a. Explanation Given information: The given function is h ( x ) = 1 ( 1 − x 3 ) 2 . Concept Used: We use the expansion of the function g ( x ) = 4 x 3 ( 1 − x 4 ) 2 = 4 x 3 + 8 x 7 + 12 x 11 + 16 x 15 + ....... = ∑ n = 1 ∞ 4 n x 4 n − 1 with center and divide it by 4 x 3 . Also, we use ratio test given by Ratio test: Assume that ρ = lim n → ∞ | a n + 1 a n | exists. Then, ∑ a n converges absolutely if ρ < 1 . Then ∑ a n diverges if ρ > 1 . The test is inconclusive if ρ = 1 . Then, radius of convergence is given by 1 ρ . Calculation: As the given function h ( x ) = 1 ( 1 − x 4 ) 2 is the multiple of the function g ( x ) = 4 x 3 ( 1 − x 4 ) 2 , so its power series can be obtained by dividing power series of g ( x ) = 4 x 3 ( 1 − x 4 ) 2 = 4 x 3 + 8 x 7 + 12 x 11 + 16 x 15 + ....... = ∑ n = 1 ∞ 4 n x 4 n − 1 by 4 x 3 b. To determine To find: The expansion of the power series of the function h ( x ) = 1 ( 1 − x 4 ) 2 by squaring the power series of another function f ( x ) = 1 ( 1 − x 4 ) which is its root.

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Videos

Question

Chapter 11.6, Problem 44E

To determine

To find:The expansion of the power series of the function h(x)=1( 1− x 3 )2 by dividing the power series of another function g(x)=4x3( 1− x 4 )2 which is its multiple and find its radius of convergence using ratio test.

To determine

To find: The expansion of the power series of the function h(x)=1( 1− x 4 )2 by squaring the power series of another function f(x)=1(1−x4) which is its root.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.6, Problem 43E

Chapter 11.6, Problem 45E

Students have asked these similar questions

7. Use the binomial series to expand the function as a power series. State the radius of convergence. (a) VI- 8x (b) (1 – x)²/3 (c) 4+ x2

Study the convergence of the series

Use the power series 1 1 + X 2(-1)"x", Ixl < 1 n = 0 to find a power series for the function, centered at 0. f(x) = arctan 7x f(x) = Determine the interval of convergence. (Enter your answer using interval notation.)

Chapter 11 Solutions

CALCULUS (CLOTH)

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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 expansion of the power series of the function h ( x ) = 1 ( 1 − x 3 ) 2 by dividing the power series of another function g ( x ) = 4 x 3 ( 1 − x 4 ) 2 which is its multiple and find its radius of convergence using ratio test.

Solution Summary: The author explains how to find the expansion of the power series of h(x)=1 (1-x

Videos

To find:The expansion of the power series of the function h(x)=1( 1− x 3 )2 by dividing the power series of another function g(x)=4x3( 1− x 4 )2 which is its multiple and find its radius of convergence using ratio test.

To find: The expansion of the power series of the function h(x)=1( 1− x 4 )2 by squaring the power series of another function f(x)=1(1−x4) which is its root.

Want to see the full answer?

Chapter 11 Solutions