Exercise 15.  Show that      converges for   .   

Solution 15.

See text and/or instructor's solution manual.

Solution.   .  

This is a geometric series  of the form      where      and converges for    by Theorem 4.12 in Section 4.3.

We know that   .

Taking the modulus we have  .   Then

    iff        iff        iff       iff        iff     .   

Therefore,        converges for   .   

We are done.   

Aside.  We can see what Mathematica does

We are really done.   

Warning.  Mathematica has plugged    into  .  

There has not been a check to see if   !

Do not blindly trust computer answer !

The point     satisfies  .

        If you substitute    into  ,  then the you will get  

                    .  

        The point    has      and does not satisfy  .

If you substitute   into  ,  then the series does not converge.

If it is difficult to imagine that    diverges, then just look at the terms  

                      

We are really really done.   

Caveat.  If we are only interested in the formula    the image of the right half-plane    is easy to find.

However, if we are only interested in the formula    the image of the right left-plane    is also easy to find.

Would you be shocked to find out that these images regions are the same ?  Hold on to your hats !

                                        The composite mapping  ,  where   ,   ,   and   .  

                                        The composite mapping  ,  where   ,   ,   and   .  

Aside.  This can be viewed as a source and sink.  We will learn more about sources and sinks in Section 11.11.

Aside. How good is your eye? Did you notice the subtle difference in the above graphs. Look again in the W-planes.

 Always remember that the exponential function is periodic.

This solution is complements of the authors.

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