Example 4.13.  Show that  .  

Solution.  If we set  ,  then  .  By Theorem 4.12, the sum is  

            .

Explore Solution 4.13.

We can use the definition of convergence of a series and find the limit of the partial sums.

Or we can see that this is an infinite geometric series with ratio  .  

The sum of the infinite geometric series is now found.

The series of absolute values converges, therefore the series converges.

Use Mathematica to construct some of the partial sums of the infinite series.

Use Mathematica to plot some of the partial sums of the infinite series.

We see that the sum of the infinite geometric series  .  

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