What is a geometric series?

A geometric series is a series of a geometric sequence. Given a geometric sequence whose th term is , we can define its geometric series (which is the sum of the first terms of the sequence) as:

According to the above formula, we need to know the first term , the last term as well as the ratio . We can see that this is not defined if the ratio is 1.

Another equation for the series is:

In this case we need to know the first term , the number of terms the ratio .

Example of a geometric series

Consider the following sequence:

Let us find the sum of the first 15 terms. Of course we could determine all 15 terms and then sum them up. Or we can use the above formula. We already know the first term and we only need to find the ratio to be able to use the formula. We find as the ratio of any two successive terms: . We obtain the series as:

Now we can easily find the sum of the first 15 terms as .