Sum to Infinity

On this page we calculate the sum to infinity of a geometric sequence.

The sum to infinity of a sequence is the limit of the sum of infinitely many terms in the sequence.


It’s possible to compute the sum to infinity for all geometric sequences that have a common ratio -1<r<1 with a very simple formula:


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


A geometric sequence begins 40, 32, 25.6, 20.48 \dots. Calculate the sum to infinity of this sequence.

In this case, a=40, r=\dfrac{32}{40}=0.8. Therefore,


To calculate the sum to infinity of any sequence, the terms of the sequence must converge to zero. Even then, it is not always possible. This is called a necessary but not sufficient condition: not all sequences that converge to zero have a defined sum to infinity Here we show why we can calculate S_{\infty} for all geometric sequences with \abs{r}<1. For other kinds of sequences, different arguments need to be made.

If the terms of any sequence do not converge to zero, then the sum to infinity doesn’t converge to a limit. We can state that the sum to infinity can’t be calculated for any sequence that diverges. The sum of a diverging sequence diverges as more terms are added:

The arithmetic sequence 4, 7, 10, 13, 16, \dots diverges (each new term is larger than its preceding term)

The sum 4 + 7 + 10 + 13 + 16+\dots also diverges.

Sum to Infinity of a Geometric Sequence

A geometric sequence has the form:

    \[a, ar, ar^2, ar^3, \dots, ar^n, \dots \]

When -1<r<1 and r\ne0, then the sequence converges to zero, regardless of the first term (Although a=0 doesn’t generate a very interesting sequence).

The sum formula

We already know how to add a defined number of terms of a geometric sequence:


where a is the first term of the sequence, r is the common ratio and n is the number of terms to be added.

If we multiply top and bottom by -1 (essentially, multiplying by 1 which has no effect to the value of the expression), we have:


This version of the summation formula is easier to use when r<1.

The sum to infinity formula

If \left|r\right|<1 then as n\rightarrow \inftyr^n \rightarrow 0

Take any number that is less than one. For example, 0.7.

On the applet below, increase the value of n to observe that as n increases, the value of r^n decreases towards zero.

Applet link

This applet offers 15 decimal places, however, if you consider


we know that this number is not zero. Very, very small, but not zero.

We say ‘tends to’ when a value that is changing approaches another value, which may be a constant or a function. We use the notation ‘\rightarrow‘ for ‘tends to’.

Therefore if \left|r\right|<1, then as n\rightarrow \infty, r^n \rightarrow 0.

The line above reads ‘if the absolute value of r is less than 1, then as n tends to infinity, r^n tends to zero.’

Another way to write this is:

    \[\text{if }\left|r\right|<1, \quad \lim_{n\rightarrow\infty}r^n=0\]

the line above reads: ‘If \left|r\right|<1, the limit of r^n as n tends to infinity, is zero’.

Explained Example

When adding together an infinite number of terms of a geometric sequence that has \left|r\right|<1, it would seem that there is a limit.

Consider the sequence 20, 10, 5, 2.5 \dots

Here is the sum of the first several terms:




It would appear that as n gets larger, S_n gets closer to 40.

According to the summation formula for a geometric sequence, the sum of the first n terms is:


However, as n approaches infinity, 0.5^n approaches zero.

Therefore, the limit of this sum is


We say



In general, because r^n\rightarrow 0 as n \rightarrow \infty, we replace r^n with zero in the formula:

    \[\text{if }\left|r\right|<1, \quad \sum_{n=1}^{\infty}a(r)^n=\dfrac{a(1-r^{n})}{1-r}=\dfrac{a(1-0)}{(1-r)}=\dfrac{a}{1-r}\]

That is:

    \[\text{if }\left|r\right|<1, \quad \sum_{n=1}^{\infty}a(r)^{n-1}=\dfrac{a}{1-r}\]



Try these problems on the CK12 website on summing infinitely many terms of a geometric sequence.

Top BC Grade 12 Number and Algebra