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About this Lesson
 Type: Video Tutorial
 Length: 4:04
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 43 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: 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
Finding Power Series Representations by Differetiation Page [1 of 1]
One of the greatest things about the calculus being applied to power series is that it gives us another means of producing power series for functions through differentiation of other power series. I want to illustrate this with an example, so you can really see the power of using calculus on power series.
Let's just recall the fact that if we look at , that actually has a power series expansion  or a Maclaurin series or Taylor series, there's a thousand names for these things  of just x^n. This is just the geometric series. Of course, this only converges though when x < 1, in absolute value. So as long as the minus 1 to 1, not including either endpoint.
So there's a great fact. Now, on that interval, on that particular interval where things converge absolutely, I could actually differentiate. Now let's actually differentiate. If I differentiate this side, what do I get? Well, if I differentiate that side, that's just [(1  x)^1], and so the derivative of that, I've got blop^1, so that's going to be 1 blop^2, multiplied by the derivative of the inside, which produces a negative sign, so I get a negative sign here. Negative times a negative is a positive. So I see this positive one and that negative being underneath, puts me down below, (1  x)^2.
So, in fact, if I take the derivative of this side we know we get . So we know by the calculus of power series that on this particular interval I can now differentiate this just term wise. If I differentiate this term wise, what do I see? Well, that would be not be equal; therefore, the sum and going from one to infinity of just nx^n  1. And how could I write that? Well, I could write that if I wanted to, if wanted to start at zero and going from zero to something, so make a little change of variables. You've got to be careful, though. If the first term here starts off with a one, then if n=0 I want that to be the first term to still be a one, so I'd better put an n + 1. I'd better up the ante everywhere. I'm just making a change of variables. So when I used to start at one, just gave me a 1 times x^0, and notice that's what I get here when I plug this in. A 1 times x^0, and then I start counting off.
So, in fact, just given this, I know that on the same interval I now see a power series for . That power series is the foundation and going from zero to infinity of (n + 1)x^n. So we actually can figure out new power series for new functions by taking a familiar one and applying calculus to it. Really wonderful application of the calculus.
Maybe I'll stop here and let you contemplate what would happen if we think about integrals and power series. It could be exciting.
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