1. Lesson 6

1.8. Explore 4

Mathematics 20-1 Module 1

Module 1: Sequences and Series

 

Infinite Geometric Series Formula

 

In Try This 2 you may have discovered that for values between −1 and 1, the expression rn approaches zero as n increases. This causes the sum of a geometric-series expression to approach It is assumed that when n reaches infinity, the sum of the infinite geometric series will be By multiplying both the numerator and the denominator of this expression by −1, you get

 

 

 

If the sum of an increasingly greater number of terms in an infinite series approaches a finite number, this number is the sum of the infinite series.


 


formula sheet

Now is a good time to add the formula for the sum of an infinite geometric series to your copy of Formula Sheet.

 


 

textbook
Did You Know?


The symbol for infinity, also known as a lemniscate, is ∞. This symbol resembles a figure-eight on its side. The symbol was suggested for this purpose by English mathematician John Wallis in 1655. The ancient Romans, however, had previously used the symbol to represent one thousand. S∞ refers to the sum of an infinite number of terms.

 

Turn to “Example 1” on page 61 of the textbook to learn how you can evaluate an infinite geometric series using the formula you just derived. Pay attention to the notation used to show that a series is convergent. In part b., find out what would happen if you tried to apply the formula to a divergent series. At what point in the process will you realize that the infinite series cannot be evaluated?