Suppose that a function f has derivatives of all orders near x = 0. By the Fundamental Theorem of Calculus,

a. Evaluate the integral using integration by parts to show that

b. Show that integrating by parts n times gives

This expression, called the Taylor series for f at x = 0.

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10) – 10) = [rwa f'(t) dt. froe-n f"(1)(x – t) dt. f(x) = f(0) + xf'(0) +