Sine series for cos(x) (FOURIER SERIES)

[konradz]

I was finally able to figure out how to find the sine series for cos(x), but only for [0,2pi]. A question i have though is what is the interval of validity? is it only [0,pi]?
Ie if I actually had to sketch the graph of the sum of the series, on all of R, would I have cosine or just a periodic extension of cosine from [0,2pi]?

 

[CompuChip]

Hey Konrad, welcome to PF.
I am afraid I don't entirely understand your question. You say that you have managed to write cos(x) as a(n infinite) sum of sines on the interval [0, 2pi].
But both cos(x) and the sines you used are periodic with period 2pi, aren't they? So if the infinite sum converges to cos(x) on an interval with a length of at least one period, then it converges to cos(x) everywhere, doesn't it?

 

[LCKurtz]

If you have expanded cos(x) in a sine series using $p = 2\pi$ in the formula
$$b_n = \frac 2 p \int_0^p \cos(x) \sin{\frac{n\pi x}{p}}\,dx$$
what you are representing is the $4\pi$ periodic odd extension of cos(x).

[edit - corrected typo: b_n not a_n]