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Calculus: Alternating Series

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:37
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 103 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 Alternating Series (3 lessons, $4.95)

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 Alternating Series
Alternating Series Page [1 of 2]
So we've looked at, so far, means and ways, tests, devices, and mechanisms that allow us to look at an infinite series and trying to determine whether that infinite series converges to diverges. Many of those tests, for example, the comparison test and the integral test and so forth, a lot of those required that the infinite series at hand, were all positive. That is to say, they were positively termed series. So every single sum end was a positive number. I now want to take a look and have us think together about what happened if, in fact, the terms aren't all positive. And that, the most basic, or most interesting case, to consider is the case where the signs alternate. We go from positive to negative, to positive to negative, to positive to negative, and so forth; they alternate plus and minus, plus and minus. Not surprisingly, these are referred to as "alternating series" because the signs alternate as you go down the line: plus, minus, plus, minus, plus, minus.
First of all, let me just show you how you can write one of these. Suppose that you have a sequence of numbers a[n], and they're all positive. Then, if they're all positive, you're saying, "Wait a minute, all positive? You're supposed to have them alternating, what's going on?" Well, I'm going to alternate them right now, in the actual infinite series, we can consider the infinite series n . And, if you think about it, what happened?
Well, when n = 1, I put in a 1 here, and I see a 2 power minus 1 to an even, is going to be positive, but the a[1], this = a[1]. Then, I add to that, what happens when n = 2? When n=2, I see (-1)^3, -1^3 is negative 1, so that's going to be -a[2]. And then when n = 3, I have (-1)^4, which is positive, so I have +a[3], and now you see the pattern indeed, just as I advertised, I have these signs alternating. So, this is an alternating series, and that's how an alternating series looks. I've got a whole bunch of positive terms, and in front of it, something that goes plus, minus, plus, minus, plus, minus.
Okay, now, how can we figure out if an alternating series converges or not? Well, here's a really, really neat thing that we can actually think about on our own, and verify it. Let's add one extra condition. Not only will all the a's be positive, let's pretend the a's are decreasing. So, the a's are decreasing. What does that mean again? It means that the a[1] is the biggest, then comes a[2] then comes a[3],^ and these are shrinking in size. They're all getting smaller and smaller and smaller in a decreasing sequence.
Now, let's further assume that, in fact, the limit of the terms themselves, actually goes to 0. So, what I'm saying here is that, in fact, not only are they shrinking, but they're, in fact, going to actually hit the target of 0. They're going to get as small as they possibly can given the fact that they all have to be positive, they get as small as they possibly can, and the limit, the approach is 0. Okay, so these are two hypotheses that I'm putting in here, right? Here's the first one, they're decreasing, and the second one that, in fact, is the limit of the terms themselves is 0.
Given those two hypotheses, what can we say about this alternating series? Well, let's just think about the sequence of partial sums, which means, let's just see what happens as we just take that term, then subtract that, then add that, then subtract that, then add that, and do that forever. What happens to that sequence of numbers? Well, it's actually pretty neat. I'll draw a number line here, and S[1], let's say S[1] is just the very first, that's a[1]. So a[1] is positive, let me put 0 by the way, 0 is like way over here. So, here's a[1], which is the first partial sum, so I'll also write that as S[1]. Now, what happens next? Well, what happens next, is then I actually subtract a[2]. Now, a[2] is a positive number, but it's actually less than a[1]. So where is it going to take me? I'm going to go here and I'm going to back up, because I'm going to subtract it, but I'm not going to go all the way to 0 because, in fact, the number is less than a[1]. So, it's only going to go part of the way, and that's going to be S[2], that's going to be the sum of the first two people. Now I go backwards and I get to here, and that's S[2]. Let me say that again, S[2] is just this thing. So, I go to a[1] and then I back up a[2], and that's going to take me not all the way to 0, because I know that a[2] is smaller than a[1], so I'm going to still be positive. So, that's S[2].
Now, what about S[3]? Well, now I take that answer and I add a[3]. So, I'm going to go now forward a little teeny bit, but I'm going to go forward a little teeny bit that's actually smaller than the amount I just went. So, that's going to push me back somewhere in here. So, I go a smaller distance, so now I go this way, and this is going to be S[3]. And then, what about S[4]? Well, now I'm going to subtract a little bit, but the way I subtract is much less than I just did. So, I start to wiggle back and forth, this is S[4]. And it's like, literally, a bouncing ping-pong ball, right? So, it starts here at a[1], then it goes to S[1], S[2], S[3], S[4], S[5], and starts to bounce around. It starts bouncing around and it's going so fast. So, if you keep doing this again and again and again, we're honing in on something. And, since every little offset that we go, every time we stick on an extra term, we know those terms are approaching 0, that means we're always adding less and less and less and less. So, therefore, it seems absolutely reasonable, that if I keep doing this I'm going to actually head toward a point. I'm sort of vibrating right around the point, and that point is the limit S. So, in fact, if we have an alternating series, and the terms themselves, without the alternating part are all positive, they're decreasing and the limit of the terms is 0, then, in fact, the associated infinite series will converge.
So, converges. So, if you have an alternating series that allows us to alternative back and forth, and then we see that, in fact, the partial sums are actually becomes a sort of vibrate right here, during the limit it actually exists and equals this number, that target and that target equals this infinite series.
So, this is the alternating series test. The alternating series test says, "If you have an alternating series and the terms without the negative signs are all positive, decreasing, and the limit of the terms themselves is 0, then this series will converge." A quick example, , n going from 1 to infinite. This is called the "alternating harmonic series," because we go from 1, then we go -1/2, + 1/3, -1/4, and so on. So, again, we see it alternate.
We're seeing it's an alternating series, so we can use the alternating series tests. The alternating series tests just says, "Make sure that everybody is positive after you strip away the stuff," and, so, we're looking a , that's just the plus or minus part that is all positive. Plainly the sequence 1 over n is decreasing, so it's positive, it's decreasing. And what's the limit as we went n go to infinite of that just that piece? It equals 0, so since that equals 0, the sequences, the terms themselves, the sequence is decreasing and they're all positive. Therefore, by the alternating series tests, this must converge.
This is a marked difference between just the good old-fashioned non-alternating thing where, in fact, we have , that's the harmonic series which diverges. So, the fact that every other time I get to subtract a number rather than continually add numbers, means that in this example, the series actually all of a sudden can be summed, and it's an alternating series.
So, alternating series, what do you do? You make sure that without the alternating part, what's left is positive, decreasing, and the limit of the terms is 0. If all those three conditions are met, the alternating series test says, "The thing converges." Neat.
See you at the next lecture.

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