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Unformatted text preview: CSCE 750, Fall 2009 — Quizzes with Answers Stephen A. Fenner September 24, 2011 1. Give an exact closed form for ∞ X k =1 2 k 3 k +1 . Simplify your answer as much as possible. Answer: We reduce the expression to a form we’ve already seen in class: ∞ X k =1 2 k 3 k +1 = 2 3 ∞ X k =1 k 3 k = 2 3 ∞ X k =1 kr k , where r = 1 / 3. We saw in class 1 that the sum on the right is ∞ X k =1 kr k = r (1 r ) 2 for all r such that  r  < 1. Thus the final answer is 2 3 · 1 / 3 (1 1 / 3) 2 = 2 3 · 1 / 3 4 / 9 = 1 2 . 2. Let f a realvalued function defined on R . Recall: • f ( n ) is strictly monotone increasing iff x < y = ⇒ f ( x ) < f ( y ) for all x,y ∈ R . • f ( n ) is strictly monotone decreasing iff x < y = ⇒ f ( x ) > f ( y ) for all x,y ∈ R . 1 If you forgot the formula, rederive it as follows: (1) start with the formula for an infinite geometric series, ∑ ∞ k =0 r k = 1 / (1 r ); (2) differentiate both sides with respect to r ; (3) multiply both sides by r . 1 Show that if f ( n ) and g ( n ) are both strictly monotone decreasing , then f ( g ( n )) is strictly monotone increasing . Answer: For all x,y ∈ R , we have x < y = ⇒ g ( x ) > g ( y ) ( g is strictly decreasing) = ⇒ f ( g ( x )) < f ( g ( y )) . ( f is strictly decreasing) Thus f ( g ( n )) is strictly increasing. 3. Use the substitution method to show that if T ( n ) = 4 T j n 2 k + n 2 , then T = O ( n 2 lg n ) . Only show the inductive step. Don’t worry about any base case(s)....
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This note was uploaded on 12/13/2011 for the course CSCE 750 taught by Professor Fenner during the Fall '11 term at South Carolina.
 Fall '11
 Fenner
 Algorithms

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