(a) Explanation Given: f ( x ) = 2 ∑ k = 1 ∞ ( − 1 ) k + 1 k sin k x Calculation: Assume that f ( x ) = 1 2 a 0 + ∑ k = 1 ∞ a k cos k x + ∑ k = 1 ∞ b k sin k x on the interval. If f is a continuous odd function, it can be shown that a k = 0 for k = 0 , 1 , 2 , ⋯ , and b k = 1 π ∫ − π π f ( x ) sin k x d x = 2 π ∫ 0 π f ( x ) sin k x d x for k = 1 , 2 , ⋯ . For this problem use the fact that a function f ( x ) can be represented in the following form f ( x ) = a 0 2 + ∑ k = 1 ∞ a k cos k x + ∑ k = 1 ∞ b k sin k x , Where, b k = 2 π ∫ 0 π f ( x ) sin k x d x for k = 1 , 2 , 3 ⋯ (b) To determine To evaluate: the convergent alternating series using the Fourier series in part a.

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Question

Chapter 9.3, Problem 54PS

(a)

To determine

To prove: the function f(x)=x on [−π,π] has the given Fourier series representation

(b)

To determine

To evaluate: the convergent alternating series using the Fourier series in part a.

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Chapter 9.3, Problem 53PS

Chapter 9.3, Problem 55PS

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Chapter 9 Solutions

Calculus

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To prove: the function f ( x ) = x on [ − π , π ] has the given Fourier series representation

Solution Summary: The author demonstrates that the function f(x)=x has the given Fourier series representation.