Infinite geometric series

Warm-up

Three quick questions from earlier lessons.

Discover

You walk half way to a wall, then half the remaining distance, and so on: $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dots$ What does this add up to?

1
It grows without bound

Key idea

When $|r| < 1$ , the terms shrink so fast that the infinite sum has a value.

If

|r| < 1

:

a + ar + ar^2 + \dots = \dfrac{a}{1 - r}

. If

|r| \ge 1

, the series has no sum.

Worked example

Write $0.\overline{36} = 0.363636\dots$ as a fraction.

0.\overline{36} = \dfrac{36}{100} + \dfrac{36}{100^2} + \dots

Blocks of two digits.

S = \dfrac{36/100}{1 - 1/100} = \dfrac{36}{99}

a = 36/100, r = 1/100.

0.\overline{36} = \dfrac{4}{11}

Divide by 9.

Your turn, guided

Your turn: $6 + 2 + \dfrac{2}{3} + \dots$

r = \dfrac{1}{3}

, and

|r| < 1

2 ÷ 6.

S = 9

6 / (1 − 1/3) = 6 / (2/3).

Practice ★

Find the sum $7 + \dfrac{ 7 }{ 5 } + \dfrac{ 7 }{ 25 } + \dots$
Find the sum $2 + \dfrac{ 2 }{ 5 } + \dfrac{ 2 }{ 25 } + \dots$
Find the sum $9 + \dfrac{ 9 }{ 5 } + \dfrac{ 9 }{ 25 } + \dots$

Watch out!

Practice ★★

Does the series $1 + \dfrac{ 6 }{ 2 } + \left(\dfrac{ 6 }{ 2 }\right)^2 + \dots$ have a sum?
1. No: the ratio is greater than 1, so the terms keep growing
2. Yes: $\dfrac{1}{1 - r}$ with $r = \dfrac{ 6 }{ 2 }$
3. Yes: every infinite geometric series has a sum

Write $0.\overline{ 4 } = 0.444\dots$ as a fraction, using $\dfrac{ 4 }{10} + \dfrac{ 4 }{100} + \dots$
Write $0.\overline{ 6 } = 0.666\dots$ as a fraction, using $\dfrac{ 6 }{10} + \dfrac{ 6 }{100} + \dots$
Write $0.\overline{ 1 } = 0.111\dots$ as a fraction, using $\dfrac{ 1 }{10} + \dfrac{ 1 }{100} + \dots$

Exit ticket

Three questions to check you have got it.

Find the sum $7 + \dfrac{ 7 }{ 5 } + \dfrac{ 7 }{ 25 } + \dots$
Find the sum $2 + \dfrac{ 2 }{ 5 } + \dfrac{ 2 }{ 25 } + \dots$
Find the sum $9 + \dfrac{ 9 }{ 5 } + \dfrac{ 9 }{ 25 } + \dots$

Summary card

Infinite geometric series

$|r| < 1$ : $S = \dfrac{a}{1 - r}$ .
$|r| \ge 1$ : no sum.
Repeating decimals are such series.

Watch out: check $|r| < 1$ before using the formula.

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