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Calculus: The Squeeze and Absolute Value Theorems

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About this Lesson

  • Type: Video Tutorial
  • Length: 7:55
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 85 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: Sequences (4 lessons, $11.88)

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
Sequences
The Squeeze and Absolute Value Theorems Page [1 of 2]
Let's try to see what the limit of this particular sequence is. I'm thinking of the sequence made up of . Now what in the world would the limit of that be? . So what that means is, I first let n be 1, then 2, then 3, then 4, and then 5 and I see where this approaches.
Well, if you let n approach infinity, this looks like - well, what's this thing approaching? Well, that thing's actually wiggling. It's wiggling all over the place - wiggle, wiggle, wiggle. It'll go as high possibly as 1. Actually, it will never equal 1, but it could go as high as near 1, and it could go as low as near -1. And it just bounces around. So who knows what that's doing, and then this n, we know what that's doing.
So what in the world is this limit? Well, the problem is this thing is just too complicated on top to actually figure out what's happening with all that bouncing and bouncing and bouncing. However, since we know that sine of anything will always be between -1 and 1, then I could notice the following, that the n^th term in this sequence - the - will always be less than or equal to - well, putting a 1 here - . Because the largest, the very largest, sine of anything can be, is 1. Remember, sine just wiggles up and down between -1 and 1. So the very biggest this could possibly be is , and the smallest it could possibly be - how low does the sine curve go? It goes as low as -1, and that's as low as it goes. So in fact, this will always be bigger than or equal to . That's just a fact.
So here I have two sequences that flank the sequence that we're after. You see that? So we have this sequence here. We don't know what's happening to the sine. It's bouncing all around. But I know that it's always going to be less than or equal to , and always greater than or equal to .
Now these two individual sequences actually are not that hard to understand. In fact, what happens as n goes to infinity of this thing? Well, this whole thing would therefore go to zero. And similarly, as n goes to infinity, this sequence would also go to zero.
So if you think of this pictorially for a second, if I just sort of plot the values of the a's as you go through. So here's a line, and here's 1, 2, 3, 4, 5 and so forth. And I plot these points. I see this point here, and then I see this point here, and this point here, and this point here, and this point here. And if I plot these, I get the negative values - this point here and this point here and this point here and this point here and so forth. And what do I know about this sequence? All I know is that at 1, it's somewhere in between these two points, so it's somewhere in here. And at 2, it's somewhere in between here. And at 3, it's somewhere in between here. And at 4, it's somewhere in between here. And at 5, it's somewhere in between here, and so forth. So what's happening?
Well, what's happening is I `m forcing this sequence that I'm after to sit in between these two sequences, both of which converge to zero. The limits of both of these things - this approaches zero as n goes to infinity, this approaches zero as n goes to infinity. So what are the possibilities for this? This has to also go to zero, because it's this mystery series, and the top is squishing down to zero, and the bottom is squishing up to zero, and this always has to fit right in between. And so it's being squashed in place, and so therefore, this mystery series must also - this mystery sequence has to go to zero.
And in fact this has a name. This is called the squeeze theorem. The squeeze theorem basically says that if you're trying to figure out the limit of a particular sequence and you're not sure what it is, but you can find a sequence that's definitely bigger than it - so there's the mystery sequence, and there's one that's bigger than it - and that bigger one converges somewhere. And then you have one that's always smaller than the mystery sequence, and that also converges to the same place as the bigger one converges to. So you have these two sequences that come together like this. And your particular mystery sequence actually lives right in between. And since these guys come together, that thing is squashed in right between them like a bug, and therefore must, in fact, have a limit, and the limit must be that common limit.
So if you can bound your sequence from above and bound your sequence from below and the limit of this thing is the same as the limit of this thing, then in fact, the limit of anything in between must exist and must be that same limit. And that's called the squeeze theorem. You're squeezing in a mystery sequence in between two sequences that you actually know.
So in fact, this is a great technique for someone that's analyzing things. For example, this question, which first seemed pretty hard, we didn't know what sine of n is doing. It turns out it doesn't make a difference, since the most it could be is 1 and the smallest it can be is -1 and and both go to zero, this thing has to go to zero. So the moral is, this thing has to go to zero, and that's because of the squeeze theorem. You're squeezing things in.
This is an important idea. We'll do this a lot. We'll be squeezing things all over the place, so this is sort of a fun fact. And in fact, if we think about this, we can actually use the squeeze idea to report something else. What if I take a sequence and I tell you that if I take the absolute value of a particular sequence, so strip away any negative signs you may see, that that limit in fact goes to zero. The only way that can happen is if the terms themselves go to zero. So if you have this, then you can conclude this.
So what that means is that if you have a sequence but it might have negative signs and positive signs and you maybe don't even know where the negative signs even are. But when you take absolute values - make all the terms positive, strip off any negative signs - if all those times - just with the positive signs - if those terms head to zero, then no matter how you put the negative signs in, this thing has to go to zero too. There's no choice. And it's a sandwichy thing kind of thing, right? Because you have this mystery sequence - there's the mystery sequence. Right? Here's the a[n]'s - positive, negative, who knows? But when you take absolute values, that will always be bigger than or equal to the mystery sequence, because all you're doing is making some negative things positive. So boom.
So you've got this very nice absolute value thing that's coming in for a landing at zero, and then actually negative absolute value, so you put a negative sign in front of that. That's going to come up from below with negatives, and that will also approach zero. So you're going to sandwich this mystery thing between the absolute value of a[n] and negative the absolute value of a[n], and they come together and approach zero. And if they're going to approach zero, and this thing is wiggling somehow in between, it's got to actually be squashed right into place and squeezed and therefore, that equals zero too.
This is a really important fact that we'll use an awful lot, that if, in fact, the limit of the absolute value of some numbers actually goes to zero, then those numbers themselves have to shrink to zero, no matter how you put plus or minus signs in. Anyway, there's a nice little application of the squeeze theorem and we'll see the squeeze theorem as an important tool in understanding sequences and see them in other stuff. I'll see you at the next lecture.

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