Explanation Result used: (1) Ratio test : Let ∑ k = 1 ∞ a k be an infinite series and let r = lim k → ∞ | a k + 1 a k | . If r < 1 , then the series ∑ k = 1 ∞ a k is converges. (2) The harmonic series is diverges. (3) The alternating harmonic series is converges. Given: The function h ( x ) = x ln ( 1 − x ) . Calculation: Obtain the power series by using the given result. Consider the function f ( x ) = − x ∑ k = 1 ∞ x k k . g ( x ) = x ln ( 1 − x ) = x ( − ∑ k = 1 ∞ x k k ) = − ∑ k = 1 ∞ x 1 + k k Therefore, the power series representation is − ∑ k = 1 ∞ x 1 + k k . Let a k = x 1 + k k . Then, a k + 1 = x 1 + ( k + 1 ) k + 1 . Use the Ratio test to compute the value | a k + 1 a k | . | a k + 1 a k | = | x k + 2 k + 1 x k + 1 k | Take lim k → ∞ | a k + 1 a k | on both sides, lim k → ∞ | a k + 1 a k | = lim k → ∞ | x k + 2 k + 1 x k + 1 k | = lim k → ∞ | x k + 2 k

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

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

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 9.2, Problem 37E

To determine

To find: The power series representation and interval of convergence by using the power series representation f(x)=ln(1−x)=−∑k=1∞xkk, for −1≤x<1.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.2, Problem 36E

Chapter 9.2, Problem 38E

Students have asked these similar questions

Use the power series X f(x) = (1-x)² Select one: a. c. x e. Ση·(-1)*·(x)" n = 1 x 1 1-X Ση·(x)+1 n = 1 En.(x)" n = 1 = d. Σn (-1)" (x)” +¹ n = 1 Σn. (x)n-1 n = 1 x Σx", x<1 to determine a power series for the function n=0

Use the power series = (-1)"x" to determine a power series centered at 0 for the function f(x) = - %3D (8x + 1)2 8r +1 O Eo(-8)"(n– 1)x"-1 O Eo(-8)"zx"-1 O Eo(8)">x" O Eo(8)"zx"-1 O Eo(-8)"x"

+(-1)". (X-2)h 2 h+1 Fex) =Ź (x-2) _(x-2) --a. 4 Findea a.values of (x) that make series converged. la. The of series. Note with an explanation of the StePs

Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The power series representation and interval of convergence by using the power series representation f ( x ) = ln ( 1 − x ) = − ∑ k = 1 ∞ x k k , for − 1 ≤ x < 1 .

Solution Summary: The author explains how to find the power series representation and interval of convergence.

Videos

To find: The power series representation and interval of convergence by using the power series representation f(x)=ln(1−x)=−∑k=1∞xkk, for −1≤x<1.

Want to see the full answer?

Chapter 9 Solutions

The power series representation and interval of convergence by using the power series representation f ( x ) = ln ( 1 − x ) = − ∑ k = 1 ∞ x k k , for − 1 ≤ x &lt; 1 .

Solution Summary: The author explains how to find the power series representation and interval of convergence.

Videos

To find: The power series representation and interval of convergence by using the power series representation f(x)=ln(1−x)=−∑k=1∞xkk, for −1≤x<1.

Want to see the full answer?

Chapter 9 Solutions

The power series representation and interval of convergence by using the power series representation f ( x ) = ln ( 1 − x ) = − ∑ k = 1 ∞ x k k , for − 1 ≤ x < 1 .