Computing Fourier Series for x^2 Between -pi and pi

[Studious_stud]

Homework Statement

Homework Equations

Equations to compute Fourier series given in question

The Attempt at a Solution

Ok so I've just done this problem and I'm having some trouble computing the Fourier series of x^{2} between -∏ < ∏

So first I calculate a_{0} and that turns out to be \frac{∏^{2}}{3}, which according to wolfram is right.

Next a_{n}, which is the one I'm having trouble with. Essentially you have to do integration by parts twice (right?) and I end up getting something quite messy, as does wolfram

http://www.wolframalpha.com/input/?i=integrate+x^2+cos+%28n+x%29+dx+from+x%3D-pi+to+pi

Is there any way to clean this up for when I plug a_{n} into the final formula?

b_{n} is 0 obviously so there's no issue there (since it's an even function times an odd function).

Thanks in advance everyone. :)

 

[vela]

You should find it simplifies down to a_n = (-1)^n\frac{4}{n^2}. Since you know n is an integer, you should find some terms vanish, etc.

You can get the answer from Wolfram Alpha by entering "FourierTrigSeries[x^2, x, 10]".

 

[Studious_stud]

You should find it simplifies down to a_n = (-1)^n\frac{4}{n^2}. Since you know n is an integer, you should find some terms vanish, etc.

You can get the answer from Wolfram Alpha by entering "FourierTrigSeries[x^2, x, 10]".

Thanks for the response.

So let me get this right... cosn∏ = (-1)^n and sinn∏ = (-1)^n?

Sorry still a bit confused!

 

[vela]

Not quite. You might want to take a look at a plot of the trig functions.

 

[Studious_stud]

Not quite. You might want to take a look at a plot of the trig functions.

Whoops, meant to say sinn∏=0, is this correct?

 

[vela]

Yes, that's right.