Intermediate Algebra (11th edition)

Finding the Sum of the Terms of an Infinite Geometric

Sequence

Evaluate the sum of the terms of the infinite geometric sequence with and

Substitute into the formula.

Infinite sum formula

Let.

Simplify the denominator.

Write as division.

Definition of division

Multiply. NOW TRY

In summation notation, the sum of an infinite geometric sequence is written as

For instance, the sum in Example 8would be written

Finding the Sum of the Terms of an Infinite Geometric Series

Evaluate

This is the infinite geometric series

with and Since we find the sum as follows.

Let.

Simplify the denominator.

= 1 Divide. NOW TRY

=

12

1

2

= a 1 =^12 , r=^12

1

2

1 -^12

S=

a 1

1 - r

a 1 = 21 r=^12. |r| 6 1,

1

2

+

1

4

+

1

8

+Á,

a

q

i= 1

a

1

2

b

i

.

EXAMPLE 9

a

q

i= 1

3 a-

1

3

b

i- 1

.

a

ˆ

i 1

ai.

=

9

4

= 3 #

3

4

= 3 ,

4

3

=

3

4

3

= a 1 =3, r=-^13

3

1 - A-^13 B

S=

a 1

1 - r

r=-^13.

a 1 = 3

EXAMPLE 8

SECTION 12.3 Geometric Sequences 697

NOW TRY
EXERCISE 8
Evaluate the sum of the terms
of the infinite geometric
sequence with and
r=^23.

a 1 =- 4

NOW TRY ANSWERS

8. 
   
   
   
   • 12 9.^158
   
   
   
   

NOW TRY

EXERCISE 9

Evaluate a

q

i= 1

a

5

8

ba

3

4

b

i

.