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The Taylor Series

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Approximating Functions With Polynomials

Problems

Problem : Compute the Taylor series for f (x) = 1/(1 + x) .

The first few derivatives of the function are

f'(x)	=
f''(x)	=
f ⁽³⁾(x)	=

so f (0) = 1 , f'(0) = - 1 , f''(0) = 2 , f ⁽³⁾(0) = - 6 . The general case is clearly that f ⁽ⁿ⁾(0) = (- 1)ⁿ n! . Hence the Taylor series for f (x) is

p _∞(x)	=	x ⁿ
	=	(- 1)ⁿ x ⁿ
	=	1 - x + x ² - x ³ + ^...

Problem : What is the Taylor series of a polynomial p(x) = a _n x ⁿ + a _n-1 x ^n-1 + ^... + a ₀ ?

Solution for Problem 2 >>

Problem : Find the Taylor series for the function g(x) = 1/ about x = 1 .

Solution for Problem 3 >>

The first couple derivatives of the function are

g'(x)	=	x ^-3/2
g''(x)	=	x ^-5/2
g ⁽³⁾(x)	=	x ^-7/2

so g(1) = 1 , g'(1) = - 1/2 , g''(1) = (- 1/2)(- 3/2) . We deduce that

g ⁽ⁿ⁾(1) =

(1)(3)^...(2n - 1) =

Hence the Taylor series for g(x) is

x ⁿ

< Previous Section
Approximating Functions With Polynomials

Next Section >
The Remainder Term

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