Month: December 2023 (Page 2 of 3)

Class Info

• Date: Wed Dec 6
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• “Series – Comparison Tests” – due Wed Dec 13

Reminder – schedule for the rest of the semester:

• Exam #3 will consist of a take-home component which will be distributed in class Monday (Dec 11) and will due the following Monday (Dec 18)
• We do not meet on Wed Dec 13 (Tues Dec 12 and Wed Dec 13 are reading days, so no classes will meet)
• There will also be an in-class component for Exam #3 which we will take that Monday Dec 18 (the take-home exercises will serve as preparation for the in-class exam); we will also review for the final exam that day
• The final exam is in-class on Wednesday Dec 20

Topics

We did some more examples of the limit-comparison test, illustrating the strategy of looking at the degrees of the numerator and denominator in order to identify a p-series to compare the given series to (in the case when the given series is a rational function of n, i.e., a ratio of two polynomials); use this strategy for the “Series – Comparison Tests” WebWork exercises.

We also introduced the Ratio Test–this is a way of identifying series that are “almost geometric” as n gets large, and using that to classify them as convergent or divergent:

The conclusion for the example above is that the given series converges, since it’s value of “rho” = 1/2, which is less than 1 (see my “Convergence/Divergence Tests” pdf for this example typed up there).

The Ratio Test is covered at the start of Sec 5.6:

We will use the ratio test to study “power series” which will be the last topic we cover, on Monday (see Sec 6.1).

Class Info

• Date: Mon Dec 4
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• “Series – Divergence Test” – due Wed Dec 6
• “Series – Comparison Tests” – due Wed Dec 13

Schedule for the rest of the semester:

I outlined the schedule for the rest of the semester:

• We have class tomorrow (Wed Dec 6) and Monday Dec 11, where we will cover additional topics on infinite series–please work on the WebWork sets on this material!
• Exam #3 will consist of a take-home component which will be distributed in class next Monday (Dec 11) and will due the following Monday (Dec 18)
• We do not meet on Wed Dec 13 (Tues Dec 12 and Wed Dec 13 are reading days, so no classes will meet)
• There will also be an in-class component which we will take that Monday Dec 18 (the take-home exericses will serve as preparation for the in-class exam); we will also review for the final exam that day
• The final exam is in-class on Wednesday Dec 20
• See the 1st board snapshot below for a day-by-day outline of the schedule

Topics

See the schedule on the left-hand side:

We briefly discussed the integral test, as a way of classifying “p-series”:

The integral test and p-series test are covered in Sec 5.3.

We then introduced the limit-comparison test, which is used to “compare” a given infinite series to one we know converges or diverges; the test consists of looking at the limit as n goes to infinity of the ratio a_n / b_n.

In this first example, we compared the given series (Sigma 1/(n^2+1) to the p-series with p=2, since they are very similar, and we know the latter converges:

Here is the presentation in the textbook (from Sec 5.4), which starts with a similar example:

We then outlined two examples from the textbook:

We will do a couple more examples of the limit-comparison test tomorrow from the WebWork, then go on to discuss alternating series.