7. Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. X g(x) = using f(x) = - (1 + 5 x 2) 2 1 + 5x2 g(x) = > k =0 The power series g(x) converges on the interval (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)Consider the function f(x) = sin (13) . a. Differentiate the Taylor series about U for f(x). b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. a. Choose the correct answer below. 11 17 3 14 20 3x 3x 3x 3x 3x , 5,_, ,_._ .v 2,_ ,_ .. A. 3x 3! 5! ' B' 3" 2! + 4! 6! -c 2 3x3 3x\" 3x20 B 5 3x\" 3x17 *"3x'2!'4!'6!"" "'3"+3!'5!+ um): c. The interval of convergence of the power series for the derivative is (Simplify your answer. Type an inequality or a compound inequality. Type an expression usmg x as the variable.) 4. Consider the function f(x) = 5 tan - 1(x). a. Differentiate the Taylor series about 0 for f(x). b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. a. Choose the correct answer below. O A. - 5+5x2 - 5x4 + 5x6 -... 5x 3 5x O B. 5x - + X + 3 7 5x3 5x/ O C. 5x + 3 + X' 5 + + .. 7 O D. 5-5x2 + 5x4 - 5x6 + ... b. f' (X ) = c. The interval of convergence of the power series of f'(x) is (Simplify your answer. Type an inequality or a compound inequality. Type an expression using x as the variable.)