Lesson 11–5

Example 1  Sum of an Infinite Geometric Series
Find the sum of each infinite geometric series, if it exists.
a. 16 – 22.4 + 31.36 – …

First, find the value of r to determine if the sum exists.
a1 = 16 and a2 = –22.4, so r =  or –1.4. Since | –1.4 | ≥ 1, the sum does not exist.

b. –5 + 1.25 – 0.3125 + …

Step 1  Find the value of r to determine if the sum exists.
a1 = –5 and a2 = 1.25, so r =  or –0.25. Since | –0.25 | < 1, the sum exists. 
Step 2  Use the formula for the sum of an infinite geometric series.

S = 	Sum formula
= 	a1 = –5, r = –0.25
=  or –4	Simplify.

Example 2  Infinite Series in Sigma Notation
Evaluate .

S = 	Sum formula
= 	a1 = , r = 
=  or –	Simplify.

Example 3  Write a Repeating Decimal as a Fraction
Write 0. as a fraction.

Method 1	Method 2	
0. = 0.587587587 …	S = 0.	Label the given 
	decimal.	
= 0.587 + 0.000587 + 0.000000587 + …	S = 0.587587587 …	Repeating decimal	
=  +  +  + …	1000S = 587.587587…	Multiply each side by 	1000.	
S = 	Sum formula	999S = 587	Subtract the second 
	equation from the 
	third.	
= 	a1 = , r = 	S = 	Divide each side by 
	999.	
= 	Subtract.		
= 	Simplify.