Evaluating an infinite series Write the Taylor series for

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567 2

Solution for problem 47E Chapter 9.4

Calculus: Early Transcendentals | 1st Edition

Textbook Solutions
2901 Step-by-step solutions solved by professors and subject experts
Get 24/7 help from StudySoup virtual teaching assistants

Calculus: Early Transcendentals | 1st Edition

4 5 1 416 Reviews

Problem 47E

Evaluating an infinite series Write the Taylor series for f(x) = 1n (1 + x) about 0 and find the interval of convergence. Assume the Taylor series converges to / on the interval of convergence. Evaluate /(I) to find the value of $\sum\limits_{k=1}^{\infty{}}\frac{(-1{)}^{k+1}}{k}\$ (the alternating harmonic series).

Step-by-Step Solution:

Solution 47E

Step 1:

Given that

f(x) = 1n (1 + x) about 0

(the alternating harmonic series).

Step2:

To find

Write the Taylor series for f(x) = 1n (1 + x) about 0 and find the interval of convergence. Assume the Taylor series converges to / on the interval of convergence. Evaluate /(I) to find the value of

Step3:

We have

f(x) = 1n (1 + x)

Taylor serie for f(x) = 1n (1 + x)

In(1+x)=x-

We know that series converges when |x|<1 and diverges when |x|>1

Now we have to check x=

So, the interval of convergence is (-1,1)

The equation

In(x+1)=

Is true for x=1

So, put x=1 we get

In(2)=

That is the sum of the alternating harmonic series is In2.

Step 2 of 1

Chapter 9.4, Problem 47E is Solved

Textbook: Calculus: Early Transcendentals

Edition: 1

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

ISBN: 9780321570567

This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. This full solution covers the following key subjects: Series, Convergence, Taylor, Find, interval. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. The answer to “Evaluating an infinite series Write the Taylor series for f(x) = 1n (1 + x) about 0 and find the interval of convergence. Assume the Taylor series converges to / on the interval of convergence. Evaluate /(I) to find the value of (the alternating harmonic series).” is broken down into a number of easy to follow steps, and 46 words. Since the solution to 47E from 9.4 chapter was answered, more than 264 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 47E 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.