Notes on Taylor Polynomials

Notes on Taylor Polynomials - Notes on Taylor Polynomials...

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Unformatted text preview: Notes on Taylor Polynomials -- Continued Definition : Suppose f is a function which has derivatives up to order N at a point a . Then for each0 n N ≤ ≤ , define the polynomial ( 29 ( 29 ( 29 ( 29 ! k n k n k f a T x x a k = ≡- ∑ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2! ! n n f a f a f a f a x a x a x a n ′′ ′ = +- +- + +- L . This polynomial is called the Taylor polynomial of degree n at a for the function f . Sometimes ( 29 n T x is referred to simply as the th n Taylor polynomial of f at a . Observations : There are several observations to be of which one needs to keep in mind while studying Taylor polynomials. Assuming that f has derivatives of all orders, the sequence ( 29 { } n n T x ∞ = is the sequence of partial sums for the Taylor Series. If ( 29 ( 29 ( 29 ( 29 ! n n n f a f x x a n ∞ = =- ∑ for all x I ∈ , where I denotes an interval contained in the interval of convergence for the power series, then ( 29 ( 29 lim n n T x f x →∞ = , x I ∈ . ( 29 ( 29 T x f a = is a constant function. ( 29 ( 29 ( 29 ( 29 1 T x f a f a x a ′ = +- is the linear approximation of f at a . It is also the equation of the tangent line to the curve ( 29 y f x = at the point x a = . In view of the last observation, one can consider the Taylor polynomials as generalizations of the notion of linear approximations. For example, ( 29 2 T x is a quadratic approximation of f at x a = . If ( 29 ( 29 ( 29 ( 29 ! k n k n k f a T x x a k = ≡- ∑ , then ( 29 n T x is a polynomial for which ( 29 ( 29 ( 29 ( 29 k k n T a f a = for all0 k n ≤ ≤ . The Taylor polynomial of degree n may exist even though the Taylor series does not exist or the radius of convergence for the Taylor series is zero. Theorem : Suppose f is a function and let ( 29 n T x denote the th n Taylor polynomial of f at a . Define ( 29 ( 29 ( 29 n n R x f x T x ≡- . If there is a number R for which ( 29 lim n n R x →∞ = for x a R- < , then ( 29 ( 29 lim n n T x f x →∞ = , that is ( 29 ( 29 ( 29 ( 29 ! n n n f a f x x a n ∞ = =- ∑ . Furthermore, if there is a number M ≥ for which ( 29 ( 29 1 n f x M + ≤ for x a R- < , then ( 29 ( 29 ( 29 ( 29 1 1 !...
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This note was uploaded on 05/06/2011 for the course MATH 256 taught by Professor Buekman during the Spring '11 term at Purdue University.

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Notes on Taylor Polynomials - Notes on Taylor Polynomials...

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