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MIT6_042JF10_assn07

MIT6_042JF10_assn07 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science October 14, 2008 Tom Leighton and Marten van Dijk Problem Set 7 Problem 1. [15 points] Express n i 2 x i i =0 as a closed-form function of n . Problem 2. [20 points] n (a) [5 pts] What is the product of the first n odd powers of two: 2 2 k 1 ? k =1 (b) [5 pts] Find a closed expression for n m 3 i + j i =0 j =0 (c) [5 pts] Find a closed expression for n n ( i + j ) i =1 j =1 (d) [5 pts] Find a closed expression for n n 2 i 3 j · i =1 j =1 Problem 3. [10 points] (a) [6 pts] Use integration to find upper and lower bounds that differ by at most 0.1 for the following sum. (You may need to add the first few terms explicitly and then use integrals to bound the sum of the remaining terms.) 1 (2 i + 1) 2 i =1
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2 Problem Set 7 (b) [4 pts] Assume n is an integer larger than 1. Which of the following inequalities, if any, hold. You may find the graph helpful. n n 1. ln( i + 1) ln( x + 2) dx i =1 0 n n 2. ln( i + 1) ln 2 + ln( x + 1) dx 1 i =1 Problem 4. [15 points] There is a bug on the edge of a
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