## Content

### 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

#### Example

Find the limiting sum for the geometric series

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

#### Solution

- Here \(a = 1\) and \(r = \dfrac{1}{3}\), so the limiting sum exists and is equal to \[ S_\infty = \dfrac{1}{1-\dfrac{1}{3}} = \dfrac{3}{2}. \]
- 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

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.