March 27 - Comparison tests

Ultimately, I want you to be able to look at a given series and quickly identify whether it converges or diverges. There will always be some problems that you have to stop and think about, but most should be pretty clear once you're comfortable with the p-series test ideas, the geometric series ratio test and upcoming - a sort of "hierarchy" of functions.

We talked some today about the theory of how you could prove that since the series Sum(1/n^2) converges (p-series test, p = 2), then the series Sum(1/(n^2 + 1)) also converges. The "+1" in the denominator is unimportant for very large values of n so it should basically behave like Sum(1/n^2). Either comparison test - direct comparison (where we note that term by term, 1 / n^2 is greater than 1 / (n^2 + 1) so the sum of 1 / n^2 will also be greater) or limit comparison (take the limit as n -> ∞ of the ratio of the terms and get a positive (finite) value ) would work.

Further notes from today are below.

March 26 - p-Series Test

We talked about several things today - the highlights were using an (improper) integral to argue that the series 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... diverges and then generalizing to any sum of the form:

Sum(1 / n^p) as n goes from 1 to ∞.

Basically, if p ≤ 1, the series diverges. If p > 1, the series converges.

Notes below.

Mar 23 - Improper Integrals

We started today with a look back at the Fundamental Theorem of Calculus. I noted that the FTOC applies to continuous functions on a closed interval [a, b]. Certain integrals will be called "improper" b/c we'll be "violating" one of those conditions. Either we will look at an integral that is discontinuous (typically an asymptote) or an interval that is not bounded - [a, ∞) or (-∞, b].

Mostly we just worked some examples in class. Having to write lim as b -> ∞ over and over gets a little old but it's necessary. Notes from class are below.

Mar 21 - Introduction to Series

We talked about series today. A series means "an infinite sum, something of the form a + b + c + ..." with the dot, dot, dot implying that the pattern continues forever.

Determining the behavior of a series will be the fundamental problem of the next two weeks. We'll learn a variety of tests to explain that the sum is findable (the series converges) or that the sum is not findable (the series diverges). Today's two main ideas are the geometric series and the behavior of the nth term.

We did geometric series last year. Identify the constant ratio for the series as r and then if | r |< 1, the series converges. Easy.

The nth term should be mostly intuitive. If I say what is the sum of "2 + 2 + 2 + ..." you would recognize that the sum will continue to increase and thus goes to infinity (the series diverges). The nth term test says that if the limit of the terms in the series aren't approaching 0, then the series diverges.

Note that we don't get to say that if the terms are approaching 0, then the series converges. Turns out that's not necessarily true. Stay tuned.

Notes from today are below.

Mar 20 - Sequences

Today (the first day after spring break...) we begin our last new unit. Ultimately, the question we are going to be solving is:

Does the given infinite sum converge?

That's it. Note that in the case of a geometric series, you already know how to answer the question (check the constant ratio) as that was covered in PreCalculus. For the next couple weeks, we'll be studying different arguments for different types of series.

The focus for today is sequences and the main issue is finding the limit of the sequence. In many cases, you already know how to do this b/c it's simply the limit as x-> ∞ that we've already done. The expressions will typically be a little more complicated (involving factorials or combinations of functions for example) and so your thinking will often have to be more nuanced.

The one new tool you'll need is called L'Hospital's Rule. It is used to evaluate limits and applies when direct substitution yields an indeterminate limit of the form 0/0 or ∞/∞.

Notes and examples from today are below.