Finding Fourier coefficients for a step function on [-π, π)

[Calu]

Homework Statement

Define $f : [−π, π) → \mathbb R$ by
$f(x)$ = $−1$ if $− π ≤ x < 0$, $1$ if $0 ≤ x < π.$
Show that the Fourier series of f is given by
$\frac{4}{π} \sum_{n=0}^\infty \frac{1}{(2k+1)} . sin(2k+1)x$

Homework Equations

The Fourier series for $f$ on the interval $[−π, π)$ is given by:

##\frac{a₀}{2} + \sum_{n=0}^\infty a_ncos(nx) + \sum_{n=0}^\infty b_nsin(nx)##
(Not quite sure why the LaTex isn't working here, I'm new at this.)
Where a₀/2, a_n, b_n are the Fourier coefficients of $f$.

The Attempt at a Solution

I have attempted to find the Fourier coefficients, however I don't think that they're correct.

I have found
a₀/2 = 1
a_n = 0.
b_n = $\frac {2}{πn} -2\frac {(-1)^n}{n}$

Could somebody tell me if these are correct? If not, I'll post up how I reached those answers.

 

[Dick]

I think you lost a $\pi$ in the second term of your expression for $b_n$.

 

[vela]

$a_0$ isn't correct either.