Overview
Many important sequences are in fact obtained through the process of adding the terms of other sequences: The terms of such a new sequence are called partial sums. When we consider the long term behavior of these partial sums, we end up considering what happens if infinitely many terms are added. We call these infinite sums series. In this section we consider a special and very important type of series called geometric series.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
 Be able to recognize that a sum is a geometric sum or series by noticing that consecutive terms have a common ratio \(r\).
 Give the general form or this type of series using the parameters \(a\) and \(r\).
 Give a formula for the partial sum \(S_n\) of a geometric series.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
 State a condition under which a geometric series converges and give a formula for the value of the series when it does converge.
 Use geometric series to solve problems in biology, finance, physics, etc.
To prepare for class

Watch the following video (by Dr. Trefor Bazett) which introduces the idea of a series:

If you have never seen the sigma sum notation before, read this short page (by MathIsFun) before doing anything else, and then play around with their “sigma calculator”.

Do the Preview Activity for this section (on WeBWorK if required by your teacher).

Read subsection 8.2.1 in Active Calculus, and:

Do Activity 8.2.2 (on paper).

Watch the following video (by GVSUmath) which explains how we compute a finite geometric sum using the formula \(\displaystyle S_n = \frac{a(1r^{n})}{1r}\):
After class

Watch the following video (by Dr. Trefor Bazett) which shows how we can find the total sum of a geometric series:

Watch the following quick recap video (by GVSUmath) about geometric series, and note how there are at least two equivalent ways to write a geometric series: either starting the indices at \(n=1\), i.e. \(\sum_{n=1}^\infty ar^{n1}\), or alternatively (and more commonly used) starting at \(n=0\), i.e. \(\sum_{n=0}^\infty ar^n\):

Watch the following video (by GVSUmath) which explains how to find the total sum of a geometric series which is not yet written in standard form:

Watch the following video (by Christine Breiner of MIT OpenCourseWare), and do the exercises in it, which explains some general tips & tricks about the sigma notation:

Watch the following video (by rootmath) which shows more examples of how one can sometimes find an explicit formula for the partial sums of a series:

Do some experimentation with the following interactive applets:
 Seeing Geometric Sequences (by Tom Owsiak)

Watch the following video (by Think Twice) which gives a beautiful visual explanation of the total sum of converging geometric series: