The sum of a geometric series

A geometric sequence multiplies by the same number each time. Its sum is

S_n = \frac{a(1 - r^n)}{1 - r} \qquad (r \neq 1)

where $a$ is the first term and $r$ is the common ratio.

The proof

Write the sum out and then multiply the whole thing by $r$ :

S_n = a + ar + ar^2 + \dots + ar^{n-1}

rS_n = ar + ar^2 + ar^3 + \dots + ar^{n}

Subtract the second line from the first, term by term. Almost everything cancels: the $ar$ in the first line meets the $ar$ in the second, and likewise for every term except the first of the first line and the last of the second.

S_n - rS_n = a - ar^n

Factorise both sides. On the left, $S_n$ is common. On the right, $a$ is.

S_n(1 - r) = a(1 - r^n)

Divide by $1 - r$ , which is allowed because $r \neq 1$ :

S_n = \frac{a(1 - r^n)}{1 - r}

Why $r = 1$ is excluded

If $r = 1$ every term is the same, so the sum is $n \times a$ . The formula would ask you to divide by $1 - 1 = 0$ , and that is the proof telling you the case was never covered: the step that cancels the terms cancels everything when $r = 1$ , leaving $0 = 0$ , which says nothing.

The sum to infinity

When $|r| < 1$ , each term is smaller than the one before and $r^n$ shrinks towards zero as $n$ grows. So for a large number of terms the $r^n$ in the formula contributes nothing:

S_\infty = \frac{a(1 - 0)}{1 - r} = \frac{a}{1 - r}

This is a genuine limiting statement rather than a rounding: the sum does not merely get close to $\dfrac{a}{1-r}$ , it is the value the partial sums approach, and they get as close as you like.

Check your understanding

Find $S_6$ for $2 + 6 + 18 + \dots$
Find $S_\infty$ for $8 + 4 + 2 + \dots$
Explain why $S_\infty$ does not exist for $r = 2$ .

Answers

$a = 2$ , $r = 3$ , so $S_6 = \dfrac{2(1 - 729)}{1 - 3} = \dfrac{-1456}{-2} = 728$ .
$a = 8$ , $r = \tfrac{1}{2}$ , so $S_\infty = \dfrac{8}{1 - \frac12} = 16$ .
With $r = 2$ the terms grow without limit, so the partial sums do too. The formula's derivation needs $r^n \to 0$ , which is false here.

The sum of a geometric series

The proof

Why r=1r = 1 is excluded

The sum to infinity

Check your understanding

Why $r = 1$ is excluded