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fourier series - sketching graphs

Show that the half range Fourier cosine series of the function f(x) = x^2 on the interval [0,1] is given by:

x^2 = \frac{1}{3} + \frac{4}{\pi^2} \displaystyle\sum_{n=1}^\infty \frac{(-1)^{n}}{n^2} cos n\pi x

Sketch f(x) and the functionto which its half-range Fourier cosine series converges on the interval [-3,3].

I can do the first past of the question fine, but I'm really poor at drawing these graph and always seem to get them wrong. Can someone please give me a general guide of what they would do to draw graphs for questions similar to this? I don't understand how to plot any points at all..

thanks in advance!

Reply 1

DFranklin

Write C(x) for your cosine series.
You know C(x) = x^2 on [0,1]
Since C(x) is a cosine series, you know C(-x)=C(x).
So now you know what C(x) is on [-1,1].
Finally, C(x) has period 2, so C(x+2)=C(x). This is enough to let you draw C(x) for the entire interval [-3,3].

Reply 2

sexyzebra

Thanks for that DFranklin, think I've got the hang of it now...would you mind checking the attached file though just to make sure it's right? I've tried doing the graph to the question below as well. I think I'll have Fourier Series in the bag if these are right :smile:

f(x) = x + \pi

for

x \le 0

and

0

for

x > 0

\frac{\pi}{4} + \displaystyle\sum_{n=1}^{\infty}(\frac{2}{\pi(2n - 1)^2}cos(2n - 1)x - \frac{1}{n}sin n x)

Sketch f(x) and the function to which its Fourier series converges on the interval -3pi to 3pi.

graphs.JPG19.2KB

Reply 3

DFranklin

Looks OK, but no guarantees. I suggest running the equations through a graphing program if you want to be sure. (You could also use Excel or similar).