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Calculus: Examples of the Integral Test

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:08
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 87 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Sequences and Series (45 lessons, $69.30)
Calculus: The Integral Test (4 lessons, $7.92)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Sequences and Series
The Integral Test
Examples of the Integral Test Page [1 of 2]
So the integral test is a great device in order to figure out if an infinite series converges or diverges when three things happen. So when is it a good time to use the integral test? When three thing happen: first of all, the terms in the infinite series are all positive, they're shrinking, so they're decreasing, and the last condition is that that function right there - the function here - is a function that can be easily integrated. If it can be easily integrated, the terms are positive and they're decreasing, then the integral test, that's the test for us.
And how does it work? You just look at the integral, which is sort of the continuous analog of a sum. This sums up quantum numbers. This sums up all the numbers. And if that integral is infinite, then we say that the infinite series diverges. This is infinite; that's infinite. And if this converges, then this must converge as well. This is really great test.
Now one point I want to tell you is that, suppose that this infinite series converges to tau. Then that means that this infinite series doest converge to some sum, but that sum is not going to necessarily equal tau. So this is sort of an important point that I just want to call to your attention, that even though that sum converges, I never, ever, ever, ever said that that sum actually is going to equal the same thing as this, because in fact, it might not be.
And you can sort of see that in the picture, right? If the green area - is finite, then certainly the red area is finite, because it all lies underneath. However, you can see these little offsets of green. So in fact, they're not necessarily the same area. So in general, I don't want you to think that when you integrate the function and you get the number like , that that means that this infinite series is . All we can see is that this infinite series does converge. That's all we can say.
Okay, let's take a look at some examples where we can actually maybe see this idea in action. So let's look at an infinite series that we've seen several times, . So that's the series that goes and it keeps going forever and ever and ever and ever. And in fact, this infinite series has a neat name. It's called the harmonic series. And notice a couple of things. First of all, all the terms are positive, and these are really shrinking. Right? As you go further and further out, these numbers are getting smaller and smaller and smaller and smaller.
So if this function is an integrable function, then this is a perfect candidate for the integral test. So the question is, can you integrate ? And the answer is, yes, I sure can. So let's use the integral test and evaluate dx. The function is the function that actually captures these points along the n = 1, n =2, n = 3 and so forth.
So what is that integral? That's , and I evaluate that from 1. And up here, I technically can't plug in infinity, but as I approach infinity. And what happens? Well, as I let this thing get larger and larger and larger, what happens to natural log? The natural log gets larger and larger and larger, so in fact, this diverges to infinity. Right? As this thing goes to infinity and plug that in for x, this thing blows up. So in fact, this is infinite. Well, if this is infinite, that means that this infinite series must be infinite as well by the integral test. If this thing diverges, that diverges. And so we see that the harmonic series actually sums to infinity.
And that's actually pretty neat, because if you remember, one of these important life lessons that we've seen in our past was the quickie test. And the quickie test said that if the terms don't go to zero, then this thing, in fact, must diverge, because if the thing converges, the terms go to zero. If the terms therefore don't go to zero, the thing must diverge.
And so the question was left, what happens if the terms go to zero? Well, here's an example where the terms actually do approach zero, individually, but the whole thing is infinite. This diverges. And so the moral of the story is, to get convergence, you not only need that the terms go to zero, but in some sense, the terms have to go to zero at a really fast rate. These terms go to zero, but they don't go to zero fast enough. Since they don't go to zero fast enough, this thing actually diverges. The area actually accrues. And so it was a nice application of the integral test.
Let's take a look at another one. Let's take a look at . Now again, I see that all these terms are positive and they're shrinking. As n gets bigger and bigger and bigger, these terms are getting smaller and smaller and smaller. And so therefore, this looks like a good candidate for the integral test if I can integrate that function.
So let's see, is that an integrable function? Well, it sure is. So a lot of these infinite series that we've looked at earlier turn out to be integrable functions. So what's the integral of this? Well, this is just x^-2, and so the integral would be - x^-1. And if I evaluate this from 1 and then approach infinity here, what do I see? Well, as I approach infinity - if I let this thing get really, really, really, really big down here, this whole thing goes to zero. So in fact, I see zero when I let this go to infinity, and minus what I get when I plug in - when I put in - 1, which is a , which is -1.
And so in fact, this limit equals 1. Ah, it's finite. And so this improper integral converges. Since this converges, that means that this must converge as well. And so in fact, this must converge and we now can immediately declare, using the integral test, that converges. So that's really, really neat. This infinite series, which we had no idea whether it converges or not, we see it converges by just a really neat trick - really neat test. We just take a look at the integral. We see the integral is finite. That means the infinite series must be finite. So it converges.
Now remember that little moral I talked about earlier. Just because this equals 1 doesn't mean this thing equals 1. In fact, this infinite series, I happen to know, definitely does not equal 1. And like I said earlier, figuring out the values, the actually sum, of an infinite series is really, really a notoriously hard problem in general. There are some examples like, if you have a telescoping series or a geometric series and some other series, where you can actually give the values. But for the most part, it's very difficult to give the values.
This is an example of a function that mathematicians, using a lot of other calculus techniques, have been able to actually find the exact value of. And just for fun, I thought I would tell you what this sum goes to - what this sum actually equals. If you add these all up, it turns out you get . Isn't that wild? Not at all obvious, but just food for thought as you take a look at these integration test methods for finding out whether an infinite series converges or diverges. See you at the next lecture.

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