Fourier Series for |x| and Finding g(x): Help Needed

In summary: Thanks!In summary, the conversation was about finding the Fourier series for the function y(x) = |x| and deducing the sum of the series 1 - \frac{1}{{3^3 }} + \frac{1}{{5^3 }} - \frac{1}{{7^3 }} + .... The participants discussed integrating the Fourier series and finding the function g(x) whose Fourier series is \frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\sin \left( {2m + 1} \right)x}}{{\left( {2m + 1} \right)^3 }}} . They also discussed setting x = \
  • #1
Benny
584
0
Hi, can someone help me out with the following question?

Q. Show that the Fourier series for the function y(x) = |x| in the range -pi <= x < pi is

[tex]
y\left( x \right) = \frac{\pi }{2} - \frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\cos \left( {2m + 1} \right)x}}{{\left( {2m + 1} \right)^2 }}}
[/tex]

By integrating term by term from 0 to x, find the function g(x) whose Fourier series is

[tex]
\frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\sin \left( {2m + 1} \right)x}}{{\left( {2m + 1} \right)^3 }}}
[/tex]

Deduce the sum S of the series: [tex]1 - \frac{1}{{3^3 }} + \frac{1}{{5^3 }} - \frac{1}{{7^3 }} + ...[/tex]

I took the Fourier series for y(x) and I integrated it as follows.

[tex]
\int\limits_0^x {\left( {\frac{\pi }{2}} \right)} dt - \frac{4}{\pi }\sum\limits_{m = 0}^\infty {\left( {\int\limits_0^x {\frac{{\cos \left( {2m + 1} \right)t}}{{\left( {2m + 1} \right)^2 }}dt} } \right)}
[/tex]

[tex]
= \frac{{\pi x}}{2} - \frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\sin \left( {2m + 1} \right)x}}{{\left( {2m + 1} \right)^3 }}}
[/tex]

I don't know what to do with it to find the function whose Fourier series is

[tex]
\frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\sin \left( {2m + 1} \right)x}}{{\left( {2m + 1} \right)^3 }}}
[/tex]

Can someone help me get started? I'm not sure what to do.

Edit: Do I just equate the integral I evaluated to the integral of |x|(considering x positive and negative separately) and solve the equation for the sine series?
 
Last edited:
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  • #2
Benny said:
Edit: Do I just equate the integral I evaluated to the integral of |x|(considering x positive and negative separately) and solve the equation for the sine series?

Yes, you integrate y(x) (or y(t)) over the interval and then you can find the function g(x).
 
  • #3
Thanks for the help. I'm just having some problems working out the sum.

From the previous working I have

[tex]
\frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\sin \left( {2m + 1} \right)x}}{{\left( {2m + 1} \right)^3 }}} = \frac{x}{2}\left( {\pi - x} \right)
[/tex] for x positive or zero.

I have another expression for x negative but it isn't needed in finding the sum so I'll leave it out.

The sum I want to find is: [tex]S = 1 - \frac{1}{{3^3 }} + \frac{1}{{5^3 }} - \frac{1}{{7^3 }} + ...[/tex]

If I set x = pi/2 then sin(2m+1)x = (-1)^m for all integers m >=0. So using the equation from previous working:

[tex]
\frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\sin \left( {2m + 1} \right)\frac{\pi }{2}}}{{\left( {2m + 1} \right)^3 }}}
[/tex]

[tex]
= \frac{4}{\pi }\sum\limits_{m = 0}^\infty {\left( { - 1} \right)^m \frac{1}{{\left( {2m + 1} \right)^3 }}}
[/tex]

[tex]
= \frac{1}{2}\left( {\frac{\pi }{2}} \right)\left( {\pi - \frac{\pi }{2}} \right)
[/tex]

[tex]
\Rightarrow S = \sum\limits_{m = 0}^\infty {\left( { - 1} \right)^m \frac{1}{{\left( {2m + 1} \right)^3 }}} = \frac{\pi }{2}
[/tex]

The answer is S = ((pi)^3)/32. I don't know what I'm leaving out. Any further help would be good thanks.
 
  • #4
Check your math in the last step. (ie, the cross multiplication)
 
  • #5
Thanks for the help. I mistakenly got rid of a factor of pi and ignored some constants. It works out now.
 

Related to Fourier Series for |x| and Finding g(x): Help Needed

1. What is a Fourier Series?

A Fourier Series is a mathematical tool used to represent a periodic function as a combination of sine and cosine waves. It allows us to break down a complex function into simpler components, making it easier to analyze and manipulate.

2. How is |x| used in a Fourier Series?

The function |x|, also known as the absolute value function, can be expressed as a Fourier Series by using its piecewise definition. This means that for x values less than 0, the function is equal to -x, and for x values greater than or equal to 0, the function is equal to x.

3. Can a Fourier Series be used to approximate any function?

Yes, a Fourier Series can be used to approximate any continuous and periodic function. However, the accuracy of the approximation depends on the number of terms used in the series.

4. How can I find the Fourier coefficients for a given function?

The Fourier coefficients can be found by using the formula c_n = (1/T) * integral from -T/2 to T/2 of f(x)*e^(-2*pi*i*n*x/T) dx, where T is the period of the function and n is an integer. This formula calculates the contribution of each sine and cosine wave to the overall function.

5. How can I use Fourier Series to find g(x) for a given function?

To find g(x), we first need to find the Fourier coefficients for the function. Then, we can use the formula g(x) = (a_0/2) + sum from n=1 to infinity of a_n*cos(n*x) + b_n*sin(n*x), where a_n and b_n are the Fourier coefficients. This formula combines the individual sine and cosine waves to recreate the original function.

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