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Unformatted text preview: 6 (20 points) . Give the ﬁrst four nonzero terms of the power series expansion of the following functions. Also state for which values of x it is valid. (10 points each) (a) f ( x ) = 1 23 x 2 (b) g ( x ) = sin( x 2 ) + xe x Problem 7 (20 points) . Let f ( x ) = e x 2x 21 x 4 . (a) Write down the ﬁrst four nonzero terms of the power series expansion of f . (8 points) (b) Use the power series to compute f (4) (0), the fourth derivative of f at zero. (6 points) (c) Use the power series to ﬁnd lim x → e x 2x 21 x 4 without using L’Hˆopital’s rule. (6 points) Problem 8 (15 points) . Prove that lim n →∞ x n n ! = 0 ....
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 Fall '07
 Masson
 Calculus, Derivative, Power Series, Taylor Series, Mathematical analysis

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