Overview
In Section 8.2, we studied geometric series, in which each of the terms had a special pattern. In the next section, we will extend that idea to series with any kinds of numbers involved. This section has a lot of details involved, so you should focus on understanding the big ideas behind series. In particular, focus on the Basic objectives below to help you organize your studying.
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.

Review: Be able to use summation notation proficiently.

State the definition of an infinite series.

Be familiar with the various notations used for infinite series and recognize that they are all equivalent.

State the definition of the \(n\)th partial sum of a series.

State the connection between the sequence of partial sums and the words “convergent” and “divergent”. Explain how all of these are related to the sum of a series.

Be familiar with the statement of the Divergence Test, and why it is useful.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Be able to find a formula for the \(n\)th partial sum of a series.

Determine if a series converges or diverges by using the sequence of partial sums.

Be able to use the Divergence Test, Limit Test, and Ratio test proficiently in order to determine if a series converges or diverges. (Note: Active Calculus also talks about an Integral Test, which you should ignore for now  we will get back to this once we have studied the topic of Integration, and in particular socalled “Improper Integrals”.)

Choose the most appropriate convergence test for a given series.
To prepare for class
Use these resources to become proficient with the basic objectives (see above) before class.

Read all of Section 8.3 in the textbook, only ignoring the subsection about the Integral Test (we will come back to this after finishing chapter 5).

Watch only the beginning (until 2:15) of the following video (by GVSU Math) which summarizes how we define the total sum of a general series, via the limit of the sequence of partial sums:

Watch the following video (by GVSU Math) which shows how we can sometimes find an explicit formula for the \(n\)th partial sum \(S_n\), on examples of socalled telescoping series:

Watch the remainder (starting at 2:15) of this video (by GVSU Math) which summarizes a few “Convergence Tests” which we can use to determine whether a given series converges or diverges (for now, ignore the “Integral Test”, from 4:23 to 5:37):

Watch the following video (by GVSU Math) which shows how to use the Divergence Test to quickly recognize “obviously” diverging series:

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

Watch the following video (by Khan Academy) which introduces a special class of series, called pseries, for which we know precisely when they converge or diverge (we’ll postpone the proof of this fact until much later:

Do some experimentation with the following interactive applet (by melbapplets)  note that in this applet, they denote all “pseries” by the name “harmonic series”, which is actually only the special case \(p=1\): Plotting sequences (and also the corresponding partial sums)
After class

Watch the following video (by GVSU Math) which shows how to use the Limit Comparison Test:

Watch the following video (by The Organic Chemistry Tutor) which shows several examples on how to use the Direct Comparison Test:

Watch the following video (by GVSU Math) which shows how to use the Ratio Test:

Watch the following video (by Numberphile, with Fields Medallist Charlie Fefferman) about 3 famous series (geometric series with \(a=1\) and \(r=1/2\), harmonic series, and pseries with \(p=2\)) and some of their history and applications:
For much later (when we have studied socalled “Improper Integrals”)

Read the subsection about the Integral Test in Section 8.3 in Active Calculus.

Watch the following video (by GVSU Math) about the Integral Test: