Author: Suman Ganguli (Page 3 of 12)

Class Info

• Date: Wed Nov 29
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• “Series – Infinite Series” – due Mon Dec 4
• “Series – Divergence Test” – due Wed Dec 6
• additional WebWorks on infinite series will be assigned as we cover the topics in class, so please keep up with the WebWork!

Topics

We summarized geometric series, and went through a couple more examples from “Series – Infinite Series” that involve a “sum rule” for infinite series:

We then introduced the Divergence Test (or “n-th term test”)–if the individual terms in the series do not go to zero, then clearly the series diverges! This is covered in Sec 5.3.

We applied it to examples of infinite series where the n-th term is a ratio of two polynomials of equal degree:

Finally, we discussed the harmonic series–this is the classic example of an infinite series where the individual terms do go to zero, but the series diverges!

We sketched one proof of divergence last week, with the integral test (which is covered in Sec 5.3, after the Divergence Test).

Another proof of divergence is by “grouping” terms in the series together, as shown in the textbook (Sec 5.2):

We looked briefly at the wikipedia page for the harmonic series, which includes the two different proofs of its divergence.

Class Info

• Date: Mon Nov 20 & Mon Nov 27
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• “Series – Infinite Series” – due Fri Dec 1
• additional WebWorks on infinite series will be assigned as we cover the topics in class, so please keep up with the WebWork!

Topics

On Monday Nov 20, we went over Exam #2, and then introduced the last topic we will cover this semester–infinite series, i.e., infinite sums. We introduced “Sigma” notation, partial sums, and the idea of convergence/divergence (covered in Sec 5.2 of the textbook):

We made a connection to improper integrals–this is the “integral test” for convergence/divergence:

We went through some of the exercises from the WebWork, introducing partial sums:

On Mon Nov 27, we continued with these topics:

We talked about how convergence of an infinite series is defined in terms of the partial sums:

We then discussed geometric series:

Here is the passage from the textbook that defines geometric series, with an example:

Class Info

• Date: Mon Nov 13
• Meeting Info: 10a-11:40a, N719

Announcements

Exam #2:

• Our 2nd midterm exam will be tomorrow, Wednesday Nov 15
• See this post for a list of topics and review exercises for the exam
• Please work through on those review exercises, which I will collect later in the semester as a HW assignment

Topics

We reviewed the topics that will be on the exam, in particular partial fractions and volumes of revolution. We reviewed Quiz #3, and set up a couple of the suggested exercises from the Final Exam Review sheet:

We also revisited the topic of improper integrals, although this will not be on tomorrow’s exam:

We closed with an application of improper integrals to “an infinite” volume of revolution–“Gabriel’s horn“: