Lecture 28 (Taylor Remainder Theorem)

Lecture 28 (Taylor Remainder Theorem) - Tuesday, March 13...

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Tuesday, March 13 Lecture 28 : Taylor's remainder theorem: Convergence of a series to its generator (Refers to Section 8.8 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Define the n th degree Taylor polynomial of the function f ( x ), find the Maclaurin or Taylor series for various functions f ( x ). 28.1 Definition Suppose is a Taylor series generated by the function f ( x ). For each n , let T n ( x ) be the partial sum whose degree is n . Note that the polynomial T n ( x ) need not be the sum of the first n terms. There is difference between the sum S n of the first n terms and the polynomial of degree n . We call T n ( x ) the n th - degree Taylor polynomial of f centered at a . For example the 3 rd degree Taylor polynomial generated by sin x centered at 0 is x x 3 /3!. Here are graphs of some of the Taylor polynomials T 1 ( x ), T 3 ( x ), T 5 ( x ), T 7 ( x ) and T 9 ( x ), for the series generated by sin x . The graphs of these give us the impression that the Maclaurin series generated by sin x “converges” to sin x . As we shall soon see this is correct way of visualizing the partial sums of a Taylor series.
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Lecture 28 (Taylor Remainder Theorem) - Tuesday, March 13...

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