12.4 Find Sums of Infinite Geometric Series

Partial sum

The sum S_n of the first n terms of an infinite series

THE SUM OF AN INFINITE GEOMETRIC SERIES

The sum of an infinite geometric series with first term a₁ and common ratio r is given by

S =

provided ½r½ < 1. If ½r½ ³ 1, the series has _no sum_ .

Example 1

Find sums of infinite geometric series

Find the sum of the infinite geometric series.

a.

b.

c. 1 - 2 + 4 - 8 + …

Solution

a. For this series, a₁ = _6_ and r = _0.6_ .

S = = _{______} = _15_

b. For this series, a₁ = _1_ and r = _{_____.}

S = = _{_________}= _{____}

c. You know that a₁ = _1_ and a₂ = _-2_. So, r = _{_____} = _-2_.
Because | _-2_ | _³_ 1, the sum _does not exist_.

Find the sum of the infinite geometric series, if it exists.

1.

does not exist

2.

54

3.

13.5

Example 2

Use an infinite series as a model

Swings A person is given one push on a swing. On the first swing, the person travels a distance of 4 feet. On each successive swing, the person travels 75% of the distance of the previous swing. What is the total distance the person swings?

Solution

The total distance traveled by the person is:

d = 4 + 4( 0.75 ) + 4( 0.75 )² + 4( 0.75 )³ + . . .

=	Write formula for sum.
= ________	Substitute for a1 and r.
= _16_	Simplify.

The swing travels a total distance of _16_ feet.

Complete the following exercise.

4. In Example 2, suppose the person travels 3 feet on the first swing. What is the total distance the person swings?

12 feet

Example 3

Write a repeating decimal as a fraction

Write 0.474747 . . . as a fraction in lowest terms.

Solution

0.474747 . . .

= 47( 0.01 ) + 47( 0.01 )² + 47( 0.01 )³ + . . .

=	Write formula for sum.
= __________	Substitute for a1 and r.
= ______	Simplify.
= _____	Write as a quotient of integers.

The repeating decimal 0.474747 . . . is _{_____} as a fraction.