   Chapter 11.11, Problem 26E

Chapter
Section
Textbook Problem

# How many terms of the Maclaurin series for ln(1 + x) do you need to use to estimate ln 1.4 to within 0.001?

To determine

The number of terms of the Maclaurin series for ln(1+x) that should be used to estimate ln1.4 to within 0.001.

Explanation

Formula used:

Taylor’s inequality:

If |f(n+1)(x)|M, then |Rn(x)|M(n+1)!|xa|n+1.

Calculation:

The given function is ln(1+x).

Thus, ln1.4 can be written as ln(1+0.4). That is the value of x as 0.4.

The first derivative of ln(1+x) is 11+x.

The second derivative is (1+x)2 and so on.

That is,

f(x)=ln(1+x)f(x)=(1+x)1f(x)=(1+x)2f(x)=2(1+x)3

Thus, the kth derivative is f(k)(x)=(1)k+1(k1)!(1+x)k.

The Maclaurin series of ln(1+x) is ln(1+x)=n=1(1)n1xnn.

Choose M as the maximum value of the n+1 derivative.

Let k=n+1.

By Taylor’s inequality,

M=|(1)n+1+1(n+11)!(1+x)(n+1)|=n!(1+x)n+1

Since the values of x varies from 0 to 0.4, n!(1+x)n+1 will be the maximum value possible for M at x=0. Then,

M=n!(1+x)n+1=n!(1+0)n+1=n!

Substitute M=n! in |Rn(x)|M(n+1)!|xa|n+1 and simplify as follows

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