f(0) = 0 f'(0) = 1 f(n+1) (0)=-n.f(n) (0) for all n 2 1 A function f has derivatives of all orders for -1 < x < 1. The d

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F 0 0 F 0 1 F N 1 0 N F N 0 For All N 2 1 A Function F Has Derivatives Of All Orders For 1 X 1 The D 1 (157.38 KiB) Viewed 24 times

f(0) = 0 f'(0) = 1 f(n+1) (0)=-n.f(n) (0) for all n 2 1 A function f has derivatives of all orders for -1 < x < 1. The derivatives of f satisfy the conditions above. The Maclaurin series for f converges to f(x) for x | < 1. (a) Show that the first four nonzero terms of the Maclaurin series for fare x - + 3 general term of the Maclaurin series for f. (b) Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at x = 1. Explain your reasoning. (c) Write the first four nonzero terms and the general term of the Maclaurin series for g(x) = f(t) dt. (d) Let P represent the nth-degree Taylor polynomial for g about x = 0 evaluated at x = the function defined in part (c). Use the alternating series error bound to show that -<3 500 and write the where g is