1. Determine the interval of convergence of the following power series. (a) (x+2)n n3n (6) (-1)" (0) I8 M8 M8 M8 M8 (-1)" (x+2)" n 3n (x + 2)n n3n (x + 2)n n3n 3n (x+2)n n3n 2. Suppose that the power series n=0 Cn(x-1)" converges for x = -1 and diverges for x = 5. (a) Does the series converge for x = 2? (b) Does the series converge for x = -5? (c) Does the series converge for x = -2? 3. Suppose that the power series dn(x+2)" converges conditionally for x = 8. n=0 (a) Does the series converge for x = 7? (b) Does the series converge for x = 9? MATH 141 1 Homework 16 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 = for x 3? Why or why not? 5. Suppose that the coefficients of the power series 1, diverge for x = 2, and converge satisfy an (x - c)n n=0 |an+1| = L. lim nx |an| (a) Find the radius of convergence of an (x c)". (b) Find a power series for n=0 d dx an(x-c)", n=0 and determine its radius of convergence. (c) Find a power series for Man (x an(x = 0)"). =0 and determine its radius of convergence. dx, 3x5 6. Find a power series representation for f(x) centered at 0. For which values of x is it valid? 1 + 2x3 7. Find a power series representation for f(x). arctan x centered at 0. For which values of x is it valid?