Example 4.3.  Show that the series    is convergent.

Solution.  Recall that the real series    and    are convergent.  Hence, Theorem 4.4 implies that the given complex series is convergent.  

Explore Solution 4.3.

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

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

Use Mathematica to compute some 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 infinite series converges, indeed  .  

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