Exercise 4.  Use the ratio test to show that the following series converge.  

4 (c).  .

Solution 4 (c).

See text and/or instructor's solution manual.

Solution   Write    and use d'Alembert's ratio test and calculate the limit value  L:  

                    

Since   ,   the series       will converge.  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Remark 1. The Taylor series for    was studied in calculus, and we have  

                    ,  

which converges for all  x.

In Section 5.1 we will find that complex series are extensions of real series and we will derive the formula

                    ,

which converges for all  z.

Remark 2. In Section 5.1 we will learn that    is the complex exponential function.  

This will give rise to the computation  

                    .   

We are really really done.   

Aside.  We can let Mathematica double check our work.

                    A few of the partial sums  ,  and the limit point  .

This solution is complements of the authors.

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