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Unformatted text preview: Infinite Series Infinite series are among the most powerful and useful tools that you’ve encountered in your introductory calculus course. It’s easy to get the impression that they are simply a clever exercise in manipulating limits and in studying convergence, but they are among the majors tools used in analyzing differential equations, in developing methods of numerical analysis, in defining new functions, in estimating the behavior of functions, and more. 2.1 The Basics There are a handful of infinite series that you should memorize and should know just as well as you do the multiplication table. The first of these is the geometric series, 1 + x + x 2 + x 3 + x 4 + ··· = ∞ X x n = 1 1- x for | x | < 1 . (2 . 1) It’s very easy derive because in this case you can sum the finite form of the series and then take a limit. Write the series out to the term x N and multiply it by (1- x ) . (1 + x + x 2 + x 3 + ··· + x N )(1- x ) = (1 + x + x 2 + x 3 + ··· + x N )- ( x + x 2 + x 3 + x 4 + ··· + x N +1 ) = 1- x N +1 (2 . 2) If | x | < 1 then as N → ∞ this last term, x N +1 , goes to zero and you have the answer. If x is outside this domain the terms of the infinite series don’t even go to zero, so there’s no chance for the series to converge to anything. The finite sum up to x N is useful on its own. For example it’s what you use to compute the payments on a loan that’s been made at some specified interest rate. You use it to find the pattern of light from a diffraction grating. N X x n = 1- x N +1 1- x (2 . 3) Some other common series that you need to know are power series for elementary functions: e x = 1 + x + x 2 2! + ··· = ∞ X x k k ! sin x = x- x 3 3! + ··· = ∞ X (- 1) k x 2 k +1 (2 k + 1)! cos x = 1- x 2 2! + ··· = ∞ X (- 1) k x 2 k (2 k )! ln(1 + x ) = x- x 2 2 + x 3 3- ··· = ∞ X 1 (- 1) k +1 x k k ( | x | < 1) (2 . 4) (1 + x ) α = 1 + αx + α ( α- 1) x 2 2! + ··· = ∞ X k =0 α ( α- 1) ··· ( α- k + 1) k ! x k ( | x | < 1) James Nearing, University of Miami 1 2—Infinite Series 2 Of course, even better than memorizing them is to understand their derivations so well that you can derive them as fast as you can write them down. For example, the cosine is the derivative of the sine, so if you know the latter series all you have to do is to differentiate it term by term to get the cosine series. The logarithm of (1 + x ) is an integral of 1 / (1 + x ) so you can get its series from that of the geometric series. The geometric series is a special case of the binomial series for α =- 1 , but it’s easier to remember the simple case separately. You can express all of them as special cases of the general Taylor series....
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This note was uploaded on 01/30/2012 for the course PHYS 315 taught by Professor Nearing during the Fall '08 term at University of Miami.
- Fall '08