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Calculus: Properties of Convergent Series

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About this Lesson

  • Type: Video Tutorial
  • Length: 7:23
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 79 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: Convergence and Divergence (2 lessons, $2.97)

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
Convergence and Divergence
Properties of Convergent Series Page [1 of 2]
Let's just think about some basic properties about infinite series. So if we have an infinite series, what can we say about combining them and putting them together? So let me just say a couple words about this. I think a lot of this is very, very intuitive. The idea of an infinite series isn't intuitive, but these properties I think are going to be okay.
Suppose I have an infinite series, , and suppose that converges, and it converges to a number, A. So that's the answer. And suppose I also know that I have an infinite series - a different one - n going from 1 to infinity, but let's call them b[n].[ ]And suppose that that actually converges - so it actually sums up to a number and I'll call that number, B.
Well then, in fact, I bet you could figure out a lot of things. For example, what about the new infinite series composed of the sum of those two? Well, it turns out it's a theorem - not surprisingly - that this infinite series will also converge. And can you guess what you think it's going to converge to? That's right, it's going to converge to the answer with the A's plus the answer of the B's. So that's exactly what you might guess.
What if I actually subtract them instead of adding them? If I subtract them rather than add them, I just subtract their answers. So that's actually not so bad at all.
What if I took the infinite series a[n], but I multiply every single term in the series by a constant number, C? So instead of being just a[1], a[2], a[3], I see C(a[1]) + C(a[2]) + C(a[3]), and so forth. Well, then it turns out that you can factor out the C, which is not surprising, and this thing converges to C times the answer we got before, just multiplying through by C. So basically, you can actually factor out a common factor, even when you have infinitely many things in your sum. You certainly knew you can factor out a common factor if you only had finitely many things in the sum. That's the idea of factoring. But it turns out you can also factor out when you have infinitely many things. Just pull out the C in front and then you're done.
Okay, so there's some sort of basic facts that are worth remembering. There's one really important fact, though, that I not only do I want to call to your attention but I really want us to think about for a second, and that's the following idea. Let's suppose this infinite series, a[n], converges. So what that means is that you can actually sum all these infinitely many numbers up and you get an answer. You actually approach a number. You actually get a number.
Well then, what can I say? Well, let's think about those terms individually, just those "sum ands", the things that we're adding up. What could happen to those things? Well, think about it. I have infinitely many numbers here, right? These a[n]'s represent infinite numbers - a[1], a[2], a[3], a[4], a[5], and so forth. They're all numbers. There are infinitely many. And when I add them all together, I actually converge and I actually approach a particular target. What can I say about those a[n]'s? Could those numbers, for example, be getting very, very large? Well, if they're getting very, very large, then as I add them up and add them up, well that sum is going to get bigger and bigger and bigger and bigger and bigger and I can't be heading toward a particular number.
If I had infinitely many terms and somehow that sum actually exists, these terms here - these a[n]'s - must be getting smaller and smaller and smaller and smaller and smaller and smaller. And I can say it this way: . So if this infinite series converges - if when you add up infinitely many - all these infinite numbers, if you add up this infinite collection of numbers and you get a number - you actually hit a target - the only way that can happen is if these numbers are actually getting closer and closer and closer to zero. So each time your actual contribution of every term is getting smaller and smaller and smaller. And so in fact, the terms must be going to zero.
Now this is actually a really important fact, although I think it makes some intuitive sense, that if you have all these things - a[1] + a[2] + a[3] + a[4], yadda yadda yadda - you get all those things and you can add all of them up - all infinitely many of them up - then somehow the contribution of each next one's got to be sort of less and less and less in the limit. And so in fact, you've got to shrink down to zero.
Now, I want to really make sure this absolutely clear what we're saying here. What I'm saying is if - if, if, if - this series converges, then this limit equals zero. There's one classic mistake that calculus students love to make. Everyone makes this mistake and I want to really highlight it right now, so that in fact, hopefully we can get it behind us. We won't make that mistake.
The "if" is here and the "then" is here. The other direction is actually not true. So what does the other direction mean? The other direction would say if the limit of the individual terms equals zero, then the series converges. That is wrong. If the series converges, then the limit of the individual terms have to equal zero. But just because the terms themselves are getting smaller and smaller and smaller, it does not imply, necessarily, that the associated infinite series is going to converge. It might converge or it might not converge. So it's really, really important here. If the series converges, then the terms approach zero individually. However, if the terms are shrinking to zero, we do not know if the associated infinite series converges or if it diverges. This is a really, really, really important point. All I want us to take away from this is that if we have something that converges then the terms must go to zero - fact. Anything else is not a fact.
So if we just know that the terms go to zero - if I hide everything else here and say I just tell you the terms go to zero, what can you conclude? The answer is, nada, nothing, zilch. We don't know anything. A classic mistake would be to say, "Oh, since the terms go to zero, then the infinite series converges." That's just wrong.
So all we know for sure - all we can verify right now together, which we just did - is that if the infinite series converges and we add them all up, then the terms themselves must be getting smaller and smaller and smaller and the limit has to go to zero.
Phew! Important fact - we'll use this a lot. All right, I'll see you at the next lecture.

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