4 The Taylor series

The Taylor series is a generalisation of the Maclaurin series being a power series developed in powers of ( x − x 0 ) rather than in powers of x . Thus

Key Point 15

Taylor Series

If the function f ( x ) can be differentiated as often as required at x = x 0 then:

f ( x ) = f ( x 0 ) + ( x − x 0 ) f ′ ( x 0 ) + ( x − x 0 ) 2 2 ! f ″ ( x 0 ) + ⋯
This is called the Taylor series of f ( x ) about the point x 0 .

The reader will see that the Maclaurin expansion is the Taylor expansion obtained if x 0 is chosen to be zero.

Task!

Obtain the Taylor series expansion of 1 1 − x about x = 2 . (That is, find a power series in powers of ( x − 2 ) .)

First, obtain the first three derivatives and the n th derivative of f ( x ) = 1 1 − x :

f ′ ( x ) = 1 ( 1 − x ) 2 , f ″ ( x ) = 2 ( 1 − x ) 3 , f ‴ ( x ) = 6 ( 1 − x ) 4 , ⋯ f ( n ) ( x ) = n ! ( 1 − x ) n + 1

Now evaluate these derivatives at x 0 = 2 :

f ′ ( 2 ) = 1 , f ″ ( 2 ) = − 2 , f ‴ ( 2 ) = 6 , f ( n ) ( 2 ) = ( − 1 ) n + 1 n !

Hence, write down the Taylor expansion of f ( x ) = 1 1 − x about x = 2 :

1 1 − x = − 1 + ( x − 2 ) − ( x − 2 ) 2 + ( x − 2 ) 3 + ⋯ + ( − 1 ) n + 1 ( x − 2 ) n + ⋯

Exercises
  1. Show that the series obtained in the last Task is convergent if x − 2 < 1.
  2. Sketch the linear, quadratic and cubic approximations to 1 1 − x obtained from the series in the last task and compare to 1 1 − x .
  1. In the following diagrams some of the terms from the Taylor series are plotted to compare with 1 ( 1 − x ) .

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