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 our 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 the 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
 Do the Preview Activity for this section (on WeBWorK if required by your teacher).
 Read subsection 8.2.2 in the textbook.
 Do Activity 8.2.2 (on paper).

Watch the following video until 6:20 which explains the sigma notation used to write out a partial sum or a series:

Read the remainder of this subsection up to and including Definition 8.2, which finally precisely defines what a geometric series is.

Watch the following video 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 quick recap video about geometric series:

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

Watch the following video which explains some general tips & tricks about the sigma notation:

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