# hw1sol - Math 146 HW 1 Solutions 1.1 Find the ordinary...

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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 = αn 2 + βn + γ . Solution. Using the exercise above A ( x ) = X n ≥ a n x n = X n ≥ ( αn 2 + βn + γ ) x n = α X n ≥ n 2 x n + β X n ≥ nx n + γ X n ≥ x n = α x 2 + x (1- x ) 3 + β x (1- x ) 2 + γ 1 1- x = ( α- β + γ ) x 2 + ( α + β- 2 γ ) x + γ (1- x ) 3 = ( α ( xD ) 2 + βxD + γ ) 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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hw1sol - Math 146 HW 1 Solutions 1.1 Find the ordinary...

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