Preview
Buy lesson
Buy lesson
(only $1.98) 
You Might Also Like

Calculus: Differentiability 
Calculus: 1stOrder Linear Differential Equations 
Calculus: Solving Separable Differential Equations 
Calculus: Power Series DifferentiationIntegration 
Calculus: Manipulation, Integration by Parts 
Calculus: Integration,Convergence,Divergence Limit 
Calculus: Fibonacci Numbers 
Calculus: Area, Integration by Substitution, Trig 
Calculus: Limits of Integration and Area 
Calculus: Applying Implicit Differentiation 
College Algebra: Solving for x in Log Equations 
College Algebra: Finding Log Function Values 
College Algebra: Exponential to Log Functions 
College Algebra: Using Exponent Properties 
College Algebra: Finding the Inverse of a Function 
College Algebra: Graphing Polynomial Functions 
College Algebra: Polynomial Zeros & Multiplicities 
College Algebra: PiecewiseDefined Functions 
College Algebra: Decoding the Circle Formula 
College Algebra: Rationalizing Denominators

Calculus: Applying Implicit Differentiation 
Calculus: Limits of Integration and Area 
Calculus: Area, Integration by Substitution, Trig 
Calculus: Fibonacci Numbers 
Calculus: Integration,Convergence,Divergence Limit 
Calculus: Manipulation, Integration by Parts 
Calculus: Power Series DifferentiationIntegration 
Calculus: Solving Separable Differential Equations 
Calculus: 1stOrder Linear Differential Equations 
Calculus: Differentiability
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: 06/27/2009
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
 2174 lessons
 Joined:
11/14/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
More..Recent Reviews
This lesson has not been reviewed.
Please purchase the lesson to review.
This lesson has not been reviewed.
Please purchase the lesson to review.
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 halfinterval 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.
Get it Now and Start Learning
Embed this video on your site
Copy and paste the following snippet:
Link to this page
Copy and paste the following snippet: