Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From lesson 4: find the sum of 8 + 4 + 2 + 1.
Answer
15. - From Year 9: write 3⁄8 as a decimal.
Answer
0.375. - From Year 10: evaluate 1 − (1⁄2)3.
Answer
7⁄8.
Words used in this lesson
- sum to infinity
- the value the partial sums of a series approach. For example: S∞.
- partial sum
- the sum of the first n terms. For example: S6.
- converge
- get closer and closer to one value. For example: sums near a limit.
- recurring decimal
- a decimal whose digits repeat for ever. For example: 0.333….
- ceiling
- a value the sums stay below. For example: the line in the figure.
Will you ever pass the end of the corridor?
A corridor in a school in Santiago is 30 metres long. You walk half of the way to the far end. Then you walk half of the distance that is left, then half of what is left after that, and you keep going. Each walk is shorter than the last, but you never stop adding walks.
Predict first: will the total distance you have walked ever be more than 30 metres? Use the figure to see what happens to partial sums when the ratio is less than 1, and then try ratios of 1 or more.
Ratio 0.5: after 3 terms the sum is 28 and the ceiling is 32, so the sum is 4 below the ceiling.
Show me the steps
- Ratio 0.5, 3 terms: the sum is 28 and the ceiling is 32, so the sum is 4 below it.
- Ratio 0.5, 10 terms: the sum is 31.97, which is about 0.03 below the ceiling 32.
- Ratio −0.5, 3 terms: the terms are 16, −8, 4 and the sum is 12, which is above the ceiling 10.67. A negative ratio makes the sums alternate around the ceiling.
- Ratio 1.1, 6 terms: the sum is 123.45 and there is no ceiling, because the terms keep growing.
If the ratio is between −1 and 1, the sums settle. The partial sum is Sn = a(1 − rn) ÷ (1 − r). When |r| is less than 1, rn shrinks towards 0 as n grows, so the sums approach a single value, the sum to infinity:
S∞ = a ÷ (1 − r).
With positive terms this value is a ceiling: the sums rise towards it and never pass it. If |r| is 1 or more, the terms do not shrink and the sums do not settle, so there is no sum to infinity.
Two common slips. Using a ÷ (1 − r) when |r| is 1 or more: for 3 + 6 + 12 + … it would give 3 ÷ (1 − 2) = −3, which is nonsense for a sum that grows. And dropping the first term: a series that begins at the 3rd term has first term a × r2, not a.
Worked examples
1. Find the sum to infinity of 12 + 6 + 3 + 1.5 + …
- The first term is 12 and the ratio is 1.5 ÷ 3 = 0.5, which is between −1 and 1.
- S∞ = 12 ÷ (1 − 0.5) = 24.
2. A ball is dropped from 10 m and each rebound is 60% of the height before. How far does it travel in total?
- After the first drop of 10 m, each bounce goes up and then down: 2 × (6 + 3.6 + 2.16 + …). Inside the bracket the first term is 6 and the ratio is 0.6.
- The bracket is 6 ÷ (1 − 0.6) = 15, so the total is 10 + 2 × 15 = 40 m.
3. Write 0.272727… as a fraction.
- It is 0.27 + 0.0027 + 0.000027 + …, with first term 0.27 and ratio 0.01.
- S∞ = 0.27 ÷ 0.99 = 27 ÷ 99 = 3⁄11.
Your go. The first step is given. Find the sum to infinity of 18 − 6 + 2 − 2⁄3 + … The first term is 18.
- The ratio is −6 ÷ 18 = −1⁄3.
- Use S∞ = a ÷ (1 − r) with r = −1⁄3, so 1 − r = 4⁄3.
Check the answer
18 ÷ (4⁄3) = 18 × 3⁄4 = 54⁄4 = 13.5.
Check before you go on
1. For which of these geometric sequences does the sum to infinity exist?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Find the sum to infinity of 45 + 15 + 5 + …
3. Find the sum to infinity of 5 + 4 + 3.2 + …
Level 2 · Building
4. A plant in a greenhouse in Nairobi grows 15 cm in its first month. Each month after that it grows the same fraction of the month before, and in the long run it grows 60 cm in total. What is that fraction, as a decimal?
5. The recurring decimal 0.454545… equals p ÷ 11. Find p.
Level 3 · Stretch
6. Find the sum to infinity of the series whose terms are 3 × 0.4k for k = 1, 2, 3, …
7. A geometric series has first term 8 and sum to infinity 20. Find its second term.
Something to think about
The decimal 0.999… with the nines going on forever is the sum 0.9 + 0.09 + 0.009 + … What does the sum to infinity say it equals, and does that feel right?
One way to think about it
The first term is 0.9 and the ratio is 0.1, so the sum to infinity is 0.9 ÷ 0.9 = 1. The partial sums never pass 1 and get as close as you like, so the value they settle on is exactly 1. The decimal and the number 1 are two names for the same value.
Exit ticket
8. The first four partial sums of a geometric series are 3, 4.5, 5.25, 5.625. What value do the partial sums approach?