Problem set 6 that differ by at most 0. 1. Show your work t Problem 4. Use integration to find lower and upper bounds on the following infinite sum 1111 To achieve this accuracy, add up the first few terms explicitly and then use integration to bound all remaining terms Solution. The sum of the first three terms is: 11149 An upper bound on the remaining terms is And a lower bound is x+1)2 Overall, we have 58491 49161 36-36 +-<S<+ These bounds differ by 1/12<0.1. The actual value of the sum is 2/6, though the proof is not easy. Problem 5. A seasoned MIT undergraduate can Complete a problem set in 2 days Write a paper in 2 days Take a 2-day road trip Study for an exam in 1 day. Play foosball for an entire day. An n-day schedule is a sequence of activities that require a total of n days fe here are three possible 7-day schedules 中 pset, foosball Paper, foosball, pset, study road t ad trip, road trip, stud� 4 Problem Set 6 Problem 4. Use integration to find lower and upper bounds on the following infinite sum that differ by at most 0.1. Show your work. 1 1 1 1 S = + + + + . . . 12 22 32 42 To achieve this accuracy, add up the first few terms explicitly and then use integration to bound all remaining terms. Solution. The sum of the first three terms is: 1 1 1 49 s = + + = 12 22 32 36 An upper bound on the remaining terms is: ∞ 1 1 dx = 3 x2 3 And a lower bound is: � ∞ 1 1 dx = (x + 1)2 3 4 Overall, we have: 58 49 1 49 1 61 + = 36 ≤ 36 + 4 ≤ S ≤ 36 3 36 These bounds differ by 1/12 < 0.1. The actual value of the sum is π2/6, though the proof is not easy. Problem 5. A seasoned MIT undergraduate can: • Complete a problem set in 2 days. • Write a paper in 2 days. • Take a 2­day road trip. • Study for an exam in 1 day. • Play foosball for an entire day. An n­day schedule is a sequence of activities that require a total of n days. For example, here are three possible 7­day schedules: pset, paper, pset, foosball paper, study, foosball, pset, study road trip, road trip, road trip, study