## 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 .

Enter a value for a :    Enter a value for r :    Enter a value for n :

Sn =    =

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?
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)

Summary

• 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 .

1