Convergence and Sum to Infinity

Q: When can I use the sum to infinity formula?

Only when the series is geometric and $-1 < r < 1$.

Q: How do I find values of x for which a series converges?

Find $r$ in terms of $x$, then solve $-1 < r < 1$.

Q: What happens when r = 0?

Every term after the first is 0, so the series converges and its sum is just the first term. The condition $-1 < r < 1$ already includes $r = 0$, so do not exclude it unless the question tells you to.

Q: Does an arithmetic series have a sum to infinity?

No. An arithmetic series with $d \neq 0$ always diverges.

If the ratio of a geometric series is between $- 1$ and $1$ , the terms shrink so fast that the infinite sum has a finite value. IEB questions test the convergence condition, often with an unknown $x$ , and use the sum to infinity in real contexts.

Condition for convergence

A geometric series converges if and only if

$- 1 < r < 1$

Example. For which values of $x$ does $k = 1 \sum \infty (x - 3)^{k}$ converge?

Here $r = x - 3$ , so $- 1 < x - 3 < 1$ , giving $2 < x < 4$ . At $x = 3$ every term is 0, so the series converges to 0; that value stays in the answer.

If $∣ r ∣ \geq 1$ , the terms do not shrink and the series diverges.

Sum to infinity

$S_{\infty} = \frac{a}{1 - r}, - 1 < r < 1$

Example. $24 + 12 + 6 + \dots$ has $a = 24$ , $r = \frac{1}{2}$ :

$S_{\infty} = \frac{24}{1 - \frac{1}{2}} = 48$

Recurring decimals. $0, 363636 \dots = 0, 36 + 0, 0036 + \dots$ with $a = 0, 36$ and $r = 0, 01$ :

$S_{\infty} = \frac{0 , 36}{0 , 99} = \frac{4}{11}$

Bouncing ball. A ball dropped from 10 m rebounds to $\frac{3}{5}$ of its height each time. It falls 10 m, then goes up and down each rebound:

$Total = 10 + 2 (\frac{6}{1 - 0 , 6}) = 40 m$

Key terms

Converge: Approach a finite value as more terms are added.
Diverge: Grow without bound, or fail to settle on one value.
Sum to infinity: The value a convergent geometric series approaches.

Common mistakes

Using $S_{\infty}$ when $∣ r ∣ \geq 1$ .
Writing the condition as $r < 1$ only and forgetting $r > - 1$ .
Counting the first drop of a bouncing ball twice.
Using the coefficient in sigma notation as $a$ .

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Calculate the sum to infinity of $24 + 12 + 6 + \dots$

Worked solution

Hidden - try the question first

Solution

$a = 24$ , $r = \frac{1}{2}$
$S_{\infty} = \frac{24}{1 - \frac{1}{2}}$
$= 48$

Question 2

5 marks total

routine

Given $k = 1 \sum \infty (x - 3)^{k}$ .

(a) For which values of $x$ does the series converge? (3)

(b) Calculate the sum to infinity if $x = 3, 5$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) $r = x - 3$
$- 1 < x - 3 < 1$
$2 < x < 4$
(b) $a = r = 0, 5$
$S_{\infty} = \frac{0 , 5}{1 - 0 , 5} = 1$

Question 3

6 marks total

complex

Use a geometric series to write $0, 363636 \dots$ (that is, $0, \dot{3} \dot{6}$ ) as a fraction in simplest form.

Worked solution

Hidden - try the question first

Solution

$a = 24$
$a = 0, 36$
$r = 0, 01$
$S_{\infty} = \frac{0 , 36}{1 - 0 , 01}$
$= \frac{36}{99}$
$= \frac{4}{11}$

Question 4

9 marks total

problem-solving

A ball is dropped from a height of 10 m. Each time it bounces, it rises to $\frac{3}{5}$ of the height it fell from.

(a) Calculate the total vertical distance the ball travels before it comes to rest. (5)

(b) After which bounce is the rebound height first less than 0,1 m? (4)

Worked solution

Hidden - try the question first

Solution

(a) First drop: 10 m
Rebound heights: $6 + 3, 6 + \dots$ , with $a = 6$ and $a = 0, 36$
Each rebound is travelled up and down: $2 \times S_{\infty}$
$2 \times \frac{6}{1 - 0 , 6} = 30$
Total $= \frac{36}{99}$ m
(b) Height after bounce $n$ : $10 (0, 6)^{n} < 0, 1$
$(0, 6)^{n} < 0, 01$
$n > \frac{lo g 0 , 01}{lo g 0 , 6} \approx 9, 02$
After the 10th bounce

FAQs

When can I use the sum to infinity formula?

Only when the series is geometric and $- 1 < r < 1$ .

How do I find values of x for which a series converges?

Find $r$ in terms of $x$ , then solve $- 1 < r < 1$ .

What happens when r = 0?

Every term after the first is 0, so the series converges and its sum is just the first term. The condition $- 1 < r < 1$ already includes $r = 0$ , so do not exclude it unless the question tells you to.

Does an arithmetic series have a sum to infinity?

No. An arithmetic series with $d \neq = 0$ always diverges.

Condition for convergence

Sum to infinity

Key terms

Common mistakes

Practice questions

FAQs

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