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

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


Find the limiting sum for the geometric series

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


  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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