Infinite Geometric Series
A geometric series has first term a and common ratio r. You can calculate the sum of a set number of terms. Below is a tool to help you visualize the sum of a set number of terms.
- Enter values for a, r and the number of terms, n.
- Look at the series displayed in the large box as a sum in decimal form.
- Below the box, you will see the sum of these first n terms, Sn,and the value of .
Infinite Series?A series is infinite if there are infinite terms, but can there be a sum if there are endless terms? When would we stop adding? Consider this:
- Go back up to the entry fields and change your values for a, r and n, but this time be sure to use a value for r that is between 0 and 1!
- Look at the numbers in the series and the sum of the series.
- Now increase the value of n a few times. The value for the sum of those terms should be getting closer to the value of !
- Does adding more terms (increasing the value of n) make any difference? What if we include hundreds, thousands or even infinite terms?
- See the figure below to visualize the siuation. The sizes of the bars represent the numbers in a geometric sequence or series. You can adjust the sliders to set the values of a and r. The value r must be between 0 and 1. As long as r is between 0 and 1, what is happening to consecutive terms? If you add these numbers, will more terms have an effect on the sum?
- Each term of a geometric series can be expressed as ar(n -1).
- The sum of n terms of a geometric series can be expressed as .
- The sum of an infinite geometric series can be calculated as long as r is between zero and one, i.e. the series must be Convergent! For these series, the sum can be expressed as .