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MATHEMATICS: ASSIGNMENT 9 There are three parts to this assignment: Part A, Part B and Part C. Part A is to be completed online, and Part B and Part C are to be handed in, both before 10:00 a.m. on Friday, February 5. Part A This part of the assignment can be found online, labelled Assignment9A, at webwork.elearning.ubc.ca. Part B 1. Evaluate 1 2 3 4 + + + + ···. 3 9 27 81 2. (a) Find a power series representation, and its radius of convergence, for arctan x. 1 1 1 2. (b) Prove that π = 4 1 − + − + · · · . 3 5 7 2. (c) How many terms of the series in part (b) are required to calculate π correctly to three decimal places? (In other words, find the smallest N such that the partial sums Sn give π correctly to three decimal places for all n ≥ N .) 3. Find a faster way to calculate π than the series in question 2. You must prove that your method is faster — bonus marks will be given for the fastest justified algorithms. Part C Recall the Prisoner’s Dilemma game: suspects R and B are arrested and given a choice: to cooperate with the other suspect, or to betray the other suspect. The payoff matrix is as follows (the payoff is given in terms of jail time): B’s choice R’s choice Cooperate Betray Cooperate R serves 6 months, B serves 6 months R serves 10 years, B goes free Betray R goes free, B serves 10 years R serves 5 years, B serves 5 years In your small groups, submit one program for playing the game an indefinite number of times. The program should be no longer than what can be typed on both sides of one sheet of paper. It should have a cool name. And it should be sufficient to play the game in the absence of any other input (excepting random number generation). After the programs are submitted, they will be played against each other in a round-robin tournament. Each match will have the same number of games. Points will be distributed as follows: 5th to 8th place: 0 points, 4th place: 1 point, 3rd place: 3 points, 2nd place: 5 points, and 1st place: 10 points and a prize. Note that these are bonus assignment points, awarded to each group member, and transferrable to other assignments. Faculty members may also submit programs. Though they can’t win points, their programs may place (and thus knock other programs out of prize positions).