Infinite geometric sums and convergence | Year 11

01 / LEARN

The idea, step by step

A sum approached by finite sums

If $∣ r ∣ < 1$ , the powers $r^{n}$ approach zero. The finite-sum formula then approaches $a / (1 - r)$ . With a nonzero first term, ratios of magnitude at least 1 do not produce a convergent geometric series.

The terms tending to zero are necessary for any convergent series, but this is not a general sufficient test. The result here is justified specifically for geometric series.

S_{\infty} = \frac{a}{1 - r} when ∣ r ∣ < 1

There is no final term

The symbol $S_{\infty}$ is the limit of partial sums, not a sum ending at a last index. The tail after $n$ terms is itself geometric, beginning with $a r^{n}$ and having sum $a r^{n} / (1 - r)$ .

An ideal bouncing model may contain infinitely many shorter journeys while a real ball eventually stops. State the model assumptions before interpreting a limiting total.

A common trap

Numerically decreasing terms always give convergence.

Instead: For example $- 2, - 4, - 8, \dots$ decreases but diverges. The ratio magnitude must be below 1.

02 / WORKED EXAMPLES

See the thinking

Example 1

An alternating infinite sum

Find $9 - 3 + 1 - 1 / 3 + \dots$ .

$r = - 1 / 3$ has magnitude below 1.
$S_{\infty} = 9 / (1 + 1 / 3) = 27 / 4$ .

27 / 4

Example 2

Controlling a remaining tail

For $1 + 1 / 2 + 1 / 4 + \dots$ , find how many terms give a remaining tail below 0.01.

The tail is $2^{1 - n}$ .
At $n = 7$ it is $1 / 64 > 0.01$ ; at $n = 8$ it is $1 / 128 < 0.01$ .

At least 8 terms.

Check your understanding

Does $4 + 6 + 9 + \dots$ converge?

Show answer

No.

$r = 3 / 2$ ; the terms do not approach zero.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $12 + 3 + 3 / 4 + \dots$ to infinity.
2 marks
Find the infinite sum for $a = 5, r = - 0.6$ .
2 marks
A geometric series starts at 8 and sums to 20. Find its ratio.
2 marks
Use partial sums to explain why $1 - 1 + 1 - 1 + \dots$ does not converge.
2 marks
A ball falls 2 m and rebounds to half its preceding height each time. Find the ideal total vertical distance.
3 marks
For $10 + 5 + 2.5 + \dots$ , find the tail after four terms.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $12 + 3 + 3 / 4 + \dots$ to infinity.
2 marks
$12 / (1 - 1 / 4) = 16$ .
$16$ .
Find the infinite sum for $a = 5, r = - 0.6$ .
2 marks
$5 / (1 + 0.6) = 3.125$ .
$25 / 8 = 3.125$ .
A geometric series starts at 8 and sums to 20. Find its ratio.
2 marks
$8 / (1 - r) = 20$ gives $r = 3 / 5$ .
$3 / 5$ .
Use partial sums to explain why $1 - 1 + 1 - 1 + \dots$ does not converge.
2 marks
They do not approach one value.
Partial sums alternate between 1 and 0.
A ball falls 2 m and rebounds to half its preceding height each time. Find the ideal total vertical distance.
3 marks
The first fall is 2 m; later paired journeys total $2 (1 + 1 / 2 + \dots) = 4$ m.
$6 m$ .
For $10 + 5 + 2.5 + \dots$ , find the tail after four terms.
2 marks
The fifth term is $10 {(1 / 2)}^{4} = 5 / 8$ ; its tail sums to $(5 / 8) / (1 / 2) = 5 / 4$ .
$1.25$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics sequence and series content with introductory induction, regrouped into an original school-sized chapter.

Official curriculum source ↗

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