TITLE: Coefficient of Fourier cosine & value of full series
QUESTION [5 upvotes]: I am working on a simple Fourier question from an introductory PDE text by John Davis.
The question begins with a graph that can be reduced into piecewise:
$$f(x) =
\begin{cases}
1,  &0 \leq x < 0.5 \\
0,  &0.5 \leq x \leq 1, \  -1 \leq x \leq -0.5\\
-1, &-0.5 < x < 0 
\end{cases}$$

Questions:
  (a) Find the value of $a_{99}$, the 99th Fourier cosine coefficient for $-3 < x < 3$ 
  (b) Find the value of the full Fourier series at $x = 0, x = 1$ and also $x = -1$.

Here are my attempts at the solutions so far: 

(a) The text provides this formula for $a_n$,
$$a_n = \frac {2}{l} \int_{0}^{l} f(x) \cos (\frac{n \pi x}{l}) dx.$$
After assigning $f(x) = \pm 1$, that is by ignoring the $f(x) = 0$ from the piecewise, putting in $l = 3, n = 99$ and $ f(x) = \pm 1$, I get the answer $a_{99} = \pm 3.7463 x 10^{-12}$ from TI-84 graphic calculator. Am I correct, especially in assigning $f(x) = \pm 1$?

(b) Here I am totally lost except getting these lengthy formulas from text, and any help would be very much appreciated:
$$\begin{align}
f(x) &= \frac{1}{2}a_0 + \sum_{n=1}^{\infty} [a_n \cos (\frac{n \pi x}{l}) + b_n \sin(\frac{n \pi x}{l})].\\
a_n &= \frac{1}{l} \int_{-l}^{l}f(x) \cos (\frac{n \pi x}{l}) dx,\\
b_n &= \frac{1}{l} \int_{-l}^{l}f(x) \sin (\frac{n \pi x}{l}) dx,
{}\end{align}$$
Thank you very much for your time and help.

REPLY [2 votes]: Your function is an odd function, hence we simply have $a_n=0$.
Using standard theorems on the pointwise convergence of Fourier series, we have that the Fourier series of $f(x)$ equals zero both in $x=0$ and in $x=\pm 1$ (even without computing it!).