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Calculus: Using the Comparison Test

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About this Lesson

  • Type: Video Tutorial
  • Length: 10:10
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 109 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 Direct Comparison Test (2 lessons, $3.96)

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 Direct Comparison Test
Using the Comparison Test Page [1 of 2]
Okay, let's look at some infinite series and see if we can figure out whether they converge or diverge.
The first one I'll take a look at will be . So that looks like a pretty complicated one. The terms certainly themselves go off to zero, so the quickie test is inconclusive. This could converge or could not converge, it depends on how fast those terms are going to zero. What technique could we use? Now, one thing we could do is actually realize that you could use the technique of integration and use the integral test, because that function actually can be integrated. It would require an integration technique that we saw earlier, which was just the technique of partial fractions. So you could break this up into a whole bunch of partial fractions and use the integral test. However, why bother? If I look at this, let me try to see the essence of this infinite series. And, by the way, this is sort of a very sort of touchy, feely, Zen kind of new-age thing that I really urge you to take on, because by taking a look at what's really going on in this infinite series, it allows us to really understand what's happening and then it allows us to often figure out whether the thing should be converging or diverging and so forth. So really take on this sort of mantra to see what is really the essence. Boil down an infinite series to its essence.
All right, so when I do that here, what do I see? Well a plus 1 is insignificant in the sense that the n's are going off to infinity, so this little plus 1 here means nothing to me. And this plus 4 doesn't do it for me, so basically the real powerhouse here is the n, the n, and the n. So really what I look at and say is, "You know what? This sort of captures the sprit of . It's not equal to . These are dramatically different. Just write them out, term by term, and you see, in fact, these are dramatically different. However, the spirit, the essence of it, the soul of this infinite series turns out to be just .
So now , I happen to know converges, because it's an example of a p-series, where the p, in this case it's 3, exceeds 1. And remember p-series , that will converge whenever p > 1 and diverges otherwise. So I know this converges. So if I can now figure out a way of comparing those two things, use the comparison test, then actually I'd be in good shape and I'd be able to compare these. So now how would I want to compare these? What's my fantasy? The fantasy is to say, "Well, what's going on?" This term here converges. If I think this term might converge, then I'd want this term to actually always live under this term. And the thinking, I always use my hands. I really use my hands all the time on this. Let me try to use my hands right now for you live. So here's a known series that's known to land. I think of converging as like landing. This is known to land, okay, fine. If I have this series that I don't know about and I think it might converge, what I would like to have it do is always live below a landing series and that would push the mystery series down, and I would, in fact, land. So my hope is that this term is going to actually be less than this term. So let's see if that really is true or not.
So I'm trying to use the comparison test and I'm hoping, please, let me use the comparison test. So my fantasy is to make this quantity actually larger. I want to dominate this mystery series by a series that I know grows so slow that, in fact, it converges. Well, what do I do? Well, this 1 is downstairs. If I were to remove the 1 and not put it in here, that would make the bottom smaller. Making the bottom of a fraction smaller makes the entire fraction larger. So, in fact, I could make this even larger by keeping this n and replacing the n + 1 by the number n. And then n + 4, I could make this term smaller by not adding the 4. If I make the denominator even smaller, it makes the whole thing even larger, so I'm left with this, and that's just , which is n^3. So, in fact, I see that each term of this series is dominated from above by this term of this series. But this series I know converges. So this converges, because it's a p-series. And so I have this dominating thing and this dominating thing actually squashes down the mystery thing, so therefore the mystery thing converges, and so this thing converges by comparison with the p-series . So there's that one.
Let's try another one together. How about this one, ? Now, I see a square root here, so that's sort of frightening, and I see all this junk here. But now again take on this sort of little mantra thing, what is the essence? What is the real heart of this infinite series?
Well, the minus 2, in fact does very little for me. Even the 3 doesn't do much for me. Basically, all I see is . So, in fact, in my mind, what I see here is . And that, by the way, can be written as , so, once again, I see a p-series, but now the p = . That's a number that's less than 1, and so we saw, using the p-series test, this thing diverges. So this diverges. So that leads to me to make a guess that, in fact, this thing might diverge as well.
If I want to verify that using the comparison test, what I have to hope for is that I can compare these things in the appropriate way, so let's think about it now. Okay, I've got this known series right here. This is the known series, and that is known to diverge. To get this unknown series to diverge as well, one way of doing that is to make sure that it always lives above the diverging one. It's trying to converge, but this diverging one won't let it, because this one's going off to infinity and the other one, they mystery one is always on top of it. So it's taking off to infinity with the known one. So if I can show that this thing is always bigger than this thing, then I'm in good shape. Let's try to do that right now.
And so let's take a look at this and see if I can make this thing even smaller. So if I want to make that smaller, what can I do? One way to make a fraction smaller is to make the denominator larger. So how could I make this larger? Well, here I see a 3n - 2. What if I just looked at 3n? That would make the bottom, well, larger by two units, and so therefore it would make the entire fraction smaller. So I could write this, and that square root I could actually break up, because, in fact, that equals . So that's not exactly this infinite series, but it's close enough, because notice that , by properties of sums, that's just a constant number. I can sort of factor it out of all the sums. So I can just factor that out, and then I see this. And now that's exactly the person that we were looking at earlier, which we know diverges. So this whole thing, if that diverges, if I multiply it by , this will diverge as well. So this diverges, p-series diverges. It's a p-series, where p = , less than one. And so what that means is the original thing, this, must diverge as well, diverges by comparison with . So that diverges as well.
So you can see that when you can get these inequalities to work out just right, if you think something diverges, you get somebody underneath you that also diverges. If you think something converges, you get something over you that also converges, then you're in good shape. That's basically the essence of the comparison test. But notice that our thinking actually was a little different. To get to that point, to get to that stage, we sort of looked for the very soul of the infinite series. And, in fact, really that's all that matters. Once I see this, in some sense I can say that these two things are really sort of the same in spirit. So, in trying to find the spirit, sometimes these inequalities might not be as effective as otherwise. And, in that case, all we want to do is look for the spirit, and then once we have the spirit, we should be able to immediately jump to the answer. In fact, the fantasy would be to see that this is the spirit of this and then just stop right there and say it diverges, without going through all these pesky inequalities that are involved in a comparison like this. And this actually leads to a technique that is known to the limit comparison test, where we just go for the soul, we just go for the spirit of it and then forget about the inequalities. I'll see you at the next lecture.

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