Example 4.6.  Show that the series    is convergent.

Solution.  We calculate  .    Using the comparison test and the fact that    converges, we determine that    converges and hence, by Corollary 4.1, so does .

Explore Solution 4.6.

Enter the formula for the series, and determine if the series converges or diverges.

Hence we see that the series converges and that  .  
Or we could use Corollary 4.1 and show that the series converges absolutely.  

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 construct some of the partial sums of the infinite series.

Use Mathematica to compute more of the partial sums of the infinite series.

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

Therefore, we see that the series converges, and we have  .  

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