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Calculus: Power Series - Integration

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About this Lesson

  • Type: Video Tutorial
  • Length: 5:40
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 60 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 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
Power Series Representations of Functions
Finding Power Series Representations by Integration Page [1 of 2]
Now if we use the idea of integration together with power series, you can see that the calculus of power series again allows us to uncover interesting facts about interesting functions and their associated power series just by using a known function and applying calculus to it. So I thought I would illustrate this idea of integrating power series through a particular example.
So, let's first of all warm up by recalling that is equal to - that's a geometric series - x^n, and this holds for . Now what if I replace the x by -x? If I replace x by -x, and let's make a substitution and put in -x wherever I see an x, and so if I do that, then I would see the following: 1 - 1, and now in place of x I'm putting -x, so I have another negative sign, which makes this equal a positive sign, and that would equal sum of n going from zero to infinity - . Then I would see a -x, all raised to the n, so that would be -1^n(x^n). So there's a new identity that didn't really use any calculus at all. I just literally replaced x by -x, and I see this identity.
Now, what if I take that identity and suddenly realize that if I were to integrate this function that that actually has a name? Let me actually integrate this. If I integrate from zero to x - - that's just this function, - then, in fact, this equals the natural log of (1+x). If you integrate this, you see the natural log of . When I evaluate it at x, I get this; when I evaluate it at zero, I get just zero. So, in fact, it's just this, and I don't need the absolute value as long as I promise you that the x will live between zero and one. So let's make that promise right now. We're only going to look at half of this interval between -1 and 1. We'll look at the half-interval between 0 and 1. So therefore I see this fact is a fact. That means that this object, when integrated, produces a natural log of (1+x). That would imply that on the interval of conversion, integrating this is the same as integrating this, which means I can integrate term one. So if I actually integrate this term wise, I will get a power series representation for the natural log of (1+x). Neat!
So let's actually do that. So if we integrate term wise, what do we see? Let me just recap what I've got here. I've got . I also saw that if I integrate with respect to t, the integral of this - so if I integrate - that produces natural log (1+x), which means that will equal what happens when I integrate each term individually. What happens when I integrate each term individually? Well, maybe we should just write down some of these terms really fast. When n = 0, I just get 1. Then I have when n = 1, I have -x; then +x^2, then -x^3, and so on.
So if I integrate, what would I see? If I integrate, what I would see is the integral of 1 is just x - + , and, in fact, I see a pattern here. So, in fact, what I see is that if I integrate this side term wise, I now see an infinite series - n going from one to infinity - to get those negative signs to be right - .
So again now I see a power series expansion for a natural log of (1+x), and that power series was given to me just by integrating a known power series. So really, though, the power of calculus together with power series allows us to actually find new representations of interesting and important functions that arise naturally from other basic identities, and then applying calculus. You can really see this idea is a powerful one...it's a powerful one. Actually, integration and power series come together to allow us to actually help, in fact, express what their integrals are.
Well, you think about that, and maybe I'll see you at the next lecture. Have fun.

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