The Taylor Series

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

Solution for Problem 1 >>

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 + ...

Close

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

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

Close