AMATH 351 Autumn 2015 Homework 6 Due: Monday, November 30, 2015 Show work for full credit! 1. Find the Taylor series of the following functions about x0 = 0. Your nal answer should be a sum (i.e., no \"...\" at the end.) Determine the radius of convergence for each. (a) ln(1 + x) (b) 1/(1 x) (c) sinh(x) 2. Rewrite the given expression as a sum whose generic term involves xn . In other words, there should only be one sum in your answer and it should look like n n=... (. . .)x . (a) n=0 an xn+2 (b) n=1 nan xn1 +x n=0 an xn 3. Find the solution to the dierential equation using a power series about x0 = 0. Find the recursion relation and nd the rst four terms in each of the two linearly independent solutions (unless the series terminates sooner.) (a) y + xy + 2y = 0 (b) (1 + x2 )y 4xy + 6y = 0 (c) (1 x)y + xy y = 0 (d) x2 y x(x + 3)y + (x + 3)y = 0 1