# hwk5 - (c Show how to obtain a better lower bound by...

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Fall 2011 CMSC 351: Homework 5 Clyde Kruskal Due at the start of class, Wednesday, October 12, 2011. Problem 1. Consider the sum n s k =1 ( k 2 - 2 k ) . Assume that you guess it is a cubic polynomial an 3 + bn 2 + cn + d . Use constructive induction to justify your guess and to ±nd the constants a , b , c , d . Problem 2. Do not use integrals for this problem. You can assume n is “nice”. Consider n s k =1 ( k 2 - 2 k ) . (a) Split the sum into two equal-sized regions to obtain upper and lower bounds for its value. (b) Show how to obtain a better upper bound by splitting the sum into two unequal- sized regions? Just get the high order term right (i.e., do not worry about ²oors and ceilings and second order terms). How does your bound compare with the upper bound obtained in Part (a)?
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Unformatted text preview: (c) Show how to obtain a better lower bound by splitting the sum into two unequal-sized regions? Just get the high order term right (i.e., do not worry about ²oors and ceilings and second order terms). How does your bound compare with the lower bound obtained in Part (a)? Problem 3. Consider n s k =1 ( k 2-2 k ) . (a) Use integrals to obtain upper and lower bound bounds. (b) How do your bounds compare with those obtained in Problem 2? (c) How do your bounds compare with the exact polynomial in Problem 1? Problem 4. Find reasonably tight upper and lower bounds for ∞ s k =1 k 3 2 k . Justify your answer. You may use a calculator....
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## This note was uploaded on 01/13/2012 for the course CMSC 351 taught by Professor Staff during the Fall '11 term at University of Louisville.

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