Example 4.5.  Show that the series is divergent.  

Solution.  Here we set    and observe that  

            .  

Thus , and Theorem 4.5 implies that the series is not convergent;  hence it is divergent.

Explore Solution 4.5.

Enter the formula for the terms of the sequence, and look at the modulus.

Hence , and the series is divergent.

Aside.  Mathematica will indicate that there is a problem with convergence.

Which is comforting to see that Mathematica knows the answer too!

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.

We see that the  series   is divergent.   

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