Notes on Taylor Series for a Function

Notes on Taylor Series for a Function - Taylor Series for a...

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Taylor Series for a Function Up until now we have studied power series as defining a function on the interval of convergence. One may turn the issue around though and ask the following question: Given a function, is there a power series that represents the function on its interval of convergence? Theorem : Let f be a given function. If there is a power series ( 29 0 n n n c x a = - for which ( 29 ( 29 0 n n n f x c x a = = - on some interval, then ( 29 ( 29 ! n n f a n c = , or equivalently, ( 29 ( 29 ! n n f a c n = for all 0 n . (Here we define ( 29 ( 29 ( 29 0 f a f a and0! 1 .) There are some observations about the above theorem that one needs to keep in mind. The equality ( 29 ( 29 0 n n n f x c x a = = - holds only on the IOC for ( 29 0 n n n c x a = - . The coefficients n c are related to the derivatives of the function f . So for the function f to have a power series representation, f has to have infinitely many derivatives at the center a . For example, ( 29 f x x = will not be represented by such a series about 0 a = because it is not differentiable at 0 a = . However ( 29 f x x = does have infinitely many derivatives at 0 a . So it may be represented by a series ( 29 0 n n n c x a = - for 0 a . Since f is the sum of the power series ( 29 0 n n n c x a = - , one can write ( 29 ( 29 ( 29 ( 29 0 ! n n n f a f x x a n = = - . Definition : Suppose f is a given function. The power series ( 29 ( 29 ( 29 0 ! n n n f a x a n = - is called the Taylor Series for f about the point a . When 0 a = , the power series ( 29 ( 29 ( 29 0 0 ! n n n f f x x n = = is sometimes called the McClaurin Series for f . Examples : Below are several examples how to find the Taylor Series for several functions.
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1) Recall the Geometric Series, namely ( 29 0 1 1 n n f x x x = = = - . Then 0 n n x = is the McClaurin Series for ( 29 1 1 f x x = - . Since the coefficients are 1 n c = for all 0 n , it follows that ( 29 ( 29 0 ! ! n n f n c n = = for all 0 n . 2) Find the McClaurin series for ( 29 x f x e = . Note that ( 29 ( 29 ( 29 , , , x x x f x e f x e f x e ′′ ′′′ = = = K and in general ( 29 ( 29 n x f x e = for all 0 n . Since the problem asks for the McClaurin series 0 a = . Then ( 29 ( 29 0 0 1 n f e = = . So the coefficients of the series are ( 29 ( 29 0 1 ! ! n n f c n n = = . It follows that 0 ! n x n x e n = = on the IOC. To find the IOC, apply the Ratio Test:
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Notes on Taylor Series for a Function - Taylor Series for a...

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