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ASSIGNMENT 2.3 There are two parts to this assignment. Part A is to be completed online before noon on Monday, January 31. Part B, which should be typeset (preferably with LATEX), is to be submitted to the assignment box by the same time. Include your full name and student ID, along with the subject and assignment number, at the top of the front page. Part A [15 marks] This part of the assignment can be found online, labelled Assignment 2.3, at www.mathxl.com. Questions about the MathXL program itself should be sent to Eric; in fact, questions sent through the site will be directed to him. To ask me questions, email me directly or come to my office. Part B [15 marks] 1. (a) Evaluate X (2 − 3x)n at the right endpoint of its interval of convergence. n n≥1 (b) Evaluate 1 2 3 4 + + + + ···. 3 9 27 81 1 1 1 2. Prove that π = 4 1 − + − + · · · . (Hint: find a power series representation, along with its interval 3 5 7 of convergence, for arctan x.) 3. Recall that in the first term we developed methods for solving differential equations of the form ay 00 + by 0 + c = 0, where a, b, and c are constant coefficients. However, most differential equations do not have constant coefficients. One example is the Bessel equation x2 y 00 + xy 0 + (x2 − 1)y = 0, (1) which arises in problems of wave propagation. Here, new methods are required. Find one nontrivial solution to (1) of the form X y= an xµ+n . n≥0