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Math 257 – Assignment 1 Due: Wednesday, January 12 1. Compare the following series to an integral and show that: a) ∞ ! ne −n2 converges b) n=1 ∞ ! n=2 1 diverges n ln(n) 2. Determine whether the following series are convergent or not: a) c) ∞ ! n2 n=0 ∞ ! n=1 3. Show that: a) n=3 (−1)n 2n + 1 ∞ ! (−1)n n=0 b) 2n 2n ∞ ! n+4 2 = 3 b) n2 ∞ ! 1 1 = n−1 3 2 n=2 4. Find the radii of convergence of the following power series: a) c) ∞ ! xn (2n)! n=1 ∞ ! b) ∞ ! n2 xn n=0 2n xn n=0 5. Find the Taylor series of f (x) = ln(1 + x) around x = 0. Determine the radius of convergence of the series and discuss its convergence behaviour at the end points of the convergence interval. 6. " Consider the equation y # + y = 0. Find a power series solution of the form y(x) = ∞ n n=0 an x . Then solve the equation directly to confirm your series solution.