Troubleshooting Fourier Series Expansion for Piecewise Function

[Telemachus]

Hi there. I have some trouble with this problem, it asks me to find the Fourier expansion series for the function
f(t)=0 if -pi<t<0, f(t)=t^2 if 0<t<pi

So I've found the coefficients $$a_0=\displaystyle\frac{1}{\pi}\displaystyle\int_{0}^{\pi}t^2dt=\displaystyle\frac{\pi^2}{3}$$

$$a_n=\displaystyle\frac{1}{n^2}\cos(n\pi)$$

$$b_n=-\displaystyle\frac{\pi\cos(n\pi)}{n}-\displaystyle\frac{4}{n^3\pi}$$

Then the Fourier series expansion:

$$f(t)\sim{\displaystyle\frac{\pi^2}{6}+\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^2}\cos(n\pi)\cos(nt)- \left( \displaystyle\frac{\pi}{n}+ \displaystyle\frac{4}{n^3\pi}\right)\sin(nt)}$$

When I plot this on mathematica I get something that doesn't look like what I'm looking for. I've tried many ways, I've done the integrals first by hand, then I did it with mathematica, the graph always seems the same, it doesn't get to zero in the interval zero to -pi as it should, and it isn't close to the plot of t^2, it doesn't even get to zero on the origin. I don't know what I'm doing wrong. I've looked at the equations carefully, I'm pretty much sure I've done things right. Whats happening?

I've also tried to make a distinction between the odd and even cases, but as I supposed it didn't affect at all, the equation as I wrote it includes both cases.

 

[micromass]

I don't obtain the same integrals as you do. At least, wolfram alpha doesn't. Can you show how you found them?

 

[Telemachus]

Alright.

$$a_n=\displaystyle\frac{1}{\pi}\displaystyle\int_{0}^{\pi}t^2\cos(nt)dt=\displaystyle\frac{1}{n^2} \cos(n\pi)$$

$$b_n=\displaystyle\frac{1}{\pi}\displaystyle\int_{0}^{\pi}t^2\sin(nt)dt=\displaystyle\frac{-\pi}{n}\cos(n\pi)-\displaystyle\frac{4}{n^3\pi}$$

Thats how I did it. Do you need the mid steps? its quiet a bit tricky. For the first integral mathematica give just the same, but with a sine with an npi inside, which is obviously zero. For the second integral it gives something quiet similar to the result I've found. Anyway, I've tried using mathematica results, and gives the same graph.

Ow. I've found a mistake in my integrals now, but I think it doesn't change much.

Ok, I've corrected the mistake, now this is what I get:

Its closer now, but stills wrong.

 

[vela]

Those results aren't correct.

 

[micromass]

I second vela. Wolfram alpha gives me the integrals:

http://www.wolframalpha.com/input/?i=Integral[sin(n*x)x^2,0,pi]

and

http://www.wolframalpha.com/input/?i=Integral[cos(n*x)x^2,0,pi]

 

[Telemachus]

You were right, I had another silly mistake :) I think its okey now:

https://www.physicsforums.com/attachment.php?attachmentid=36003&stc=1&d=1306628272

It doesn't make any difference in separating the summations on odd and even numbers, right? or does it have something to do with the speed of convergence?

 

[micromass]

Ah, it's good to hear that Fourier hasn't been disproved

 

[Telemachus]

Thank you both :D