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### The limiting sum of a geometric series

We have seen that the sum of the first $$n$$ terms of a geometric series with first term $$a$$ and common ratio $$r$$ is

$S_n = \dfrac{a(1 - r^n)}{1 - r}, \quad\text{for } r\neq 1.$

In the case when $$r$$ has magnitude less than 1, the term $$r^n$$ approaches 0 as $$n$$ becomes very large. So, in this case, the sequence of partial sums $$S_1,S_2,S_3,\dots$$ has a limit:

$\lim_{n\to \infty} S_n = \lim_{n\to \infty} \dfrac{a(1 - r^n)}{1 - r} = \dfrac{a}{1 - r}.$

The value of this limit is called the limiting sum of the infinite geometric series. The values of the partial sums $$S_n$$ of the series get as close as we like to the limiting sum, provided $$n$$ is large enough.

The limiting sum is usually referred to as the sum to infinity of the series and denoted by $$S_\infty$$. Thus, for a geometric series with common ratio $$r$$ such that $$|r|<1$$, we have

$S_\infty = \lim_{n\to \infty} S_n = \dfrac{a}{1 - r}.$

#### Example

Find the limiting sum for the geometric series

1. $$1 + \dfrac{1}{3} + \dfrac{1}{9} + \dotsb$$
2. $$8 - 6 + \dfrac{9}{2} - \dotsb$$.

#### Solution

1. Here $$a = 1$$ and $$r = \dfrac{1}{3}$$, so the limiting sum exists and is equal to $S_\infty = \dfrac{1}{1-\dfrac{1}{3}} = \dfrac{3}{2}.$
2. Here $$a = 8$$ and $$r = -\dfrac{3}{4}$$, so the limiting sum exists and is equal to $S_\infty = \dfrac{8}{1+\dfrac{3}{4}} = \dfrac{32}{7}.$

Exercise 12

Explain why the geometric series

$1 + \dfrac{1}{1 + \sqrt{2}} + \bigl(3 - 2\sqrt{2}\bigr) + \dotsb$

has a limiting sum, and find its value.

Exercise 13

By writing the recurring decimal $$0.\overline{12} = 0.121212\dots$$ as

$\dfrac{12}{10^2} + \dfrac{12}{10^4} + \dotsb,$

express $$0.\overline{12}$$ as a rational number in simplest form.

In general, the limiting sum of an infinite series $$a_1 + a_2 + a_3 + \dotsb$$ is the limit, if it exists, of the sequence of partial sums $$S_1, S_2, S_3, \dots$$, where

\begin{align*} S_1 &= a_1 \\ S_2 &= a_1 + a_2 \\ S_3 &= a_1 + a_2 + a_3 \end{align*}

and so on. Infinite series are often written in the form

$\sum_{n=1}^\infty a_n.$

If the series has a limiting sum $$L$$, we say that the infinite series converges to $$L$$. This can be written as

$\sum_{n=1}^\infty a_n = L \qquad\text{or}\qquad a_1 + a_2 + a_3 + \dotsb = L.$

These two expressions mean that the limit of the sequence of partial sums exists and is equal to the real number $$L$$; they can only be used if the infinite series converges.

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