lec37 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Lecture 37 18.01 Fall 2006 Lecture 37: Taylor Series General Power Series What is cos x anyway? Recall: geometric series 1 1 + a + a 2 + = for a < 1 ··· 1 a | | General power series is an infinite sum: f ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ··· represents f when x < R where R = radiusofconvergence. Thismeansthatfor x < R, | a n x 0 | | n | | n | → as n → ∞ (“geometrically”). On the other hand, if 1 | x | > R , then 1 | a n x | does not tend to 0 . For example, inthecaseofthegeometricseries, if | a | = 2 , then | a n | = 2 n . Since the higher-order terms get increasingly small if | a | < 1 , the “tail” of the series is negligible. n Example 1. If a = 1 , | a | = 1 does not tend to 0 . 1 1 + 1 1 + ··· The sum bounces back and forth between 0 and 1 . Therefore it does not approach 0 . Outside the interval 1 < a < 1 , the series diverges. Basic Tools Rules of polynomials apply to series within the radius of convergence. Substitution/Algebra 1 = 1 + x + x 2 + 1 x ··· Example 2. x = -u.
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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lec37 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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