Math 138: Assignment 9 due Friday, July 17, 2015 at 12:00 pm NEW: Assignment 9 will be submitted ONLINE using Crowdmark Write solutions in the spaces below and upload ALL pages using the link provided. 4 4 4 4 1. Use the power series representation for tan1 x to show that = 4 + + . . . 3 5 7 9 1 2. Starting with the geometric series = 1x n x , |x| < 1, nd the sum of n=0 n=2 (1)n n(n 1) . 2n 3. Find the radius and interval of convergence of the following power series. 3)n (a) (1) 2n + 1 n=1 (d) n=2 n (x 3n (x 1)2n ln n (b) n=1 4n xn n4 (e) n=1 xn n2 + 1 (c) n=1 (4x 1)n 3n n More space for question 3. (Submit this page even if you do not write anything here) 4. (a) Find the power series representation of 3 and state the radius of convergence. x2 + x 2 4. (b) Find the power series representation of x2 arctan (x2 ) and state the radius of convergence. x2 5. (a) Write down a power series representation for ln 1 + 4 convergence. and state the radius of 1/2 ln 1 + 5. (b) Estimate the integral 0 x2 4 dx with error at most 104 . n+1 (x 1)n . n! n=0 x (Hint: Consider rst nding a closed form expression for g(x) = 1 f (s)ds). 6. Determine a closed form expression for f (x) =