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 with a very simple formula:

where is the first term and is the common ratio.

### Example

A geometric sequence begins . Calculate the sum to infinity of this sequence.

In this case, , . 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 infinityhttps://www.traditionrolex.com/33. Here we show why we can calculate for all geometric sequences with . 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 diverges (each new term is larger than its preceding term)

The sum also diverges.

# Sum to Infinity of a Geometric Sequence

A geometric sequence has the form:

When and , then the sequence converges to zero, regardless of the first term (Although 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 is the first term of the sequence, is the common ratio and is the number of terms to be added.

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

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

**The sum to infinity formula**

If then as ,

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

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

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 ‘‘ for ‘tends to’.

Therefore if , then as , .

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

Another way to write this is:

the line above reads: ‘If , the limit of as tends to infinity, is zero’.

### Explained Example

When adding together an infinite number of terms of a geometric sequence that has , it would seem that there is a limit.

Consider the sequence

Here is the sum of the first several terms:

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

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

However, as approaches infinity, approaches zero.

Therefore, the limit of this sum is

We say

## Conclusion

In general, because as , we replace with zero in the formula:

That is:

## Practice

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