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Unformatted text preview: MATH 239 MidTerm Exam, Nov. 11, 2004 2 This is the midterm examination from Fall 2004. It was both long and challenging. We will try to make our midterm shorter and more routine. 1. (a) [3 marks] Write each of the following rational functions f ( x ) in form of a formal power series f ( x ) = i a i x i . 1 (1 2 x ) 3 , x 3 (1 + 4 x 2 ) 5 . (b) [1 mark] Determine the coefficient [ x 5 ] x 2 (1+ x ) 7 . (c) [3 marks] Write the following formal power series in closed form: X i  1 2 i x 3 i , X i i + 2 i 5 i x i . 2. (a) [2 marks] Use the Binomial Theorem to prove that n i =0 ( n i ) = 2 n . (b) [3 marks] Prove that n X r =0 r X s =0 n r r s = 3 n . (Hint: Use the Binomial Theorem to expand (1 + 2) n .) 3. An even composition of n is a composition of n into an even number of parts such that each part is a positive even number. For example, the even compositions of 8 are (2 , 6) , (4 , 4) , (6 , 2) , (2 , 2 , 2 , 2)....
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This note was uploaded on 07/28/2009 for the course MATH math239 taught by Professor ... during the Spring '06 term at Waterloo.
 Spring '06
 ...
 Rational Functions

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