2 (d).  It turns out that the third series    converges everywhere on  ,  except at the point  .  

This is not easy to prove.  Give it a look.

Solution 2 (d).

See text and/or instructor's solution manual.

Solution   Set    and the series becomes   ,  and this is the harmonic series which was studied in calculus and is known to diverge.  

When  ,   but  ,  use summation by parts as follows:

With   ,  and for  ,  we have  

                      

For  ,  and  ,  we have  ,  and it follows that  

                      

Now we set  ,  and use the inequality  ,  for all  n.  

Go back to our previous result,  ,  and calculate it's absolute value:  

                       

The last term can be made arbitrarily small by taking  p  large, so by the Cauchy criterion      converges for  ,  and  .  

Remark 1.  The topic of summation by parts is probably not covered in most calculus courses and this might be your first introduction to it.

Remark 2.  Do not be discouraged there are numerous useful facts that we do not have time to cover in the standard calculus course.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell