Fourier Series of x/3: Finding Coefficients and Solutions

[Kuma]

Homework Statement

I'm trying to find the Fourier series of x/3.

Homework Equations

The Attempt at a Solution

So i believe that it is an odd function so a0, ak = 0. To find bk:

bk = 1/3pi int (x sin kx dx) from -pi to pi. From integration by parts
= 1/3pi ([x sin kx from -pi to pi]+ 1/k int (cos kx dx) from -pi to pi)
= 1/3pi [(pi sin k pi + pi sin -k pi) +1/k^2 (sin k pi - sin -k pi)]
I think that works out to be 0? Is that right?

 

[A. Bahat]

Well, by inspection $b_k$ cannot all be zero, because then the whole function would be zero.

 

[Kuma]

Sorry I made a big mistake. Forgot to mention the period which is from 0 to 2pi. Also my integration by parts was wrong which resulted in bk being 0 when it shouldn't be.

First question, how do i do it with a period from 0 to 2pi. Is my integrand just from 0 to 2pi?
The formula for bk is 1/L*integral from L to -L f(x) sin k pi x/L dx

What is L in this case? 2pi??