me function f(x) = 2n+1 (-1)" x²+1 1 n 2 = X- 2x has derivatives of all orders, and the Taylor series for f about x = 0 is x + 2 X³ X5 X² --+ 2 4 8 + a. Using the Ratio Test, determine the interval of convergence of the Taylor series about x = 0 for f. b. Let g(x) be the function such that g(0) = 0 and whose derivative is f(x) = 2x x + 2 Write the first four nonzero terms of the Taylor series for g(x) about x = 0. Use the first two nonzero terms of this series to estimate g (¹) c. The series found in part (b) is an alternating series whose terms decrease in absolute value to 0. Show that the 1 approximation found in part (b) differs from g 1500 (1) by less than

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