Explanation Given: The function is f ( x ) = x 2 + 1 , 0 < x < 3 . Calculation: The function f ( x ) = x 2 + 1 is a parabola with vertex at ( 0 , 1 ) and symmetric to y -axis. Here, the function f ( x ) = x 2 + 1 is defined in the interval of ( 0 , 3 ) . Therefore, the graph of the Fourier expansion of the function f ( x ) = x 2 + 1 with periodic length of 3 is a replica of the graph f ( x ) = x 2 + 1 with interval ( 0 , 3 ) . So, extend the function for all real value by the use of periodicity f ( x + 3 ) = f ( x ) . The graph of a function f ( x ) = x 2 + 1 with interval ( 0 , 3 ) and Fourier series extension around the point 0 is shown in Figure 1. It is observed from the Figure 1 that value f ( x ) at point x = 0 from left tends to 10 and from right tends to 1. Therefore, the convergence of Fourier series for f ( x ) = x 2 + 1 at x = 0 is calculated as follows: f ( 0 ) = f ( 0 + ) + f ( 0 − ) 2 = 1 + 10 2 = 5.5 The convergence of Fourier series for f ( x ) = x 2 + 1 at x = 0 is 5.5. The graph of the Fourier Cosine series f ( x ) = x 2 + 1 for periodic interval ( 0 , 3 ) around the point 0 is shown in Figure 2

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 11, Problem 7RE

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The values at which the Fourier series, cosine series and sine series converges at x=0 for the function f(x)=x2+1, 0<x<3.

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

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The values at which the Fourier series, cosine series and sine series converges at x = 0 for the function f ( x ) = x 2 + 1 , 0 < x < 3 .

Solution Summary: The author explains the Fourier series, cosine series and sine series converge at x=0 for the function 'f(x)'

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The values at which the Fourier series, cosine series and sine series converges at x=0 for the function f(x)=x2+1, 0<x<3.

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Chapter 11 Solutions

The values at which the Fourier series, cosine series and sine series converges at x = 0 for the function f ( x ) = x 2 + 1 , 0 &lt; x &lt; 3 .

Solution Summary: The author explains the Fourier series, cosine series and sine series converge at x=0 for the function 'f(x)'

Videos

The values at which the Fourier series, cosine series and sine series converges at x=0 for the function f(x)=x2+1, 0<x<3.

Want to see the full answer?

Chapter 11 Solutions

The values at which the Fourier series, cosine series and sine series converges at x = 0 for the function f ( x ) = x 2 + 1 , 0 < x < 3 .