12.15 The Fourier series for the function y(x) = \x( in the range-mx < π is4cos(2m + 1)xy(x) = 2-π(2m + 1)2By integrating this equation term by term from 0 to x, find the function g(x) whoseFourier series is4 sin(2m+ 1)x2m + 1)-0 (InUsing these results, determine, as far as possible by inspection, the form of thefunctions of which the following are the Fourier series:cos θ + 0 cos 3 θ +cos 504 . . . ;2527125し、9I You may find it helpful to first set x-0 in the quoted result and so obtain valuesfor S.-Σ(2m + 1)-2 and other sums derivable from it.]

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Asked Mar 6, 2019
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12.15 The Fourier series for the function y(x) = \x( in the range-m
x < π is
4cos(2m + 1)x
y(x) = 2-π
(2m + 1)2
By integrating this equation term by term from 0 to x, find the function g(x) whose
Fourier series is
4 sin(2m+ 1)x
2m + 1)
-0 (In
Using these results, determine, as far as possible by inspection, the form of the
functions of which the following are the Fourier series:
cos θ + 0 cos 3 θ +
cos 504 . . . ;
25
27
125
し、9
I You may find it helpful to first set x-0 in the quoted result and so obtain values
for S.-Σ(2m + 1)-2 and other sums derivable from it.]
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12.15 The Fourier series for the function y(x) = \x( in the range-m x < π is 4cos(2m + 1)x y(x) = 2-π (2m + 1)2 By integrating this equation term by term from 0 to x, find the function g(x) whose Fourier series is 4 sin(2m+ 1)x 2m + 1) -0 (In Using these results, determine, as far as possible by inspection, the form of the functions of which the following are the Fourier series: cos θ + 0 cos 3 θ + cos 504 . . . ; 25 27 125 し、9 I You may find it helpful to first set x-0 in the quoted result and so obtain values for S.-Σ(2m + 1)-2 and other sums derivable from it.]

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Expert Answer

Step 1

To determine the related functions and the Fourier series based on the given Fourier series

Step 2

Carry out the steps as shown. 

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Step 3

Observe that we have a sine series on the right hand side of the equation (2). So we extend the function (the one obtain...

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