Chapter 08 — Sequences and Series — Part 5: Sum to Infinity of a Geometric Series

CHAPTER 08: SEQUENCES AND SERIES

PART 5: SUM TO INFINITY OF A GEOMETRIC SERIES

In Part 4, we studied the sum of a finite number of terms in a geometric series. In this lesson, we extend that idea to a series containing infinitely many terms.

We shall establish the condition under which an infinite geometric series has a finite sum, derive the formula, and apply it to examination-style problems.

1. Introduction

A geometric series is formed by adding the terms of a geometric progression. Each term is obtained by multiplying the preceding term by a constant number called the common ratio.

Consider the series:

8 + 4 + 2 + 1 + ½ + ¼ + ···

The first term is 8 and the common ratio is ½. Notice that each successive term becomes smaller.

Although there are infinitely many terms, the running total approaches a fixed value. This limiting value is called the sum to infinity.

2. Meaning of Sum to Infinity

For a geometric series with first term a and common ratio r, the series is:

a + ar + ar² + ar³ + ar⁴ + ···

The sum of the first n terms is denoted by S_n. When the number of terms increases indefinitely, we investigate whether S_n approaches a finite limiting value.

If it does, that limiting value is called S_∞.

Important distinction

A series does not automatically have a finite sum just because it contains infinitely many terms. Its common ratio must satisfy a particular condition.

3. Derivation of the Sum to Infinity Formula

From Part 4, the sum of the first n terms of a geometric series is:

S_n = a(1 − rⁿ)/(1 − r),   r ≠ 1

For an infinite geometric series, we consider what happens as n becomes very large.

When −1 < r < 1, the powers of r approach zero:

rⁿ → 0 as n → ∞

Therefore:

S_∞ = lim_n→∞ S_n

= lim_n→∞ a(1 − rⁿ)/(1 − r)

Since rⁿ approaches zero:

S_∞ = a(1 − 0)/(1 − r)

S_∞ = a/(1 − r)

This is the formula for the sum to infinity of a convergent geometric series.

4. The Condition for Convergence

The formula is valid only when the common ratio satisfies:

|r| < 1

Equivalently:

−1 < r < 1

This condition is called the condition for convergence.

Common ratio	Behaviour	Result
0 < r < 1	Terms decrease towards zero.	Converges
−1 < r < 0	Terms alternate in sign and decrease in magnitude.	Converges
r = 1	Terms remain constant.	Diverges
r > 1	Terms increase in magnitude.	Diverges
r ≤ −1	Terms do not approach zero.	Diverges

Examination reminder:

Always check |r| < 1 before applying the sum to infinity formula. A negative common ratio is allowed, provided its absolute value is less than one.

5. Worked Examples

Example 1: Finding the sum to infinity

Find the sum to infinity of:

12 + 6 + 3 + 3/2 + ···

Solution

First term:

a = 12

Common ratio:

r = 6/12 = 1/2

Since |1/2| < 1, the series converges.

S_∞ = 12/(1 − 1/2)

= 12/(1/2)

= 24

Example 2: A negative common ratio

Find the sum to infinity of:

18 − 6 + 2 − 2/3 + ···

Solution

a = 18

r = −6/18 = −1/3

Since |−1/3| < 1, the series converges.

S_∞ = 18/[1 − (−1/3)]

= 18/(4/3)

= 27/2

Example 3: Common ratio expressed as a decimal

Find the sum to infinity of:

40 + 10 + 2.5 + 0.625 + ···

Solution

a = 40

r = 10/40 = 1/4

Since |1/4| < 1:

S_∞ = 40/(1 − 1/4)

= 40/(3/4)

= 160/3

Example 4: Finding the first term

The sum to infinity of a geometric series is 30. Its common ratio is 2/5. Find its first term.

Solution

Given:

S_∞ = 30,   r = 2/5

Using:

S_∞ = a/(1 − r)

30 = a/(1 − 2/5)

30 = a/(3/5)

a = 30 × 3/5

a = 18

Example 5: Finding the common ratio

A geometric series has first term 15 and sum to infinity 25. Find its common ratio.

Solution

25 = 15/(1 − r)

25(1 − r) = 15

1 − r = 15/25

1 − r = 3/5

r = 2/5

Example 6: Finding a range of values of x

A geometric series is given by:

6 + 3x + (3/2)x² + (3/4)x³ + ···

(a) Find the range of values of x for which the series converges.

(b) Find the sum to infinity in terms of x.

Solution

First term:

a = 6

Common ratio:

r = 3x/6 = x/2

For convergence:

|r| < 1

|x/2| < 1

|x| < 2

−2 < x < 2

Now calculate the sum:

S_∞ = 6/(1 − x/2)

S_∞ = 12/(2 − x)

This result is valid only for −2 < x < 2.

