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About this Lesson
 Type: Video Tutorial
 Length: 11:28
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 124 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 UTAustin 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, padic 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.
About this Author
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Sequences and Series
Infinite Series
Introduction to Infinite Series Page [1 of 3]
So we've seen a lot of things to deal with the idea of a sequence. So a sequence is just a collection of numbers in order, going down and marching through like little soldiers and that's fine. So we've seen plenty of those. In fact, plenty o' beans. So much for infinite sequences. You might be wondering, though, why are those things even interesting or useful or important to us? Well, they're useful and interesting and important to us, because what I want to think about now is return back to days of yore, when we were just little kids. And remember you'd go to school and you'd learn about addition, and this was great. And I don't know about you, but with me, you can the boy out of the country, but you can't take the country out of the boy. And when I think back to addition, I think back to what I used to think of when there would be addition. You would take 1 + 1 and that would equal 2. Right? And so forth.
Now in fact, addition actually has a very, very fancy name. It's actually a specific example of an idea called a binary operation. That means "bi", two  you need two things, and if you take two things, you can combine them, with this operation, and get something else, like 5 + 3, and you can get 8. Not a big deal.
Now once you have the idea of 1 + 1 = 2, you can then start to generalize, and that's always sort of fun. "How about this? 1 + 1 + 1 = 3. Right. So that's pretty good." And then after a while, you get to the point where you say to yourself, "All right, let's go for the gold. What is 1 + 1 + 1 + 1 + 1," but keep going forever. Now this actually becomes a big problem, and let me show you why. You see, on the one hand, you might say, well if 1 + 1 + 1... that just goes on forever  that's infinite. Don't even worry about that. Let's just think about the idea of this being a binary operation  an operation that works on two people.
How do you go from the two people to the three people? Well, that's actually pretty much okay. What you do is, you first just add these two people up, and then you take the answer and then you combine this. So you can actually add as many numbers as you want as long as there's only finitely many of them. But what happens if you have infinitely many terms? Then it turns out the definition of "plus" is sort of a problem. If your plus is for finite many things, and you're trying to shove in infinite many things. It's like pickled pig's feet in a glass jar. You just can't fit them all in. It's all sort of crumbly and icky.
So the question is, what does it mean to add something up infinitely often? And let me actually show you that this is genuinely an issue, not just with this stupid example, but with actually real, real annoying examples. And in fact, the idea of adding things up infinitely often  which by the way, has now come to be known as infinite series. So in fact, an infinite series is just a collection of terms that are all being added together and I have infinitely many of them. This turned out to be a huge thorn in the sides of mathematicians and scientists for years and years and years, just until recently, in fact, just up until a couple hundred years. People really figured out what was going on. In the mathematical timeframe, that's absolutely nothing. That's like yesterday.
So let's consider the following example. 1 + 1 + 1 + 1 + 1 and so on, forever and ever and ever  what does that thing equal? Well let's take a look at it and see. On the one hand I see a 1, and then if I add 1, I get zero, so that's zero. And then if I add 1 again, I have 1. I made a mistake. Where is the whiteout when you need it, because you make a mistake and it's so embarrassing. Here you are, you're trying to do a good job. People are relying on you.
So we have 1 + 1, that's now zero, and then plus 1 is now 1, and then minus 1 is now zero, and then plus 1 is zero and so forth. So what does this equal? Well, let's just call it a big question mark. Now here's how the debate can go. You could say, well look, 1 + 1 is zero. 1 + 1 is zero. 1 + 1 is zero, and so forth, and if you add up a whole bunch of zeros, forever and ever and ever, that equals zero.
On the other hand, someone can come along and say, "Okay now, hold on there, big fella'" Let's take that 1 right here and keep that way out in front, and now look at what's left. Here I see a 1 + 1. That's zero. Then I see a 1 + 1. That's zero. Then I see a 1 + 1 forever, and so when I add them up now, I get zero, zero, zero plus 1. It equals 1. So therefore, zero equals 1. Well this does not make mathematicians happy at all. Well, they hate that. When 0 = 1, this is not good. So what's the problem?
