The limiting sum of a geometric series

When an infinite geometric series has a limiting sum, the condition −1 < r < 1, the formula S = a/(1 − r), and problems including finding the values of x for which a limiting sum exists.

Adding infinitely many terms

Walk halfway to a wall, then half the remaining distance, then half again. The distances $\tfrac12 + \tfrac14 + \tfrac18 + \cdots$ never stop, but their total gets as close as you like to $1$. We say the series has limiting sum $1$.

Limiting sum of a GP

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

Partial sums closing in on the limit

Each step is one more partial sum S₁, S₂, S₃, … The dashed line is the limiting sum. With a negative ratio, the sums jump above and below it.

Try this

Make $r$ negative. How do the steps change?
Move $r$ close to $0.8$. Why does it take longer to get close?

Finding limiting sums

Example 1●

Find the limiting sum of $18 + 6 + 2 + \cdots$

$a = 18$, $r = \dfrac13$, which is between $-1$ and $1$.

$S_\infty = \dfrac{18}{1 - \frac13} = \dfrac{18}{\frac23} = 27$

Example 2●●

Find the limiting sum of $8 - 4 + 2 - \cdots$

$a = 8$, $r = -\dfrac12$: $\;S_\infty = \dfrac{8}{1 + \frac12} = \dfrac{16}{3}$.

Example 3: a series involving x●●●

For which $x$ does $1 + (x - 1) + (x - 1)^2 + \cdots$ have a limiting sum? Find it when $x = 1.5$.

$r = x - 1$, so we need $-1 < x - 1 < 1$, that is $0 < x < 2$.

$S_\infty = \dfrac{1}{1 - (x - 1)} = \dfrac{1}{2 - x}$, which is $2$ when $x = 1.5$.

Example 4: working backwards●●

A GP has first term $4$ and limiting sum $12$. Find $r$.

$\dfrac{4}{1 - r} = 12$, so $1 - r = \dfrac13$ and $r = \dfrac23$.

Flashcards

Limiting sum

$S_\infty = \dfrac{a}{1 - r}$

When does it exist?

Only when $-1 < r < 1$.

$ frac12 + frac14 + frac18 + cdots$

$1$

Practice

Find the limiting sum of $18 + 6 + 2 + \cdots$
Find the limiting sum of $8 - 4 + 2 - \cdots$
A GP has first term $4$ and limiting sum $12$. Find $r$.
For which $x$ does $1 + (x - 1) + (x - 1)^2 + \cdots$ have a limiting sum?
Which series has a limiting sum?
Find the limiting sum of $0.9 + 0.09 + 0.009 + \cdots$
Find the limiting sum of $5 + \dfrac52 + \dfrac54 + \cdots$
A GP has ratio $\dfrac14$ and limiting sum $8$. Find its first term.
Find the limiting sum of $1 + (x - 1) + (x - 1)^2 + \cdots$ when $x = 1.5$.