1. Determine the interval of convergence of the following power series. 8 (a) (ar + 2)" SCI)(x + 2) 2n DO (e) S' (x + 2) 3n n=1 n 3n n 3n n=1 n 3n n=1 (b) (-1)n ( 2+ 2) n (d) > (ar + 2) 2n n=1 n 3n n 3n 2. Suppose that the power series En(x - 1)" converges for r = -1 and diverges for r = 5. n=0 Does the series converge for x = 2, for x = -5, for x = -2? 3. Suppose that the power series > dn(x + 2)" converges conditionally for * = 8. n=0 Does the series converge for x = 7, for x = 9? Hint: Does the ratio test ever guarantee conditional convergence? What must be true about the ratio test limit if the series converges conditionally? 4. Is it possible for a power series centered at 0 to converge for x = 1, diverge for x = 2, and converge for x = 3? Why or why not? 5. Find a power series representation for the following series centered at 0. For which values of a are they valid? 37-5 (a) f(x) = (b) f(x) = arctan r 1 + 2x3