Infinite Series Advanced Activity

 

In this activity you will use a computer as well as theorems to analyze three series.  This will illustrate both the power and limitations of computers and the importance of theory backing up computations.

 

You will probably need to use a standard programming language such as C or Pascal for this project.  Calculators and high-level programs such as Excel, Maple, and Matlab will probably be too slow to successfully carry out the needed calculations.  If you need help with the programming see your instructor.

 

Consider the infinite series

 

and its partial sums

.

1.      Calculate . 

 

2.       Looking at the sequence of partial sums calculated in the first question, would you say that the infinite series converges?  If so, what is its value?  If you need to calculate more partial sums to answer these questions, do so. 

 

3.      Now answer the question of convergence using appropriate theorems from calculus.

 

Consider the series

 

and its partial sums

.

4.      Calculate  with n sufficiently large for you to determine whether or not this infinite series converges.  You will probably need to use very large values of n --- perhaps in the millions!  You won’t want to display all of your values for  because that would make your program run to slowly.  Look up the mod function and use an “if-then” statement to print only selected values.  The expression “n mod k” represents the remainder after n is divided by k.  For example, 11 mod 4 equals 3 and 12 mod 4 equals 0.  If you display  only when n mod 1000000 = 0, then you will display  only when n is a multiple of 1000000.

5.        Now answer the convergence question using theorems from calculus.

 

Now consider the series

 

and its partial sums

.

6.      Repeat the last two questions for this series.

 

 

Finally, consider the series  where .

7.      Calculate  carrying 32 or fewer significant digits in your calculations.  Do these partial sums suggest that the series converges?  If so, to what value?

 

8.      Use L’Hôpital’s rule to find .  What does this tell you about convergence?  To better understand why computers and calculators have problems with this expression,  click here.

 

9.      Summarize your conclusions.  Can a computer be used in a reliable way to check to see whether or not a series converges?  Why or why not?  Is it important to have theorems with proofs?