9 (c).  Can we conclude that  .   Justify your answer.

Solution 9 (c).

See text and/or instructor's solution manual.

      Can not we conclude that  ,  because the integrand has the form

                    ,

but    has a zero of order  3  at the origin, and does not satisfy the hypothesis of Theorem 8.7,

which states that  Q(z)  must have a zero of order at most 1 at the origin.  

We are done.   

Aside.  We can let Mathematica double check our work.

Note.  There is no difficulty with "tail end at infinity."   

However, the integrand    is not continuous at  ,  so the difficulty occurs at the origin.  

For  ,   ,   and  

                      

Hence the integral diverges.

Maple can check our work too!

     > int( x^(1/3)/(x^3*(x+1)), x=0..infinity );

               

                    The area under the curve    over the interval    is infinite.

                    The area under the curve    over the interval    is finite.

                    The area under the curve    over the interval    is infinite.

This solution is complements of the authors.

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