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**Unformatted text preview: **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 + a 1 x + a 2 x 2 + a 3 x 3 + represents f when x < R where R = radius of convergence. This means that for x < R, | a n x | | n | | n | as n (geometrically). On the other hand, if 1 | x | > R , then 1 | a n x | does not tend to . For example, in the case of the geometric series, 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 . 1- 1 + 1- 1 + The sum bounces back and forth between and 1 . Therefore it does not approach . Outside the interval- 1 < a < 1 , the series diverges. Basic Tools Rules of polynomials apply to series within the radius of convergence....

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