Infinite Series Activity

 

In this project you will use a calculator or computer to explore the relationship of partial sums and sums of infinite series.

 

Consider the series

 

and its partial sums

.

1.      Calculate . 

 

2.      Do these partial sums seem to be approaching any specific value in the limit?  If so, what is this value?

 

3.      Use your partial sums to estimate the value of .

 

Now consider the alternating series

 

and its partial sums

.

 

4.      Calculate . 

 

5.      Do these partial sums seem to be approaching any specific value in the limit?  If so, what is this value?

 

6.      Use your partial sums to estimate the value of the alternating series.

 

7.      How do the partial sums  compare to the value of the series?

 

8.      How do the partial sums  compare to the value of the series?

 

9.      Do your observations for the last two questions apply to all alternating series?  Check the theorems and proofs in your calculus textbook.