Fourier Series Problem

1. Oct 22, 2014

Calu

1. The problem statement, all variables and given/known data
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
$\frac{4}{π} \sum_{n=0}^\infty \frac{1}{(2k+1)} . sin(2k+1)x$

2. Relevant equations

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

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

3. 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
an = 0.
bn = $\frac {2}{πn} -2\frac {(-1)^n}{n}$

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

2. Oct 22, 2014

Dick

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

3. Oct 23, 2014

vela

Staff Emeritus
$a_0$ isn't correct either.