence  .  

Explore Solution 4.22.

Enter the formula for the coefficients.  There is a formula for the even subscripts and a different formula for the odd subscripts.

Use the Cauchy-Hadamard formula and find the radius of convergence R.

The series    has radius of convergence    by the Cauchy-Hadamard formula because   .  

We can let Mathematica find the sum of this series.

Thus we see that f[z] is composed of two "infinite geometric series"  , whose radius of convergence are  , respectively. Hence the radius of convergence of f[z] must be  .  

We can use Mathematica to compute the first few terms in the Maclaurin series for f[z] and compare them with the formula that was given.

We can plot some of the partial sums and see that they converge. Convergence will be faster if we choose a smaller disk   the following graphs use the smaller disk with  .  

(c) 2006 John H. Mathews, Russell W. Howell