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Taylor Series

# Taylor Series - Z e x-e-x x 3(a Find the Taylor series...

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11.10 Taylor Series Math 296 1 . Famous series Find the MacLaurin series for the following functions. (That is, find the Taylor series about a = 0.) (a) f ( x ) = e x (b) f ( x ) = sin x (We did the other famous one in class last time: cos x = 1 - x 2 2! + x 4 4! - x 6 6! + · · · = n =0 ( - 1) n x 2 n (2 n )! .) 2 . Use the known MacLaurin series for e x , cos x , sin x , or 1 1 - x to find the MacLaurin series for: (a) f ( x ) = x sin( x 2 ) (b) f ( x ) = e x - e - x (simplify!) (c) Use the result of part (b) to evaluate the following indefinite integral as a series.

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Unformatted text preview: Z e x-e-x x 3 . (a) Find the Taylor series centered at 1 for the function 1 x 2 . (b) Find the Taylor series centered at π 2 for the function cos(2 x ). 4 . (a) Find the Taylor series for the function ln(1 + x ) centered at 0. (b) Now ﬁnd the power series representation using techniques from the Section 11.9 instead. Compare your answers to the one for part (b)!...
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