Example 4.4.  Show that the series    is divergent.

Solution.  We know that the real series    is divergent.  Hence, Theorem 4.4 implies that the given complex series is divergent.

Explore Solution 4.4.

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

Remark 1.  Since the answer returned for the imaginary part was  ,  this means that a sum was not found.
It is known that the partial sums of the harmonic series grow slowly without bound.  
For example, adding 10,  100,  1000  and  10000 terms yields:

Therefore, the series slowly diverges to .  

Remark 2.  The integral test could also be used.

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

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