Please #3, 7, 11, 15

1. If the radius of convergence of the power series _,=( Cnx" 17-22 Find a power series representation for the function and is 10, what is the radius of convergence of the series determine the radius of convergence. 2n=1 ncnx"-1? Why? 17. f(x) = 2. Suppose you know that the series 2, o b,, x" converges for (1 + 4x)2 18. f (x) = (2* x) |x| 0 27. - 18 dt 28 . 13-14 Express the function as the sum of a power series by first 29 . x2 In(1 + x) dx 30. tan x x X using partial fractions. Find the interval of convergence. 13. f (x) = 2x - 4 14. f (x) = - 2x + 3 31-34 Use a power series to approximate the definite integral to x2 - 4x + 3 x2 + 3x + 2 six decimal places. 15. (a) Use differentiation to find a power series representation for 31. Jo 1 + x3 dx 32. arctan ~ dx f ( x) = 33. x In (1 + x 2 ) dx 34. x2 (1 + x)2 1 + x4 dx What is the radius of convergence? (b) Use part (a) to find a power series for 35. Use the result of Example 6 to compute arctan 0.2 correct to five decimal places. f ( x) = - (1 + x) 3 36. Use the result of Example 5 to compute In 1. 1 correct to four decimal places. (c) Use part (b) to find a power series for 37. (a) Show that the function x 2 f ( x) = (1 + x) 3 f (x) = > *" 16. (a) Use Equation 1 to find a power series representation for f(x) = In(1 - x). What is the radius of convergence? is a solution of the differential equation (b) Use part (a) to find a power series for f(x) = x In(1 - x). f' ( x) = f(x) (c) By putting x = 2 in your result from part (a), express In 2 as the sum of an infinite series. (b) Show that f (x) = ex