Example 4.16.  Show that   converges for all  z  in the disk  .  

Explore Solution 4.16.

Enter the formula for the terms in the series.

Use d'Alembert's ratio test. First, find the ratio of consecutive terms.

When  L < 1, the series will converge.  Solve   and obtain the disk  .  

Use Mathematica to find the sum of the infinite series.

We can investigate the convergence by plotting several partial sums of this series.

Since convergence will be more rapid in a smaller disk  ,  the following plot will be a smaller disk with  .  

Convergence will be slower in a larger disk  ,  the following plot will be from a larger disk with  .  

We see that the sequence of functions {S[x,n]} is converging to f[z].

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