Day 32 in Mat229

• Median: 5.5
• #2: it's important to note that you don't need to determine the value of the integral in order to decide if it's convergent or not!

  And there are different means of showing convergence.

  But one thing that you can't do is forget those rules of integration we learned in the context of proper integrals!:)

  1. All dying exponentials are convergent on \([a,\infty)\), where \(a\) is a real number. All of them can be written as \(e^{-\alpha x}\), with \(\alpha \gt 0\). Consider the interval \([0,\infty)\): \[ \int_{0}^\infty e^{-\alpha x}dx = \frac{1}{\alpha}\int_{0}^\infty e^{-u}du \]
  2. This one gives us a chance to show off our L'Hôpital chops....

     But graphing the area concerned provides a connection to the integrals of part a. above, and makes the question of convergence easy (by symmetry).

  3. This one you can bound with an integral of a power of \(x\).

  4. Most of you got to thinking about \(\infty\) and forgot about the other endpoint: the integrand is infinite at \(x=1\): \[ \int_{1}^\infty \frac{d x}{x^2-x} \] So we split it into two: \[ \int_{1}^\infty \frac{d x}{x^2-x} = \int_{1}^2 \frac{d x}{x^2-x} + \int_{2}^\infty \frac{d x}{x^2-x} \] And while this integrand is convergent for the interval \([2,\infty)\), it is divergent on \([1,2]\).

     Another comparison shows that this is so.

• Improper integrals: 347-350, 353-354, 355-363 (odd), 372-373, 394-395.
• Sequences: 2, 3, 5, 8, 13, 21 (why did I choose those numbers?:); 23-26, 27-34, 46-49, 59, 60
• Series: 67-69, 71-73, 79-80, 89-91, 93-94, 105, 116

Once again you'll be allowed a cheat sheet (one page, front and back).

In the previous section, we determined the convergence or divergence of several series by explicitly calculating the limit of the sequence of partial sums \({S_k}\). In practice, explicitly calculating this limit can be difficult or impossible. Luckily, several tests exist that allow us to determine convergence or divergence for many types of series. In this section, we discuss two of these tests: the divergence test and the integral test.

The idea of the divergence test is quite simple: if the terms of a series don't converge to 0 as they traipse off to infinity, then the series must diverge.

Before we get to that, however, I think that we left off as we discussed geometric series, so let's go back to page 4, and pick up from there.

So you're kind of confronting two somewhat unfamiliar animals simultaneously. What ties them together is this idea of running off to infinity, and trying to figure out how to successfully add up an infinite amount of things.

Great thought experiments, but perhaps intellectually troubling. That's perhaps why calculus is difficult: there are lots of intellectually troubling things going on. You don't want to turn it into abstract nonsense, however: if you have any questions, please pop by for an office visit!

1. Integral Test (.nb)
2. Integral Test (.pdf)