Explain why or why not Determine whether the | Ch 9.4 - 59E

Problem 59E Chapter 9.4

Calculus: Early Transcendentals | 1st Edition

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Problem 59E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. To evaluate# , one could expand the integrand in a Taylor series and integrate term by term.

b. To approximate π/3, one could substitute into the Taylor series for tan−1 x.

c. .

Step-by-Step Solution:

Solution 59E

Step 1:

(a)

Given statement is true

We have the integral $\int\limits_{0}^{2}\frac{dx}{1-x}$ is not defined that is $\int\limits_{0}^{2}\frac{dx}{1-x}$ diverges

To evaluate $\int\limits_{0}^{2}\frac{dx}{1-x}$ , one could expand the integrand in a Taylor series and integrate term by term.

Here the integrand is $\frac{1}{1-x}$

The taylor series of $\frac{1}{1-x}$ is given by

$\frac{1}{1-x}=1+x+{x}^{2}+{x}^{3}+{x}^{4}+.....=\sum\limits_{n=0}^{\infty{}}{x}^{n}$

Therefore

$\int\limits_{0}^{2}\frac{1}{1-x}dx=\int\limits_{0}^{2}(1+x+{x}^{2}+{x}^{3}+{x}^{4}+.....)dx$

$=[x+2x+3{x}^{2}+4{x}^{3}+....{]}_{0}^{2}$

$=[2+4+12+32+......]$

It is also diverges

Step 2:

(b)given Statement is true

To approximate π/3, one could substitute $x=\sqrt{3}$ into the Taylor series for $tan{\ }^{-1}x$ .

We know the Taylor series of $tan{\ }^{-1}x$ . is given by

$tan{\ }^{-1}x$ = $\sum\limits_{n=0}^{\infty{}}(-1{)}^{n}\frac{{x}^{2n+1}}{2n+1}=x-\frac{1}{3}{x}^{3}+\frac{1}{5}{x}^{5}-\frac{1}{7}{x}^{7}+......$

Hence $tan{\ }^{-1}\sqrt{3}=\sqrt{3}-\frac{1}{3}{(\sqrt{3})}^{3}+\frac{1}{5}{(\sqrt{3})}^{5}-\frac{1}{7}{(\sqrt{3})}^{7}+.....$ .

$\approx{}1.04720=\frac{\pi{}}{3}$

$\Rightarrow{}tan{\ }^{-1}\sqrt{3}=\frac{\pi{}}{3}$

Step 3 of 3

Chapter 9.4, Problem 59E is Solved

Textbook: Calculus: Early Transcendentals

Edition: 1

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567

Since the solution to 59E from 9.4 chapter was answered, more than 238 students have viewed the full step-by-step answer. The answer to “Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. To evaluate# , one could expand the integrand in a Taylor series and integrate term by term.b. To approximate ?/3, one could substitute into the Taylor series for tan?1 x.c. .” is broken down into a number of easy to follow steps, and 49 words. The full step-by-step solution to problem: 59E from chapter: 9.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. This full solution covers the following key subjects: could, term, Taylor, Series, integrand. This expansive textbook survival guide covers 85 chapters, and 5218 solutions.