Help on Fourier Series of Sin(x/3)

THE QUESTION STATES:

Let f be the 2 periodic function defined on [−pi,pi ) by

f(x) = sin(x/3)

Find the Fourier coefficients of f.

My attempt at the question,

After much work using trigonometry and integration by parts I have deduced

bn = (1/2pi) [(3/(1-3n))sin(((1-3n)/3)x) - (3/(1+3n))sin(((1+3n)/3)x)]

Now I only need to put the limits in. The trouble is when i put the limits [-pi,pi) into the equation I get 0 which is not the answer!

and the next part of the question states:

show for all x E (-pi,pi)

sin (x/3) = (((9)(3^.5))/pi)(((-1)^n)n)/(1-9n^2))

I think I may need to use Dirichlets theorem however I need the answer to the part of the question before!