Sum to infinity of a geometric progression

6 minAS Level1.6.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can say when a geometric progression is convergent ( $|r| < 1$ ) and find its sum to infinity $S_\infty = \dfrac{a}{1 - r}$ .

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$S_n$	the sum of the first $n$ terms
$S_\infty$	the sum to infinity: what $S_n$ settles down to as you add more and more terms
$a$ , $r$	the first term and the common ratio
$\lvert r \rvert$	the size of $r$ , ignoring its sign; $\lvert r \rvert < 1$ means $-1 < r < 1$
$\to$	"gets closer and closer to"

Infinitely many slices, one pizza

Eat half a pizza. Then half of what's left. Then half of that.

You'll go on forever (in theory), but you'll never eat more than one pizza:

$\tfrac12 + \tfrac14 + \tfrac18 + \tfrac1{16} + \dots = 1$

Adding infinitely many numbers can give a finite answer, as long as the numbers shrink fast enough. For a GP, "fast enough" has a very clean test.

Watch rⁿ disappear

You already know $S_n = \dfrac{a(1 - r^n)}{1 - r}$ .

If $-1 < r < 1$ , then $r^n$ gets closer and closer to 0 as $n$ grows. For example:

$0.5^{10} \approx 0.001$
$0.5^{20} \approx 0.000\,001$

So the $r^n$ in the formula fades away, and

$S_n \to \frac{a(1 - 0)}{1 - r} = \frac{a}{1 - r}.$

If $|r| \geqslant 1$ , $r^n$ doesn't fade (try $2^{10} = 1024$ ), so there's no finite limit.

Definition

Convergent GP and sum to infinity

A geometric progression is convergent if and only if $|r| < 1$ , that is $-1 < r < 1$ . Its sum to infinity is

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

If $|r| \geqslant 1$ the progression is not convergent and the sum to infinity does not exist.

What goes wrong when |r| ≥ 1

$r$	Example	What the partial sums do
$r = 1$	$2 + 2 + 2 + \dots$	grow forever: $S_n = 2n$
$r = -1$	$2 - 2 + 2 - \dots$	flip between 2 and 0, never settle
$r > 1$	$1 + 2 + 4 + \dots$	grow faster and faster
$r < -1$	$1 - 2 + 4 - \dots$	swing wider and wider

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.