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Unformatted text preview: Math 146: HW 1 Solutions 1.1 Find the ordinary power series generating functions of each of the following sequences in simple, closed, form. In each case the sequence is defined for all n 0. (c) a n = n 2 . Solution. Let A ( x ) = n a n x n = n n 2 x n . Recall n nx n = x (1 x ) 2 . Then x ( d dx ) X n nx n = x ( d dx ) x (1 x ) 2 X n n 2 x n = x 2 + x (1 x ) 3 Thus A ( x ) = x 2 + x (1 x ) 3 = ( xD ) 2 1 1 x , where D = d dx . (d) a n = n 2 + n + . Solution. Using the exercise above A ( x ) = X n a n x n = X n ( n 2 + n + ) x n = X n n 2 x n + X n nx n + X n x n = x 2 + x (1 x ) 3 + x (1 x ) 2 + 1 1 x = (  + ) x 2 + ( +  2 ) x + (1 x ) 3 = ( ( xD ) 2 + xD + ) 1 1 x . (g) a n = 5 7 n 3 4 n . Solution. Let A ( x ) = n a n x n . Then A ( x ) = X n a n x n = X n (5 7 n 3 4 n ) x n = 5 X n 7 n x n 3 X n 4 n x n = 5 1 7 x 3 1...
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 Spring '10
 GregKuperberg

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