   Chapter 11.10, Problem 8E

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 value of a.8. f(x) = ln x, a = 1

To determine

To find: The first four nonzero terms of the series for f(x) centered at 1.

Explanation

Result used:

If f has a power series expansion at a, f(x)=n=0f(n)(a)n!(xa)n , f(x)=f(a)+f(a)1!(xa)+f(a)2!(xa)2+f(a)3!(xa)3+

Calculation:

Consider the function f(x)=lnx centered at a=1 .

Obtain the first four nonzero terms of the series as follows,

The function f(x)=lnx at a=1 is computed as follows,

f(1)=ln1=0

The first derivative of f(x) is computed as follows,

f(x)=ddx(lnx)=1x

f(x)=1x (1)

Substitute 1 for x,

f(1)=11=1

The second derivative of f(x) is computed as follows,

f(2)(x)=d2dx2(f(x))=ddx(f(x))=ddx(1x)        (by equation(1))=1x2

Substitute 1 for x,

f(2)(x)=1(1)2=11=1

f(2)(x)=1 (2)

The third derivative of f(x) is computed as follows,

f(3)(x)=d3dx3(f(x))=ddx(f(2)(x))=ddx(1x2)    (by equation(2))=ddx(x2)

Simplify further and obtain f(3)(x) ,

f(3)(x)=(2)(x)21=2x3

f(3)(x)=2x3 (3)

Substitute 1 for x,

f(3)(1)=2(1)3=21=2

The fourth derivative of f(x) is computed as follows,

f(4)(x)=d4

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