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# lec37 - Lecture 37 18.01 Fall 2006 Lecture 37 Taylor Series...

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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 = radius of convergence. This means that for 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, 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 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. 1 1 + u = 1 - u + u 2 - u 3 + · · · Example 3. x = - v 2 .

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