6. Applications of Sum to Infinity

Application 1: Repeated reduction

A quantity starts at 100 units. Each successive contribution is half the preceding contribution. Find the total contribution if the process continues indefinitely.

Solution

100 + 50 + 25 + 12.5 + ···

a = 100,   r = 1/2

S_∞ = 100/(1 − 1/2)

= 200 units

Application 2: Recurring decimals

Express 0.3333... as a fraction.

Solution

Write the recurring decimal as:

0.3 + 0.03 + 0.003 + ···

a = 0.3,   r = 0.1

S_∞ = 0.3/(1 − 0.1)

= 0.3/0.9

= 1/3

Application 3: Another recurring decimal

Express 0.272727... as a fraction.

Solution

0.272727... = 0.27 + 0.0027 + 0.000027 + ···

a = 0.27,   r = 0.01

S_∞ = 0.27/(1 − 0.01)

= 0.27/0.99

= 3/11

7. Common Examination Mistakes

1. Forgetting the convergence condition. Always check |r| < 1.
2. Using the wrong sign. If r is negative, remember that 1 − (−r) becomes addition.
3. Confusing finite and infinite sums. S_n refers to a finite number of terms, whereas S_∞ is a limiting sum.
4. Finding the ratio incorrectly. Divide a term by the term immediately before it.
5. Ignoring restrictions on x. If r contains x, solve the convergence inequality and state the permitted interval.

8. Exercises

Attempt each question independently before opening its solution. Show all your working clearly.

Question 1

Find the sum to infinity of:

10 + 5 + 5/2 + 5/4 + ···

Show worked solution

a = 10, r = 1/2.

S_∞ = 10/(1 − 1/2)

= 20

Question 2

Find the sum to infinity of:

24 − 8 + 8/3 − 8/9 + ···

Show worked solution

a = 24, r = −1/3.

S_∞ = 24/[1 − (−1/3)]

= 24/(4/3)

= 18

Question 3

The sum to infinity of a geometric series is 48. If its first term is 12, find its common ratio.

Show worked solution

48 = 12/(1 − r)

48(1 − r) = 12

1 − r = 1/4

r = 3/4

Question 4

A geometric series has first term 20 and sum to infinity 16. Find its common ratio.

Show worked solution

16 = 20/(1 − r)

16(1 − r) = 20

1 − r = 5/4

r = −1/4

The ratio is valid because |−1/4| < 1.

Question 5

Determine whether each of the following series has a finite sum to infinity. Give a reason.

(a) 5 + 10 + 20 + 40 + ···

(b) 9 − 3 + 1 − 1/3 + ···

(c) 4 + 4 + 4 + 4 + ···

Show worked solution

(a) r = 2. Since |2| is not less than 1, the series diverges.

(b) r = −1/3. Since |−1/3| < 1, the series converges.

(c) r = 1. Since |1| is not less than 1, the series diverges.

Question 6

Find the sum to infinity of:

7 + 7x + 7x² + 7x³ + ···

State the range of values of x for which the sum exists.

Show worked solution

a = 7, r = x.

For convergence:

|x| < 1

−1 < x < 1

Therefore:

S_∞ = 7/(1 − x)

Question 7

A geometric series has first term 9 and common ratio −1/2. Find its sum to infinity.

Show worked solution

S_∞ = 9/[1 − (−1/2)]

= 9/(3/2)

= 6

Question 8

Express 0.272727... as a fraction.

Show worked solution

0.272727... = 0.27 + 0.0027 + ···

a = 0.27 and r = 0.01.

S_∞ = 0.27/(1 − 0.01)

= 0.27/0.99

= 3/11

Question 9

The sum to infinity of a geometric series is 36. Its common ratio is 1/4. Find its first term.

Show worked solution

36 = a/(1 − 1/4)

36 = a/(3/4)

a = 36 × 3/4

a = 27

Question 10

A geometric series is:

5 + 5x + 5x² + 5x³ + ···

Find the value of x for which its sum to infinity is 20.

Show worked solution

20 = 5/(1 − x)

20(1 − x) = 5

1 − x = 1/4

x = 3/4

This satisfies the convergence condition −1 < x < 1.

9. Lesson Summary

Sum to infinity:

S_∞ = a/(1 − r)

Condition for convergence:

|r| < 1

• A geometric series is obtained by adding the terms of a geometric progression.
• An infinite geometric series may have a finite limiting sum.
• The common ratio must lie strictly between −1 and 1.
• Negative common ratios are permitted if their absolute values are less than one.
• When the ratio contains an unknown variable, find its permitted range before stating the sum.

End of Part 5

You have now completed the study of the sum to infinity of a geometric series.

Proceed to Part 6 to continue developing your understanding of sequences and series.

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