Well the problem is we need to make sense out of what these dot, dot, dots mean. And we need to make sense what it means to add up infinitely many things, and that's not clearly defined right now. It's not clearly defined, because of this problem. Is this thing equal to zero? Is this thing equal to 1? Is this thing equal to something else? Who knows? Well, this really is the inspiration for a lot of people, including mathematicians, to go off and try to figure out how do you actually add things up infinitely often. And that's the real thrust of the subject of infinite series.
Let me try to introduce just some notation. There are no ideas now. It's just a matter of literally trying to see how we can write these things out in a manageable way. If you're going to write down infinitely many things, you want to be able to sort of hold onto them.
So suppose that I wanted to  this is just notation  add up a[1] + a[2]+ a[3] + a[4] and so forth. Well instead of writing it that way, and the dot, dot, dot's means it goes on forever, there's a shorthand notation that we adopt, and let me show you that right now. We use a capital Greek letter sigma (), and sigma stands for "summation". So it's actually just a sign that means summation. So I write a sigma, and then underneath it, I put down where n is going to begin. In this case, the little subscript will begin at 1, and it's going to go forever. So I'll put on top here, the ending point, which is, in this case, infinity. And then I write a[n].
So this is a very complicated and new looking notation or new symbol, and let me tell you what it means. This just means I'm going to sum up all the a[n]'s where the n goes from 1 up to infinity. So if I write that out, what that means is, first I let n = 1, and I put in n = 1 here, and so I get a[1]. Then I add to it what I get when I increase n by 1, so now I have a[2. ]And then I add to it what I get when I put in the 3, so a[3], and so forth.
So that's exactly what I have up here. This is just the notation, which means this. Let me just do a little example here just to make sure that notation is absolutely clear in our minds. Suppose I put . What would you do here? Well that sigma means it's going to be a sum. So I start off with n = 1, and then I keep incrementing until I get, in this case, up to 3. This is not an infinite sum now. This is going to be a finite sum. So when n = 1, I put in a . Then I add to it what I get when n = 2. So that would be . And then I add to it what I get when n = 3. That's . And then if I raise n equals to 4, that actually exceeds this, so I stop. Boom. With an infinite sum, you never stop, you just keep going. So in this case, you can add this up and you can see + + , and you can add that up and see what number that is. So that's what this notation means. And if I put the infinity symbol here, that means this goes on forever, and this is genuinely an infinite series. This is called an infinite series.
Okay, let's look at some examples of infinite series, just to get you familiar with the notation. , what does that look like? Well, let's write out the first two terms of that. When n = 1, I'm left with , and then I add the summation . I increase this index to 2 and look at . And then plus and so on and it goes on forever. So that's what this means. It just means adding up all those reciprocal numbers.
Remember, when you read this, by the way, I don't even say "sigma", I'd say "summation"  summation as n goes from 1 to infinity of . So if you see the symbol in a child's story you're reading to your nephew or niece or something and you come to this, the way you would read this out loud is, "The summation as n goes from 1 to infinity of ," and that's what it means.
For example, what about this? Summation  I'm going to use a different letter here  t goes from 1 to infinity of . Well that equals + + and so on forever, and that's what that means. And if you have summation, let's say k  it doesn't make a difference what letter you use here. It's just a placeholder. k goes from 1 to infinity of . That would be + + , and that goes on forever and ever and ever.
So anyway, that's this notation now. I'm just trying to introduce this sigma notation for compactifying an infinitely long sum into something I can actually write down. So this is just a symbol rather than an idea.
But the question still remains, what does it mean? What does it mean to add up infinitely many things? And how can we figure out what, in fact, an infinite sum is if we can make head nor tail of it. And what about this devil's sum  1  1 + 1  1 + 1  1 + 1  that really drove mathematicians just bonkers. So this is the cliffhanger I'm going to leave you with right now is to think about how can we possibly make sense out of an infinite series, and how can we then figure out if, in fact, these things actually can be added together and if we can get an answer. So a couple questions here: what does it mean to add up infinitely many things, and when, in fact, will we get an answer and what is the answer? Turns out those are three different questions and they have three different answers. I'll you see you up at the next lecture.
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