Extra Example 1.  Find the radius of convergence of the infinite series  .

Explore Extra Solution 1.

The sequence of coefficients is  .   
The coefficients with even subscripts are , and the coefficients with odd subscripts are  .  
So that we have

            and    .  

It follows that   .   
If  ,  then  ,  and the series converges.   
If  ,  the series diverges.  

We are done.

Aside.  The given series is the sum of two geometric series:      and   ,  and are known to have radii of convergence and , respectively.  Hence their sum has radii of convergence .

              

The even and odd series are:

                



              

Therefore, S(z) is the sum of two geometric series:

            

which might be printed by Mathematica as  

We are done.

Aside.  We can investigate what happens as .

Caveat.  In some cases involving complex numbers, earlier versions of Mathematica will report a sum when the series actually diverges.  In Mathematica 4.1 the following series converges.

The above series actually diverges because   for all n.

Most likely this occurs because Mathematica 4.2 is summing the series and then making a substitution.

Now try another value.

Indeed, the above series diverges because   for all n.

This bug has been correct in Mathematica 5.1 and the above series does not converge.

However, you have been given a warning!

Caveat.  The mathematical analysis proves that the series diverges.  But for some reason Mathematica 4.1 will compute a sum.  For sure it is a mistake which is equivalent to substituting into the formula  .  This bug has been corrected in Mathematica 5.1.

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