1 10 7 Taylor Series Goal Given any differentiable function f find a power series X cn x a n which is equal to f x n 0 for all x on its interval of convergence Easier Look Let a 0 This is also called a Maclaurin Series We begin by assuming the first equation below is true for all x on the interval of convergence of the series f x X cn xn c0 c1 x c2 x2 c3 x3 n 0 1 If TN is the N th partial sum of the Taylor series of f also called the nth degree Taylor Polynomial of f and RN is the remainder i e the sum of the rest of the terms we can prove the series converges to f x for all x in the interval of convergence by proving RN x 0 Example Find the Maclaurin Series for f x sin x and explain why the series converges to sin x 2 As we can see it is easy to estimate RN x if the Taylor series is alternating using the Alternating Series Estimation Theorem If not we can use the following Taylor s Inequality If f N 1 x M for x c r then the remainder RN x of the N th degree Taylor polynomial of f satisfies the inequality Example Find the Taylor series for f x ex centered at x 2 You do not have to prove that RN 0 3 IMPORTANT MACLAURIN SERIES TO KNOW 1 ex 2 1 1 x 3 sin x Use a Taylor Series to write x 2 e t dt as a power series 0 4 Some interesting results of Taylor Series 5
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