   Chapter 11.10, Problem 6E

Chapter
Section
Textbook Problem

Use the definition of a Taylor series to find the first four nonzero terms of the series for f ( x ) centered at the given values of a. f ( x ) = 1 1 + x ,     a = 2

To determine

To find:

The first four nonzero terms of Taylor series

Explanation

1) Concept:

Taylor series of the function   f at   a is

fx=n=0fnan!(x-a)n=fa+f'a1!(x-a)+f''a2!(x-a)2+f'''a3!(x-a)3

2) Given:

fx=11+x,   a=2

3) Calculation:

As   a=2, Taylor series is given by:

fx=n=03fn2n!(x-2)n=f2+f'21!x-2+f''22!x-22+f'''23!x-23+

Let’s find the coefficients of this series.

fx=11+x

So,

f2=13

Differentiate   f(x) with respect to x

f(x) can be written as   fx=1+x-1

f'x=-1·1+x-1-1

fx=-11+x2

So,

f2=-19

Now differentiate   f'(x) with respect to x to get f''x

f''x=-1·-21+x-2-1=21+x-3

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