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

[Benny]

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

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

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

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

Deduce the sum S of the series: $$1 - \frac{1}{{3^3 }} + \frac{1}{{5^3 }} - \frac{1}{{7^3 }} + ...$$

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

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

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

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

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

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?

 

[siddharth]

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).

 

[Benny]

Thanks for the help. I'm just having some problems working out the sum.

From the previous working I have

$$\frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\sin \left( {2m + 1} \right)x}}{{\left( {2m + 1} \right)^3 }}} = \frac{x}{2}\left( {\pi - x} \right)$$ 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: $$S = 1 - \frac{1}{{3^3 }} + \frac{1}{{5^3 }} - \frac{1}{{7^3 }} + ...$$

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:

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

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

$$= \frac{1}{2}\left( {\frac{\pi }{2}} \right)\left( {\pi - \frac{\pi }{2}} \right)$$

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

The answer is S = ((pi)^3)/32. I don't know what I'm leaving out. Any further help would be good thanks.

 

[siddharth]

Check your math in the last step. (ie, the cross multiplication)

 

[Benny]

Thanks for the help. I mistakenly got rid of a factor of pi and ignored some constants. It works out now.