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# lec38 - 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 38 18.01 Fall 2006 Lecture 38: Final Review Review: Differentiating and Integrating Series. If f ( x ) = a n x n , then n =0 n +1 a n x f ( x ) = na n x n 1 and f ( x ) dx = C + n + 1 n =1 n =0 Example 1: Normal (or Gaussian) Distribution. x x e t 2 dt = 1 t 2 + ( 2! t 2 ) 2 + ( 3! t 2 ) 3 + · · · dt 0 0 x t 4 t 6 t 8 = 1 t 2 + 2! 3! + 4! ... dt 0 x 3 1 x 5 1 x 7 = x 3 + 2! 5 3! 7 + ... x 2 Even though e t dt isn’t an elementary function, we can still compute it. Elementary functions 0 are still a little bit better, though. For example: sin x = x x 3! 3 + x 5! 5 − · · · = sin π 2 = π 2 ( π/ 3! 2) 3 + ( π/ 5! 2) 5 − · · · But to compute sin( π/ 2) numerically is a waste of time. We know that the sum if something very simple, namely, π sin = 1 2 It’s not obvious from the series expansion that sin x deals with angles. Series are sometimes com- plicated and unintuitive.
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lec38 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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