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Exam 3, Fall '04

Exam 3, Fall '04 - Math 12 Tufts University Department of...

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Math 12 Tufts University Department of Mathematics Exam 3 November 15, 2004 11:50–1:20 No books, notes, or calculators . Cross out what you do not want us to grade. You must give reasons for full credit. You are required to sign your exam book. With your signature you are pledging that you have neither given nor received assistance on this exam. You may assume known the facts below; R is the radius of convergence. e x = n =0 x n n ! , R = 1 1 - x = n =0 x n , R = 1 1 . (12 points) For each of the following power series, find the radius of convergence and the interval of convergence. (a) n =0 8 n n x 3 n (b) n =0 (2 n + 1)! 3 n ( x + 2) n . 2 . (20 points) Using the definition of a Taylor series, find the Taylor series for f ( x ) centered at a = 1 for the following functions f ( x ). [Do not use other known series for your derivation]. (a) f ( x ) = x 3 - 2 x + 1 (b) f ( x ) = 1 (2 x + 1) 2 . 3 . (12 points) Using an infinite series given at the start of the exam, (a) evaluate n =0 (ln 2) n n !
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