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Calculus: The Summation of Infinite Series

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About this Lesson

  • Type: Video Tutorial
  • Length: 11:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 121 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: Infinite Series (4 lessons, $6.93)

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
Infinite Series
Summation of Infinite Series Page [1 of 2]
So let's think about what an infinite series is and what does it mean to add up infinitely many things - something that seems a little bit unwieldy upon first inspection. So what I want to do is I want to take a collection of numbers, and I'll just denote them in this fashion. a[1,] that just signifies the very first number. a[2] is the second number and so forth. So this is just the notation to allow me to tell you which number I'm thinking about in my summation. And I can write that in a shorthand way, using the sigma notion writing it as . So this is a shorthand way of just writing out this infinitely long process. And the question is, how do we make sense out of the dot, dot, dot? That is to say, how do we make sense out of adding up things forever without stopping?
Well, the way we realize what this thing means is to adopt the strategy that we've used a lot together, and that is, when faced with a hard problem, don't do it. Instead, do an easier problem and try to inch up to it. And that's exactly a technique that's going to allow us to uncover what this could mean.
So what's the easier problem? Well, an easier problem is just to add two things together. So let's do that. Seems easy to me. So in fact, let me actually call that the fist sum. So I'll call that s[1]. So s[1 ]will just the sum of the first two numbers. That's no problem. We know how to add two numbers. We did that a long time ago. What now should I let s[2] be? That will be the second sum. So that will be just taking the first three numbers and adding them up. Again, not a problem. Just add three numbers up, and you'll get an answer. What would s[3] be? I'll take the next sum, so I have a[1] + a[2] + a[3] + a[4. ]So I take the first four numbers and add them all up. So in this case, I do one sum, in this case, I do two sums, in s[3], I do three sums, and so on. Each of these, by the way, we can absolutely do.
In fact, let me try to write down a general term. So again, trying to use the notation and the ideas that we've been developing here. What would s[m] be? Well that would be the m^th sum, which means that I basically will do m additions. So I'd see a[1] + a[2] + a[3], and if you look at the pattern here, on the first sum, I end with an a[2]. On the second sum, I end with an a[3]. On the third sum, I end with an a[4]. So on the m^th sum, I should add with one more than m. So that would be a[m] + 1.
And that actually can be computed too, because there's only finitely many numbers there. And even though m might be pretty big, like a thousand, I can still add up a thousand numbers. That's not the problem. The big problem is when there's dot, dot, dots and nothing else. You just go off to the horizon.
So in fact, we can actually compute all these things. Well, let's actually look at the s's now. That's actually a sequence of numbers. That's now a sequence of numbers,{s[1], s[2], s[3]...s[m]}, and it keeps going. And that sequence of numbers I call the sequence of partial sums, because it's sort of part of the sum. And then as I go down, I bite off bigger and bigger parts of the sum. So this is called the sequence - now the sum, the infinite series - but just looking at those numbers, s[m], that's called the sequence of partial sums. So those are just a whole bunch of numbers.
And now what I'm going to say is, well, are those numbers heading toward a particular target? So I'm going to ask what is the limit of this sequence? Are we going to head towards something? If the limit of the sequence of partial sums exists, then I say that this infinite series actually can be summed up, and that sum will equal whatever limit we hit here. In that case, I say this infinite series converges. So an infinite series converges if I can actually add up all the terms.
And now, much more rigorously, let me say it this way. If the limit as m goes to infinity of this sequence, s[m] - the sequence of partial sums, the sums I get by doing this - if that limit exists and equals s, then I say that the associated infinite series - adding up all the terms of a's - that converges, and in that case, it actually sums up to the target s that we found.
If this limit in fact does not exist - if this limit is not a number, doesn't exist - then I say that this infinite series diverges, cannot be summed up. So really the question of convergence of divergence of an infinite series, which is just a way of saying, can the numbers be added up or not, really revolves around the idea of inching up to the infinite series by just adding one term at a time, one term at a time, one term at a time, and ask, as you add one term at a time, are the successions of partial sums heading toward a particular target, or are you drifting around in some sort of amorphic form. If you're heading toward a particular target as you go and drift down and add and add more terms, then in fact, this infinite series can be summed up. Or we say that infinite series converges, and it converges to that target that we are heading toward. And otherwise, we say the series diverges.
So let's look at an example. So let's go back to the example that we're looking at, 1 - 1 + 1 - 1 + 1 and so forth. How could I write that in shorthand, by the way? In shorthand, I can write that . Let's just check that. When n = 1, this is -1^2, which is 1. Then I add to that when n = 2. When n = 2, this is -1^3, which is minus 1. So this toggles between -1 and 1 in succession, and I add them up.
Let's look at what the sequence of partial sums is. So what's s[1]? Well s[1] is the sum of the first two people, so that's zero. What's s[2]? s[2], I add the first three people. That's going to be 1 - 1 + 1. That equals 1. What's s[3]? s[3 ]is the sum of the first four people. So what's 1 - 1 + 1 - 1? That's zero. What's s[4]? That's the sum of the first five people - 1 - 1 + 1 - 1 + 1, and that's it, so the first one, two, three, four, five. So what's that? Well, that's going to be just 1. And you can see the pattern that's forming. We have zero and then 1 and then zero and 1 and zero and 1, and that continues forever. Every even subscript will have a value of 1. Every odd subscript will have a value of zero. So this sequence is the sequence 0, 1, 0, 1, 0, 1, 0, 1, forever. Now is that sequence honing in on one particular value? The answer is, no, I'm not honing in on something. I'm bouncing like a bouncing ball - like a ping-pong ball - between 1 and 0 and 1 and 0, sort of like a metronome just ticking back and forth. But for a limit of a sequence to exist, I've got to be sort of honing in on a particular value. But I'm bouncing back and forth and so therefore, this infinite series does not exist, because the limit of the sequence of the partial sums does not exist. Instead of honing in on a number, I'm bouncing like a metronome between zero and 1. And so that means, therefore, this infinite series diverges. It cannot be added up. It does not add up to a number.
So that finally resolved that issue, which of course, everyone was thrilled by. Mathematicians rejoiced. But one question remains. If I actually want to figure out if an infinite series actually converges or can be summed or not, I've got to look at the limit of this particular sequence, this sequence of partial sums.
So this raises the question, how do you actually find the sequence of partial sums? And how do you then find the limit? The answer is, in general, in almost all cases, you can't actually do that. You can't actually find the sequence of partial sums. It's just too complicated. Look, it's a formula for what you get if you keep adding these things together, together and together. So it's a very, very hard formula to actually figure out. So in most cases, we can't do that, which means, sadly, in most cases, we can't necessarily figure out what a convergent infinite series converges to.
Now this is really awful news, because you're saying, "Gee, if we're going to add up infinitely many things like this, we're going to certainly want to know what we add up to." It turns out that our knowledge is just so minimal that we have to be satisfied with something less. In particular, we have to satisfied with just the question of whether in fact there is an answer or not, whether this does sum to a number or whether it diverges. So most of the things that we'll be doing together will actually involve just the question, does this thing actually sum to a number or not? And if it sums to a number, most of the time, we won't be able to say what the number is exactly that it's summing to. But we will be able to say it hit the number.
Anyway, there are special cases, in fact, where you can actually find he s[m]'s - for example, in the devil's series issue here. And in those cases, then you can actually take the limit and find out that actual sum exactly. Well, we'll think more about these infinite series issues as we get very serious with this topic. I'll see you at the next lecture.

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