   Chapter 11.11, Problem 25E

Chapter
Section
Textbook Problem

# Use Taylor’s Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.

To determine

The number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.

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 ex.

Thus, all the derivatives of ex is ex itself.

Suppose that, the nth derivative of ex is ex.

To estimate the value of e0.1, substitute the value of x as 0.1.

Note that Taylor series is called Maclaurin series when a=0.

Usually, Maclaurin series will starts with n=2.

Increase the value of n until the reminder is less than 0.00001.

By Taylor’s inequality,

If |f(n)(0.1)|=e0.1M, then |Rn(x)|M(n+1)!|0.10|n+1.

Substitute the value of n as 2. Then,

|R2(x)|M(2+1)!|0

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