Example 4.21.  The infinite series    has radius of convergence Cauchy's root test because  

            ,  

hence   .  

Explore Solution 4.21.

Enter the formula for the coefficients.  

Use the Cauchy root test and find the limit and then the radius of convergence R.

The series    will converge in the disk  .  

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  .  

We see that the series    is converging in the smaller disk  .  

Aside.  We can let Mathematica try to determine the "sum" of this series.

Unfortunately, there is no known closed formula.  This is an instance where the infinite series is "the answer."

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