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About this Lesson
 Type: Video Tutorial
 Length: 8:00
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 86 MB
 Posted: 06/27/2009
This lesson is part of the following series:
Calculus (279 lessons, $198.00)
Calculus: Sequences and Series (45 lessons, $69.30)
Calculus: Power Series Function Representations (4 lessons, $5.94)
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
 Thinkwell
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Sequences and Series
Power Series Representations of Functions
Differentation and Integration of Power Series Page [1 of 2]
Let's think about a generic power series. I'll call it f(x), and it will have that power series feel to it, and . So the an is the coefficient, c is the center point where this thing is based around, and this is the whole function. You can find out the interval convergence if you want, and so forth.
Now, what about considering calculus on power series? So what is the calculus of power series? Well, actually this is really neat, because if, in fact, you converge inside that interval, so on the interior of the interval where you have absolute convergence, it turns out that on that interior you actually are differentiable functions. So this s actually can be differentiated. And conversely, you can actually integrate this function. So for all the values of x that are actually within the interval of convergence, and on the interior let's not worry about the endpoints we're sure we have absolute convergence, we can differentiate this and integrate this and we can use the tricks that we think should work. In fact, they will work on the inside.
So what kind of techniques would those be? Well, suppose you want to differentiate. If you want to differentiate a power series, all you've got to do is differentiate each term individually and then put them all together, string them all together. So, in particular, f 1(x) will equal the sum. Now, where do we go from? Notice the very first term here, when n = 0, that's going to be a0, this to the zero power. So this is just a constant, and a derivative of a constant is zero. So that term won't appear anymore.
So I start my sum at one, since the first term is a constant and we'll just leave that when I take a derivative. You go from one to infinity, and it's exactly what you think. I bring down the n, because the x is a variable, so I bring down the nan (x  c)n. Technically we're really using a little teenyweenie chain rule, because I see blop to the n power. The derivative of blop to the n is n blop to the n  1.
All I did here was to say I've got blop to the n, the derivative of blop to the n is n blop to the n  1, and then technically I've got to multiply that by the derivative of the inside, but that's just 1. So this really is okay. Taking the derivatives you have to that. Not a big deal.
What if you want to integrate? Suppose you want to integrate this function. What I'll do is I want to integrate this starting with  suppose that I'm actually going to center this thing around 0. Let me actually center this around 0, f(x) = the sum, n going from 0 to infinity, xn just for simplicity. Then if I want to integrate from 0 to x, f(t)dt, what I would do is basically integrate this side with a t here; and therefore, I could integrate term by term. If you integrate each of these terms individually and evaluate at t, what I would see is . And again, both of these things hold inside the integral of conversions. Outside you might not be able to actually perform calculus, because these things might not even converge or might not converge in a good way.
Anyway, inside there we can integrate term wise, just taking this plus c, and also I need an infinite series there. I start at zero and I go to infinity. So you just add up what you get by integrating each term individually. You just add one to the exponent and divide by it. That's all there is to it. Now, let's actually use these ideas to give a new proof of the fact that the derivative of ex is itself ex.
Let's give a power series proof, so let's use power series to actually learn about calculus, even though we already know it. It's a good check. As an illustration, let's remember that ex = and that's true for all x. So the radius of convergence is infinite. This is true for all x. We don't have to worry about only on certain areas. Great everywhere.
Now what I want to do is take the derivative. So if we differentiate this, (ex) that will equal of this infinite series. Now we see how to deal with infinite series in terms of calculus. Well, this is what I'm trying to figure out. My hope is that the derivative of ex is itself. Let's see if we can verify that by saying the derivative of an infinite series, n will not go from 1 to infinity, the first term this is to constant, so its derivative is 0, and I just differentiate everything here. So I see . Doesn't look like ex, so maybe something's wrong.
Well, let's write n! as n multiplied by (n  1)!. So we peel off a factor of n, and I'm left with (n  1)!. So I have n going from 1 to infinity, Happily the n's cancel out. That's good news. But I'm still not quite there yet, because I have the sum, . Well, what is that? Let's think about where n starts, n starts at 1. So when I plug in one into top here, I see a power of 0 and I have 0!. Then when n = 2, I see 2  1. So that's 1 and 1!. Then I see 2 and 2!, 3 and 3!. In fact, this is actually equal to as long as I start my count at 0, the first term. Because when I let n = 1 and plug it in, I have 0 and 0!, Then when n is 2, I see 1 and 1!, which is when n = 1 here, 1 and 1!. Just by a little change of variables, I see I have this, and that is known to be ex. And so now we've actually verified that the derivative of ex is itself, and we've verified that actually using power series, and knowing how to differentiate a power series.
So you can see this differentiation technique, it's only consistent with a lot of the mathematics we've already seen. We'll take a look at more of these types of issues up next.
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